Niels Bohr - Collected Works (Volume 6 - Foundations of Quantum Physics I; 1926 - 1932)

NIELS BOHR C O L L E C T E D WORKS GENERAL EDITOR

ERIK RUDINGER T H E NIELS BOHR ARCHIVE, COPENHAGEN

VOLUME 6

FOUNDATIONS OF Q U A N T U M PHYSICS I ( 1 9 2 6 - 1932) EDITED BY

J 0 R G E N KALCKAR T H E N I E L S BOHR INSTITUTE, COPENHAGEN

1985

NORTH-HOLLAND A M S T E R D A M * N E W YORK O X F O R D * T O K Y O

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FOREWORD TO VOLUMES 6 AND 7

Volumes 6 and 7 of Niels Bohr’s Collected Works, which present the successive stages of his profound analysis of the observational situation in quantum physics and the radical epistemological lesson that he derived therefrom, confront the editor with some problems of their own. Bohr’s writings on these issues are notoriously difficult to fathom, partly due to the intricacies of the subject itself and partly to Bohr’s extremely condensed style and his subtle, but often indirect manner of argumentation. For one who in his youth enjoyed the unforgettable experience of learning about the fundamental issues of the quanta1 description of nature through conversations with Bohr himself, it was therefore tempting in many places to try to assist the reader by suggesting an interpretation of Bohr’s meaning. It was felt, however, that this temptation had to be resisted, since otherwise the nature of this edition might give rise to the misunderstanding that one tried here to “canonize” a definite interpretation. Here, of all places, Bohr should be allowed to speak for himself, without exegesis imposed by the editor. Even so, the discerning reader will not fail to notice the not inessential difference in shadings between Bohr’s views and those advocated by other proponents of the so-called “Copenhagen interpretation”, notably Heisenberg and Rosenfeld. Another decision which had to be faced concerned the question of ref,Prences to later discussions of these issues by other authors. Clearly, however, a reasonably thorough confrontation of the various points of view would swell the proportions of these volumes to an intolerable degree. Even works dealing with the historical development of the subject at hand are only commented on in a few rare cases. The reader should bear in mind that within the scope of an edition of Bohr’s Collected Works, the purpose of the introductory discussions is simply to assist the reader in comprehending the development of Bohr’s thoughts through the years, rather than to attempt anything so ambitious as a detailed physical, philosophical or historical analysis. Whenever possible we have tried to illuminate the development by quotations

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from contemporary letters between Bohr and his circle of colleagues and pupils, sometimes restricting our own narrative to a few lines connecting one letter with the next. Occasionally we have also quoted from the interviews conducted by Thomas Kuhn and his collaborators during the nineteen sixties, the transcripts of which are collected in the Archive for the History of Quantum Physics (AHQP). Obviously, these recollections of events many years back, vividly as they often are told, must nevertheless be treated with caution. A critical reader may perhaps wonder why Bohr’s correspondence with Pauli is quoted in such abundance, in view of the fact that a complete edition of Pauli’s scientific correspondence will soon be available. There is a twofold answer to this question. First, as regards the subject and period covered by these volumes, Bohr’s correspondence with Pauli is scientifically by far the richest and most illuminating - even compared to the Heisenberg correspondence which comes next to it in importance. Second, but even more decisive is the vivid picture we receive from this correspondence of the warm human relationship between Bohr and Pauli. In general we miss in Bohr’s written legacy the magic charm and spontaneity which made any conversation with him such an unforgettable experience. Indeed, only in the correspondence with Pauli do we see him divest himself of the formality otherwise characteristic of the style of his writings. But here we catch at least a glimpse: We are allowed to share his delight in Pauli’s witty sarcasms, which never really hide a feeling of deep friendship. And we see before us Bohr’s smile while he prepares his repartee with the innocent jokes and delicate puns that he liked so much and which he in conversations used true to his own doctrine that “some things are too serious to be spoken of except by way of joking”. As I here repeat this often quoted maxim of Bohr’s, I suddenly recall how one day - inspired by a conversation about the old Goethe, who after completing Faust sealed it in a paper parcel which was only to be opened after his death - we looked up Goethe’s last great letter to v. Humboldt. Imagine Bohr’s pleasure and emotion when we found that Goethe here referred to Faust, this work of an entire life, by the words “these very serious jests”.

*** The editing of these volumes and the writing of the introductions have been carried out from a physicist’s point of view. Accordingly, the interest has been focused on the physical basis for the development and refinement of the complementarity argument, which Bohr accomplished through an ever wider and deeper probing of the quanta1 description until its very essence was brought clearly to light. One must keep in mind that notwithstanding the importance which Bohr attached to an extension of the complementarity argument to areas outside physics, its firm basis always remained the formalism of quantum theory, since

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it is within this framework that the implications of the argument stand out in their full poignancy. Clearly, in connection with the editing of these volumes historical questions often arose, in which I had the need of guidance. In these matters I have been fortunate enough to be able to rely on Erik Rudinger’s wide experience and patient assistance. In the early phase of the work I benefitted much from discussions with Klaus Stolzenburg, whose thesis has also provided me with many valuable insights. Nevertheless, a historian - not to speak of a philosopher would no doubt have placed the emphasis differently. The proportions of the introductions and the amount of material quoted vary considerably from one article t o another. I saw no reason to conceal that I found some articles of more interest than others, even though the judgement in such respect must remain somewhat subjective. For example, nearly half of volume 6 is dedicated to the Como Lecture and its prehistory, for the reason that we here witness the genesis of the complementarity argument. In the translation of the Danish and German letters we have endeavoured to strike a balance between achieving an idiomatic English version and preserving the characteristic original style. Especially in the case of Bohr’s own letters, however, we have let the latter consideration weigh more heavily than the former, and I have even occasionally permitted myself to try to simulate Bohr’s own use of the English language. We wish to express our gratitude to Aage Bohr for many a thoughtful advice inspiring us to important amendments of the manuscript during its successive stages, and not least for his continued interest and encouragement. I am grateful to Jens Lindhard for several illuminating discussions on the Szilard machine, alluded to in the introduction t o part I11 of volume 6. We are thankful to James G. O’Hara for his numerous, valuable suggestions for improvement of the language of the introductions as well as the translations of the letters. We extend our thanks to Asger Aaboe, David C. Cassidy and Ulrich Roseberg for their helpful comments, to Carsten Jensen for his help with the proofs and the index, and not least to Helle Bonaparte and Lise Madsen for their patient efficiency in preparing the manuscript, Finally, we thank the desk editor of North-Holland Physics Publishing, Mrs Jane Kuurman, for her excellent assistance. Jerrgen Kalckar

VII

The Institute wishes to express its sincere gratitude to the Carlsberg Foundation for the continued generous support which has been vital for the realization of the present edition of Niels Bohr’s Collected Works. The Institute also wishes to acknowledge valuable grants from the Leon Rosenfeld Scholarship Fund and from the Niels Bohr Fund administered by the Royal Danish Academy. The Niels Bohr Institute

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ABBREVIATED TITLES OF PERIODICALS

XI1

Ann. d. Phys.

Annalen der Physik (Leipzig)

Comptes rendus, Acad. Sci. Paris

Comptes rendus hebdomadaires des seances de I’Academie des sciences (Paris)

Fys. Tidsskr.

Fysisk Tidsskrift (Ksbenhavn)

J. Chem. Soc. London

Journal of the Chemical Society (London)

Mat.-Fys. Medd. Dan. Vidensk. Selsk.

Matematisk-fysiske Meddelelser udgivet af Det Kongelige Danske Videnskabernes Selskab (K~benhavn)

Naturwiss.

Die Naturwissenschaften (Berlin)

Overs. Dan. Vidensk. Selsk. Forh.

Oversigt over Det Kongelige Danske Videnskabernes Selskabs Forhandlinger (Ksbenhavn) (until May 193 l )

Overs. Dan. Vidensk. Selsk. Virks.

Oversigt over Det Kongelige Danske Videnskabernes Selskabs Virksomhed (Ksbenhavn) (from June 193 l )

Phil. Mag.

Philosophical Magazine (London)

Proc. Camb. Phil. Soc.

Proceedings of the Cambridge Philosophical Society

Proc. Roy. Soc.

Proceedings of the Royal Society of London

Proc. Nat. Ac. Proc. Nat. Acad. Washington

Proceedings of the National Academy of Sciences of the United States of America (Washington D.C.)

Sitzungsber. d. preuJ3. Akad. d. Wiss. Sitzungsber. Preuss. Akad. Wiss.

Sitzungsberichte der Preuljischen Akademie der Wissenschaften (Berlin)

ABBREVIATED TITLES OF PERIODICALS

Stud. Hist. Phil. Sci.

Studies in history and philosophy of science (London)

Z. Phys.

Zeitschrift fur Physik (Braunschweig)

2. Physik 2s. f. Phys.

ZS. f. Phys. Zeits. f. Phys. Zeitsch. f. Phys. Zeitschr. f . Phys.

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ABBREVIATIONS

AHQP AIP Bohr MSS BSC Mf MS PWB I

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Archive for History of Quantum Physics American Institute of Physics, New York Bohr Manuscripts Bohr Scientific Correspondence Microfilm Manuscript Pauli Wissenschaftlicher Briefwechsel, Bd. I (see p. [lo], ref. 5)

ACKNOWLEDGEMENTS

N. Bohr, “Faraday Lecture: Chemistry and the Quantum Theory of Atomic Constitution”, J. Chem. SOC.London, 1932, pp. 349-384 is reprinted by permission of the publisher, The Royal Society of Chemistry. N. Bohr, “The quantum postulate and the recent development of atomic theory”, Atti del Congress0 Internazionale dei Fisici 11-20 Settembre 1927, Como-Pavia-Roma, Volume Secondo, Nicola Zanichelli, Bologna 1928, pp. 565-588, Discussion Remarks, ibid., pp. 589-598, are reprinted by permission of the publisher Nicola Zanichelli. N. Bohr, “Wirkungsquantum und Naturbeschreibung”, Naturwiss. 17 (1929) 483-486 is reprinted by permission of the publisher, Max-Planck Gesellschaft.

N. Bohr, “Atomic Theory and the Description of Nature”, Cambridge University Press, 1934 (1961), pp. 1-24, 92-101, 102-119, is reprinted by permission of the publisher, Cambridge University Press. N. Bohr, “On Atomic Stability”, Brit. Ass. Adv. Sci., Report of the Centenary Meeting, London - 1931, September 23-30, London 1932, p. 333, is reprinted by permission of the publisher, The British Association for the Advancement of Science. W . Heisenberg, “Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik”, Z. Phys. 43 (1927) 172-198 is included in this volume by permission of Werner Heisenberg and of the publisher, SpringerVerlag.

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ACKNOWLEDGEMENTS

The following articles are reprinted by permission from Nature, 0 Macmillan Journals Limited: “Atomic theory and wave mechanics”, 119 (1927) 262; “The quantum postulate and the recent development of atomic theory”, 121 (1928) 78 (abstract) and 121 (Suppl.) (1928) 580-590; “Quantum theory and relativity”, 123 (1929) 434; “Philosophical aspects of atomic theory”, 125 (1930) 958; “The use of the concepts of space and time in atomic theory”, 127 (1931) 43; “Maxwell and Modern Theoretical Physics”, 128 (Suppl.) (1931) 691-692. The following abstracts from Overs. Dan. Vidensk. Selsk. Forh. are reprinted by permission of the Royal Danish Academy of Sciences and Letters: N. Bohr, “Atomteori og Bdgemekanik” (Juni 1926 - Maj 1927) 28-29; “Kvantepostulatet og Atomteoriens seneste Udvikling” (Juni 1927 - Maj 1928) 27; “Kvanteteori og Relativitet” (Juni 1928 - Maj 1929) 24; “Om Benyttelsen af Begreberne Rum og Tid i Atomteorien” (Juni 1930 - Maj 1931) 26.

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GENERAL INTRODUCTION TO VOLUMES 6 AND 7 A Glimpse of the Young Niels Bohr and his World of Thought bY JQRGEN KALCKAR

Volumes 6 and 7 of Niels Bohr’s Collected Works contain his contributions to the clarification of the foundations of quantum theory and the general epistemological lesson which he derived from this analysis. We are here confronted with issues which Bohr had especially at heart, and which he again and again subjected to renewed probing and elaboration, from the highly original ponderings of his youth to the serene mastery of his later years. Nowhere else in Bohr’s lifework do we come closer to the man Niels Bohr. For this reason we felt it natural to introduce these volumes by collecting a few characteristic traits that taken together might illustrate the personal background for Niels Bohr’s general philosophical attitude. The reader looking for a more systematic account may consult LCon Rosenfeld’s Biographical Sketch in volume 1, which also contains notes on some of the people mentioned on the following pages. Face to face with the wonderful unity in Niels Bohr’s personality, with the perfect harmony between the scientific thinker and the living and feeling human being, we cannot but wonder: Out of what unique and delicate concomitance of inborn gifts and outer circumstances did this unity evolve? However unfathomable the enigma of the unfolding of genius, it has enticed imagination and challenged inquiry since time immemorial. In the case of Niels Bohr, our curiosity is not least kindled by the peculiar epistemological timbre that permeates his entire life-work. In fact, it may be not too much of an exaggeration to describe Bohr as a born philosopher of nature, who found in physics a marvellously powerful instrument for probing into the foundations of human knowledge and man’s description of the world. Niels Bohr’s perspicacious father, the distinguished physiologist Christian Bohr, was very early aware of the depth and originality of Niels’ talents - even though his brother Harald was generally considered the brighter of the two.

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Niels Bohr’s parents.

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Legend has it that Niels was only three (!) years old when he made the famous reply to his father, who was explaining to him the ingenuity of the construction of a tree: how the trunk divides into branches, which agaifi divide into thinner branches and so on: “But father, if it had been otherwise, there would have been no trees”. The truly remarkable feature of this answer is not so much the precociousness of the little boy, but rather its kinship to the views of the mature thinker, that “if this is so, it could not have been otherwise”, i.e., that every element in nature is essentially unique. Many of us remember how Bohr told that as far back as he could remember he had liked “to dream of great interrelationships”. It is perhaps not without interest to compare this piece of childhood recollection with a note left by Christian Bohr’, in which he recalls: ”When I speak of that period of my earliest childhood which I can clearly recollect myself, then, like the whole of my later life, it was characterized to the highest degree by one single gift, if I may call it such, which goes back as far as I can remember, and which was never out of my mind for a single week, I dare say hardly a single day. I owe it to this gift that my life has retained some coherence and that in no period, in spite of quite a few less fortunate tendencies, did I ever depart from serious unselfish striving. I refer to the love of natural science, or more precisely certain aspects of the study of nature. I am quite sure that I had this love in my ninth year in essentially the same form as it still dominates my life. If I had to describe it more closely, I think it would be best to call it an instinct; regard for my position in life or the like has certainly not been involved in it; neither was it any definite purpose which obsessed me. It was not until much later in my life that I became grateful that this love made me work along a line - that of scientific research and education - which from the ethical point of view I place on the highest level. And I remember well that when this became clear to me I had the most definite conception of what a misfortune it would have been for me if this instinctive urge had not been of such a kind that it inevitably must lead to an aim that I could respect.” The exact extent of his father’s influence on Bohr’s thinking cannot be gauged of course, but one may venture the opinion that it represents by far the strongest single external impetus. It must be borne in mind that in spite of the scope and intensity of his philosophical ponderings, Bohr - in contrast to scholars like

’

Note in Christian Bohr’s handwriting, in the possession of the Bohr family. Quoted in Niels Bohr, His life and work as seen by his friends and colleagues (ed. S . Rozental), North-Holland Publ. Co., Amsterdam 1967, p. 11. The translation is slightly revised.

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Heisenberg and von Weizsacker - never acquired any traditional academic background in philosophy, and that he certainly did not think highly of the teachings of “professional” philosophers. I remember how he liked to joke about the snares of a language that permitted a grammatical construction like “das Es”, thereby suggesting “it” as a subject for philosophical inquiry. He used t o say that since those distant days when human beings began to use words like “here” and “there”, and “before” and “now”, there had been no further advance as regards epistemological insight into the nature of space and time until Einstein’s theory of relativity. In contrast, Bohr was deeply attracted by the true “philosophers of life” like Confucius, Lao-Tse, Socrates or Spinoza, whom he ranked with the great poets and with religious thinkers like Buddha or the old Jewish prophets, in their common endeavours to interpret the conditions of human life. Often in conversations he would measure one with the other and would often incline to credit the poet with the deeper understanding. It is typical in this respect that Bohr’s knowledge of the austere teachings of Hegelian philosophy as far as we know was derived from the smiling wisdom of Poul Martin Mdler’s “Adventures of a Danish Student”, this gem in Danish literature. Thus, the general epistemological view, which the young Bohr was developing, was shaped by very little systematic external influence. This is, of course, not to say that he may not have picked up a good many seeds of wisdom from reading and - not least - from discussions within his father’s circle. In Bohr’s Memorial Lecture on Harald H0ffding2 he recalls: “My first reminiscences about H ~ f f d i n gdate back to some evening gatherings, described by him in his ‘Recollections’. About a generation ago a small circle of scientists met regularly in their homes and discussed all sorts of questions that had captured their interest. The other members of this circle were H ~ f f d i n g ’ close s friends from their student years, Christian Christiansen and Vilhelm Thomsen as well as my father, who was quite a bit younger, but who during the years got on more and more intimate terms of friendship with H ~ f f d i n g .From the time that we were old enough to benefit from listening to the conversations and until the gatherings were interrupted on our part by the early death of my father, we brothers were allowed to be present when the meetings were held in our home, and from there we have some of our earliest and deepest impressions. During the often very vivid discussions, Christiansen especially enjoyed teasing Herffding in his humorous manner about general philosophy’s aloofness from the world. But like everybody else he appreciated quite well the full extent to which H ~ ffding’spower of comprehension and ’Overs. Dan. Vidensk. Selsk. Virks., Juni 1931 - Maj 1932, pp. 1 3 1 - 1 3 6 .

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desire for a general synthesis was so to speak the nourishing soil from which the ideas of the others germinated, stamped by their different studies and outlook.” The quotation from H~ffding’s recollection^"^ alluded to by Bohr contains a portrait sketch of Bohr’s father from which we may glimpse familiar features of the son: “The regular meetings, from which I derived much pleasure, began roughly at the time I have been speaking about. They started when I used to join Christian Bohr, the physiologist, after the meetings at the Royal Danish Academy, and we would then carry on the discussion in a cafe. ... As a physiologist and a disciple of the Leipzig scientist Ludwig, he followed a line that requires the strict application of physical and chemical methods to physiology. Outside the laboratory he was a keen worshipper of Goethe. When he spoke of practical situations or of views of life, he liked to do so in the form of paradoxes and these were improvised as a rule. A conversation was given new life when he joined in. Our gatherings at the cafe after the meetings at the Academy soon included a third member, the physicist Christiansen. He and Bohr had many interests in common, as Bohr’s physiological method led him to detailed studies in physics. ... This trio, which had been formed, soon tired of caft life, and it was therefore arranged that we should in turn go to each other’s home on those Friday evenings when the Academy was meeting. A fourth man now joined us, the famous philologist Vilhelm Thomsen.” When looking back on the formative years of the young Niels Bohr, it is altogether essential to realize that notwithstanding his deeply attentive openness, Bohr was already then much too original and independent a thinker to be easily influenced by the various philosophical viewpoints that he might encounter. But all his life he had a most sensitive ear for striking formulations and a keen appreciation of the subtleties of language. Thus, whenever he came across a phrase that struck him, he would add it to his stock and use it with slight variations time and again, even when after many years he would no longer be certain where it originally belonged. The r6le of the “wisdom of the East” may provide an illustration of the point at issue. Bohr himself has told us how deep and lasting an impression he received from the meeting with the old cultures of Japan and China during his voyage Harald H ~ f f d i n g Erindringer, , Gyldendalske Boghandel following quotation is from pp. 171-172.

-

Nordisk Forlag, Copenhagen 1928. The

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around the world in 1937. However, it is really not possible to trace any influence on his thinking and writing from this encounter. In fact, the beautiful simile about our being actors as well as spectators in the drama of existence, which Bohr later used to associate with ancient oriental thinkers, already appears in the lecture to the Scandinavian Scientists from 1929 (videp. [253]). It is possible that Bohr actually picked it up from the “Adventures of a Danish Student”, where the licentiate says4: “Thus on many occasions man divides himself into two persons, one of whom seeks to deceive the other, while a third one, who is really the same as the two others, is highly surprised at this confusion. In short, thinking becomes dramatic and silently plays the most tangled intrigues with itself. But the spectator anew becomes an actor. I shall insert a long chapter on this in my work.” Similar considerations apply to Bohr’s admiration for Kierkegaard, which was altogether artistically oriented, centred on the subtleties of style and language. This was certainly the case in his later years when I discussed this theme with him, but I have learned from conversations with Mrs Margrethe Bohr that this was true even in the early days when she first met her husband. In his careful study5, David Favrholdt sifts through the arguments of Jammer6, who believed he had found a strong influence on Bohr’s thinking from Kierkegaard and H ~ f f d i n g . Favrholdt finds no evidence for any such influence. As regards Bohr’s attitude to Kierkegaard in his later years, I remember him joking about Kierkegaard’s simile, comparing man’s position in existence to a leap into water seven thousand fathoms deep. “You can drown in seven fathoms of water”, said Bohr, “the question is, whether or not you can swim, and then it is immaterial whether the water be seven, seven thousand, or seven million fathoms deep”. He also liked to quote the closing sentences from “Stages on Life’s Way” where Kierkegaard praises the richness and beauty of his mother tongue. Bohr made the point that the very art and profundity of these lines turn them into a paean not merely to the Danish, but to all human language, reminding us of its infinite capability for refinement’. Here translated after Poul Martin Maller, En Dansk Students Eventyr, in: Efterladre Skrifter. I Udvalg ved Chr. Winther, Reitzel, Copenhagen 1873, Vol. 1, p. 243. David Favrholdt, Niels Bohr and Danish Philosophy, Danish Yearbook of Philosophy 13 (1978) 206-220. Max Jammer, The Conceptual Development of Quantum Mechanics, McGraw-Hill, New York 1966, pp. 172 ff. Bohr refers to this point in his article Dansk Kultur, an introduction to Danmarks Kultur ved Aar 1940, Det danske Forlag, Copenhagen 1941-1943. This article is reproduced in Vol. 10.

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Harald Herffding.

Harald and Niels Bohr.

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Let us now once more briefly return to the philosophical interests of the young Niels Bohr. In 1905 a few students, who had met in H~ffding’scolloquia (which formed part of his introductory course in philosophy, “Filosofikum”), decided to form a discussion group to debate philosophical and scientific problems. They called the group “Ekliptika”, because the number of members was limited to twelve. The basis of the nocturnal discussions was questions of philosophy and epistemology, raised in H ~ f f d i n g ’ scolloquia. During the winter term the group met several times a month at the CafC a Porta at Kongens Nytorv and the discussions were prolonged into the small hours. In an article written on the occasion of Niels Bohr’s 70th birthday in 1955, a member of the circle, the art historian Vilhelm Slomann, described in retrospect these “Ekliptika” evenings as follows*: “When the discussions were beginning to tail off, it often happened that one of them [i.e., Niels and Harald] said a few generous words about the lecture and continued in a low voice, at a furious pace and with vigorous intensity, but was often interrupted by his brother. Their way of thinking seemed to be co-ordinated; one improved on the other’s or his own expressions, or defended in a heated, yet at the same time good-humoured manner his choice of words. Ideas changed their tone and became polished; it was not a defence of preconceived opinions, but something new which came into being. This way of thinking h deux was so deeply ingrained in the brothers that nobody else could join in. The chairman used to put his pencil down quietly and let them carry on; but when everybody moved in closer to them he might say ineffectively, ‘Louder, Niels’.” During his years of study at the University, Bohr’s philosophical ponderings crystallized into a comprehensive epistemological view, which - even before the completion of his studies - he planned to publish as a book. When an interviewer asked him what place this work then had in his existence, he gave the characteristic reply: “It was, in some way, my life!”g. As far as we know, Bohr’s youthful philosophical ideas revolved around the problem raised by the movability of the partition between subject and object - a theme which was to recur in full orchestration in his mature writings on epistemology. In those early days Bohr did not look to physics but to mathematics for logical analogies, and he found them in Riemann’s theory of Vilhelm Slomann, Minder om samvzr med Niels Bohr [Recollections of Niels Bohr], Politiken, 7 October 1955. Archive for the History of Quantum Physics (AHQP), interview with Niels Bohr, 17 November 1962. Transcript, p. 2.

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Margrethe Nerrlund and Niels Bohr.

Niels Bohr.

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multivalued complex functionslO. He pictured the singular r81e of the subject in the description as an analogy to a branch point in the complex plane. Just as the uniqueness of the function depends on a specification of the sheet of the Riemann surface under consideration, the unambiguous communication of experience drawn from introspection demands the fixation of the position of the partition between subject and object. Although Bohr did not return to this analogy, it strikes us today as certainly having a Samifiar ring. It is a remarkable testimony to the inner preparedness for an epistemological “lesson” with which Bohr approached physics. It is this type of anticipation that lies behind the famous remark of his friend, the psychologist Edgar Rubin, when Bohr lectured him in the late twenties on the “lesson” of quantum mechanics: “But Niels! You told us all of that twenty years ago!”

*** Against the background outlined here one may perhaps grasp how Bohr came to experience the break-through in quantum physics as an immense liberation of the mind. It was as if he, like the great artists, had created a world from within, so deeply was this widening of the epistemological horizon in consonance with his own innermost feelings. Never did he suffer from the common spiritual dichotomy leaving intellectual insights apart from the longings of the soul. Indeed, it may be appropriate here, of all places, to state with confidence that whatever admiration we may justly feel when measuring Niels Bohr’s achievements in science and philosophy, the true miracle remains the unity, the fullness and harmony of his human personality. He was certainly not spared his part of the sufferings that are the lot of mankind. But they only added to his human growth, so that in the end he emerged as a living symbol of all that is lovable in man.

lo

Cf. Leon Rosenfeld, Niels Bohr’s Contribution to Epistemology, Physics Today, Vol. 16, no. 10,

1963, pp. 41-54.

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CHRONOLOGY OF EVENTS

Except where otherwise indicated, the dates refer to the receipt of the papers by the journals in question 1925 January

Kramers and Heisenberg, Dispersion theory

July

Heisenberg, Matrix mechanics

August 3 1

Bohr, Lecture “Atomic Theory and Mechanics”

SeptemberNovember

Born, Heisenberg and Jordan; Dirac, Formal developments of quantum mechanics

November

Uhlenbeck and Goudsmit, Spinning electron

December

“Atomic Theory and Mechanics” published in Nature

1926

January

Pauli, Dirac, The hydrogen spectrum derived from matrix mechanics

January- June

Schrodinger, Wave mechanics (four papers)

February

Thomas, Spinning electron

March

Schrodinger, Equivalence between matrix and wave mechanics

April- July

Jordan, Transformation theory

May

Heisenberg takes up his lectureship in Copenhagen

P A R T I: T H E E M E R G E N C E O F T H E C O M P L E M E N T A R I T Y A R G U M E N T

July- August

Born, Dirac, Statistical interpretation of wave mechanics

October

Schrodinger visits Copenhagen

December

Dirac, Transformation theory

December 17

Bohr, Lecture in the Danish Academy on “Atomic Theory and Wave Mechanics”

1927 February

Dirac, Quantum electrodynamics

March

Heisenberg, Uncertainty relations

April

Experiments by Davisson and Germer published

May 28

Campbell, “Philosophical Foundations Theory”. Answer by Jordan

July

Bohr, Earliest drafts of manuscript to Como Lecture and answer to Campbell

September 13

Bohr, Manuscript: “Fundamental Problems of the Quantum Theory”

September 16

Bohr, Lecture in Como

October 12-13

Bohr, Manuscript: “The Quantum Postulate and the Recent Development of Atomic Theory” (sent to Darwin)

October 24-29

5th Solvay Conference. Bohr lectures. Discussions with Einstein

November 11

Bohr, Lecture in the Danish Academy with the same title as the Como Lecture. Continued work on preparation of the manuscript

of

Quantum

1928 January-February

Dirac, Relativistic electron theory

End of March

Last proofs for publication of the paper in Naturwissenschaften and Nature

April 13

“Das Quantenpostulat und die neuere Entwicklung der Atomistik” , Naturwissenschaften

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

April 14

“The Quantum Postulate and the Recent Development of Atomic Theory”, Nature

July

Transactions of the Como Conference

INTRODUCTION bY

J0RGEN KALCKAR

1. COPENHAGEN DISCUSSIONS PRIOR TO THE ESTABLISHMENT

OF

THE UNCERTAINTY RELATIONS

In Volume 5 we left Niels Bohr in the spring of 1926 busily engaged in discussions on the Thomas effect for the electron spin. Also, during these months Schrodinger published his famous series of papers on “wave mechanics” which intensely occupied Bohr (cf. p. [8]), and at the beginning of May Heisenberg arrived in Copenhagen to take up his lectureship. Shortly afterwards, in June, Bohr fell ill, probably from overwork, and had to abandon his plan to join Heisenberg on a holiday trip to Norway’. In August, however, Bohr was well enough to go to Oxford to participate in a meeting of the British Association, and when Schrodinger visited Copenhagen, he was the one to fall victim to Bohr’s recovered strength. But more about this shortly. The two and a half years from Bohr’s lecture on “Atomic Theory and Mechanics” in August 1925 to the final publication in April 1928 of his next major work “The Quantum Postulate and the Recent Development of Atomic Theory” saw the beginning as well as the completion of the entire proud edifice of quantum mechanics. The chronological table on pages [3] to [5] has been included to facilitate the reader’s orientation among the rapidly moving events. As a glance at the chronological table will show, the Como Lecture was delivered at a time when not only the formal framework of non-relativistic quantum theory had been completed, but when also the decisive experimental demonstration of the “wave aspect” of matter by Davisson and Germer and by G. P. Thomson had become known. Before the final publication of the lecture, Dirac had inaugurated a new era with his relativistic quantum theory of the electron.

’

Cf. the letter from Heisenberg to Mrs Margrethe Bohr, dated Lillehammer, 12 June 1926. BSC, microfilm no. 1 1 .

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

In view of the almost tumultuous progress during these three years, the following brief outline of the development of Bohr’s views from 1926 to 1928 - based on letters and reminiscences - may serve to provide the reader with a general background for appreciating the final versions of the Como Lecture. It need hardly be emphasized that the scope of the present work allows only a sketch of the most salient features2. On the road towards the completion of the Como Lecture there are three occasions which in particular invite our attention: The debate with Schrodinger during his visit to Copenhagen in the autumn of 1926, the discussions with Heisenberg in connection with his establishment of the uncertainty relations, and, last but not least, the discussions with Einstein during the Solvay Conference in 1927. Along with these conversations, there is the almost continuous exchange of ideas with Pauli, which we should also like to illustrate by excerpts from their letters. As is evident from the final version of “Atomic Theory and Mechanics”, Bohr never entertained any doubts that with Heisenberg’s matrix mechanics the decisive break-through towards the creation of a consistent “quantum mechanics” had been achieved. It is amusing to observe the difference in attitude expressed in the following two quotations from Bohr and Einstein, respectively: Bohr to Rutherford, 27 Jan 26 Full text on p. 14571

Einstein to Besso, 25 Dec 25 German

“In fact, due to the last work of Heisenberg prospects have with a stroke been realized, which although only vague[ly] grasped have for a long time been the centre of our wishes.” “The most interesting recent theoretical achievement is the Heisenberg-Born-Jordan theory of quantum states. A real sorcerer’s multiplication table, in which infinite determinants (matrices) replace the Cartesian coordinates. It is extremely ingenious, and thanks to its great complication sufficiently protected against d i ~ p r o o f . ” ~ As already alluded to, it was during the early months of 1926 that Schrodinger published his classic series of papers on “wave mechanics” and proved its equivalence to “matrix mechanics”. Bohr greeted enthusiastically the new scheme which so strongly brought the “wave-particle-dualism” (as it was dubbed in the early days) into focus. In a letter to Fowler of April 14, 1926, he writes: From the vast literature we shall here only direct the attention of the interested reader to a perhaps lesser known work, the valuable thesis by Klaus Stolzenburg: Die Entwicklung des Bohrschen Kornplernentaritatsgedankens in den Jahren 1924 bis 1929, Universitat Stuttgart (Technische Hochschule), 1977. Albert Einstein, Michele Besso, Correspondunce 1903-1955, (ed. Pierre Speziali), Hermann, Paris 1972 (reprinted 1979).

PART I: THE EMERGENCE OP THE COMPLEMENTARITY ARGUMENT

“It looks indeed that we are on a high road to advance now. At present we are here having a visit of Pauli, who has just in these days succeeded in proving that the beautiful method of determining energy values for stationary states proposed by Schrodinger in the last issue of the Annalen der Physik will always lead to results identical with those derived by the methods of Heisenberg, Born and Dirac. It looks even that Schrodinger’s method may offei a simplification in the calculations especially as regards the determination of the transition probabilities.”

Bohr

10 R H. Fowler, 14 April 26 Full text On p. [421]

Thus, contrary to the Gottingen school’s inclination to regard the wave theory merely as a formal artifice, Bohr would stress the physical significance of this description. In this connection Friedrich Hund in an interview4 with Thomas Kuhn recalls a conversation with Bohr, which he tentatively dates to November 1926 or February 1927: “In Copenhagen I showed Bohr the paper in which I first applied the Schrodinger equation to a molecule, you know this one-dimensional model with the ‘node rule’, the interpolation between separated and united atoms. Originally it contained, I believe, some sentences like this: ‘I apply here the Schrodinger version of the new quantum mechanics, because it especially suits this particular problem’, or probably even more cautiously expressed. And then Bohr got somewhat annoyed and said: ‘You are doing Schrodinger quite an injustice, there is something much deeper behind it.’ Now I do not know exactly what he said, but it would have been something like this: that he just considered the foundation of the quantum theory on a particle-conception and a wave-conception as very essential, and liked to make both these two pillars of quantum theory equally strong.” However, even though Bohr would defend the physical significance of the wave theory, he dissented emphatically from Schrodinger’s attempt at a “literal” interpretation of the “matter-waves” so as to avoid the discontinuous “quantum jumps” (to use the phraseology of that time). When Schrodinger visited Copenhagen in October this difference of opinion came to the fore. The background to Schrodinger’s visit was the following: In July, Heisenberg had gone to Munich on vacation to stay with his parents5. Just at that time, Translated from the German transcript, 2nd interview on 25 June 1963, p. 8 . Interviews belonging to the Archive for the History of Quantum Physics (AHQP). Vide W. Heisenberg, Der Teil und dus Gunze, Piper Verlag, Munich 1962. English translation, Physics and Beyond, Harper and Row Publ., New York 1971. The passages quoted below (p. [ l l ] ) are however translated anew by the editor, as the rendering in the published English translation was found to be quite misleading when compared to the German original. Concerning Heisenberg’s visit

AHQP,

E;:r6jwlth

Hund3

German

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

Sommerfeld had invited Schrodinger to give some seminars on his theory and Heisenberg attended these lectures. Whereas he was impressed by the mathematical simplicity of wave mechanics, he was highly dissatisfied with Schrodinger’s interpretation of the formalism. In a letter to Pauli he remarked: Heisenberg to Pauli, 28 July 26 PWB I, letter [1421 German

“Just as nice as Schrodinger is as a person, just as strange I find his physics. When you hear him, you believe yourself 26 years younger. In fact, Schrodinger throws overboard everything [that tastes] ‘quantum theoretical’: Photo-electric effect, Franck collisions, Stern-Gerlach effect etc. Then it is not difficult to make a theory. But it just does not agree with experience.” During the discussion which followed Schrodinger’s lectures, Heisenberg tried in vain to point out the fallacies in Schrodinger’s argumentation. He was silenced by an acidulous rejoinder from Wien. In his book, Heisenberg tells how he perhaps the same evening - somewhat depressed wrote to Bohr about the unfortunate outcome of the discussion. Although such a letter has not been found in the Niels Bohr Archive, it may well be that Bohr’s letter to Schrodinger in September, inviting him to lecture in Copenhagen to the Physical Society (“Fysisk Forening”), was a consequence of Heisenberg’s experience. The visit took place during the first week of October, and the heated discussions with Schrodinger on this occasion remained vivid in Bohr’s memory throughout his life. They belonged to those episodes that he was very fond of recounting. He would relate how Schrodinger not only attempted a literal interpretation of the stationary states as “standing matter waves”, but even pictured the transitions between such states as a continuous process during which one vibration gradually died away and another took over. But then Bohr would confront Schrodinger with the fact that without the discontinuous transitions, it was not even possible to maintain Einstein’s derivation of Planck’s radiation formula. Bohr’s insistence led Schrodinger to despair. He explained that if one had to put up with these damned quantum jumps (“verdammte Quantenspringerei”), he was sorry that he ever got himself mixed up with quantum theory. To this Bohr answered: “But we are all of us so grateful that you actually did!”, adding that the formalism of wave mechanics represented an immense step forward. In his beautiful book Heisenberg has given a colourful account of the debate6: to Munich, see also the letter to Pauli quoted below. The full letter in German is reproduced in Wolfgang Pauli, Wissenschaftlicher Briefwechsel mit Bohr, Einstein, Heisenberg u.a., Band I: 1919-1929 (eds. A. Hermann, K. v. Meyenn and V. F. Weisskopf), Springer-Verlag, New York 1979. In the following this volume is referred to as PWB I followed by the number of the letter. Ref. 5 . Cf. also AHQP, interview with Heisenberg, 25 February 1963. Transcript, pp. 10-11.

P A R T I: T H E EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

Erwin Schrodinger (Courtesy AIP Niels Bohr Library with the kind assistance of Jost Lemmerich).

“The discussions between Bohr and Schrodinger began already at the railway station in Copenhagen and were continued each day from early morning until late at night. Schrodinger stayed in Bohr’s house and so for this reason alone there could hardly be an interruption in the conversations. And although Bohr was otherwise most considerate and amiable in his dealings with people, he now appeared to me almost as an unrelenting fanatic, who was not prepared to make a single concession to his discussion partner or to tolerate the slightest obscurity. It will hardly be possible to convey the intensity of passion with which the discussions were conducted on both sides, or the deep-rooted convictions which one could perceive equally with Bohr and with Schrodinger in every spoken sentence.. . . ...So the discussion continued for many hours throughout day and night without a consensus being reached. After a couple of days, Schrodinger fell

P A R T I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

ill, perhaps as a result of the enormous strain. He had to stay in bed with a feverish cold. Mrs Bohr nursed him and brought tea and cakes, but Niels Bohr sat on the bedside and spoke earnestly to Schrodinger: ‘But surely you must realize that ...’.” On his return to Berlin, Schrodinger summarized his impressions of the visit to Copenhagen in a letter to Bohr: Schrodinger to Bohr, 23 Oct 26 German text on p. [459]

“It is no small debt of gratitude that I feel that I owe you and your wife as well as all the others, who treated me with such kindness and friendship, such consideration and helpfulness during my stay in Copenhagen. For more than one reason this week will remain unforgettable in my memory - I need hardly elaborate more on this. The impression of the unique beauty of the city with its lovely and magnificent surroundings would in itself have been a lasting and permanent one. The lovely, sunny, hospitable home alone, with its kind household, which received me, a stranger, as an old friend and accorded me every comfort, this was an experience which the heart will never forget. But now this city, this house, this family - they are those of the great Niels Bohr, it is to him that I owe all this friendliness. I may talk with him for hours on the problems that I have so much at heart, and learn from him about his present attitude towards the numerous attempts to extend a little further the wide and solid foundation that he has created for modern physics. For a physicist, if he is really a physicist with all his heart, this is a truly unforgettable experience! It is possible that the stubbornness, with which in our dialogues I continued to adhere to my ‘wishes’ for a physics of the future, in the end may have left you with the impression that the general and specific objections that you raised against my views had not made any real impression on me. That is certainly not the case. In a certain sense I can say: the psychological effect of these objections - in particular the numerous specific cases in which for the present my views apparently [‘offenbar’] can hardly be reconciled with experience is probably even greaterfor me than for you. And this for the very reason that you, as it seems to me, have found a certain preliminary position in the view that all the apparently visualizable’ pictures are really only to be regarded symbolically, above all - as pointed out by Dirac in his most recent publication and also advocated by Born - that the c’s, the amplitudes or coefficients of the individual proper vibrations merely convey statistical statements on the

’The German word is “anschaulich” which is difficult to render in English. It means approximately something that we can imagine in pictures. Henceforth this word (and the Danish equivalent “anskuelig”) will be translated as “visualizable”.

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

behaviour of a large number of identical systems, and do not describe the behaviour of a single system. However, I am quite unable to set my mind at rest with this preliminary solution. It appears to me in general just as inapplicable as my own. When a light wave strikes a large number of atoms (like a gas), then every single atom must after all emit a weak secondary wave, otherwise one cannot understand the attenuation and dispersion of the light wave. On the other hand, if the light wave has just the resonance frequency, then only a few single atoms may in fact suffer a considerable change (‘being raised to the higher state’). Apparently there is a contradiction here and you say: here the words and concepts used until now no longer suffice. I do not feel satisfied with this ascertainment [‘Konstatierung’], and from it I cannot deduce that I am justified in continuing to operate with contradictory statements. One may weaken the statements, by saying, e.g., that the collection of atoms ‘in certain respects behaves as if ...’ and ‘in certain respects so as if ...’, but this is so to speak merely a juridical expedient that cannot be converted into clear reasoning. I do not consider it inconceivable to construct pictures that actually reproduce the above circumstances. The radiation damping has hitherto not really been taken into account in any of the new theories. In reality it represents, however, a necessary supplement to any theory, also to the original one, operating with electron orbits, as you have often emphasized. For many purposes it may be ignored, and, as a matter of fact, one always does so, inasmuch as phenomena that really can only be directly deduced from the radiation damping (line width, decay time), are either indirectly derived, by roundabout methods, or merely qualitatively through the correspondence principle by going back to the classical description. - Now, perhaps the radiation damping, the reaction on the system from the wave emitted by itself, should still be taken into account in a way quite different from what I originally thought, namely not by adding a (non-linear) term in the wave equation which would anyhow be a severe blemish, but in a quite different manner, perhaps - to take an example - through coupling to another system, the ‘ether’, possessing a continuous spectrum of eigenvalues from zero to infinity. I have not yet any definite ideas in this direction, and I should not impose my phantasies on you. What I vaguely see before my eyes is only the thesis: Even if a hundred attempts have failed, one ought not to give up hope of arriving at the goal, I don’t say through classical pictures, but through logically consistent conceptions of the true nature of the space-time events. It is extremely likely that this is possible.”

P A R T I : THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

Following a second letter from Schrodinger, Bohr replied without going into detail about Schrodinger’s worries: Bohr to Schrodinger, 2 Dec 26 German text on p. (4621

“Since your visit we have here often and thoroughly discussed the questions at issue. In these very days Klein is in the process of finishing a paper on the possibility of exploiting wave mechanics for the understanding of atomic processes involving discontinuitiesg. For you this paper is hardly going to bring much new, but I think that you will enjoy seeing how well wave mechanics is suited to exhibit the correspondence between classical electrodynamics and quantum theory. Actually it is possible, on the basis of wave mechanics, to build up a correspondence theory just as closed as matrix mechanics, which on the other hand may be regarded as a correspondence theory based on the corpuscular mechanics. In this connection it is interesting to see how the concept of wave or corpuscle presents itself as the more suitable concept, according to the point in the description where the assumption of discontinuities explicitly appears. In my opinion this is easily understood, since the definition of every concept or rather every word presupposes the continuity of the phenomena and hence becomes ambiguous as soon as this presupposition cannot be upheld. Howewer, no doubt this is merely the abomination of the subterranean that you find disgusting, and I need hardly stress with what great interest I follow your endeavours to realize your brighter hopes. If you are not able completely to kill the ghosts in ordinary space and time, then perhaps a settlement may be reached in the future in a five-dimensional world.’’ Already on October 26, Bohr had written to Fowler and told about the discussions with Schrodinger:

Bohr to R.H. Fowler, 26 Oct 26 Full text on p. [423]

“We had great pleasure of the visit of Schrodinger. The discussions centered themselves gradually on the problem of the physical reality of the postulates of the atomic theory. We all agreed that a continuity theory in the form indicated in his last paper at a number of points leads to expectations fundamentally different from those of the usual discontinuity theory. Schrodinger himself continued in his hope that the idea of stationary states and transitions was altogether avoidable, but I think that we succeeded at least in convincing him that for the fulfilment of this hope he must be prepared to pay a cost, as regards reformation of fundamental concepts, formidable in comparison with that hitherto contemplated by the supporters of the idea of a continuity theory of atomic phenomena. I understood that Schrodinger had 0 . Klein, Elektrodynamik und Wellenmechanik vom Standpunkt des Korrespondenzprinzips, Z. Phys. 41 (1927) 407-442.

P A R T I : T H E E M E R G E N C E OF T H E C O M P L E M E N T A R I T Y A R G U M E N T

been working under the impression that the essential characteristics of the matrix mechanics was the final recognition of the impossibility of ascribing a physical reality to a single stationary state, but I think that this is a confounding of the means and aims of Heisenberg’s theory. Just in the wave mechanics we possess now the means of picturing a single stationary state which suits all purposes consistent with the postulates of the quantum theory. In fact, this is the very reason for the advantage which the wave-mechanics in certain respects exhibits when compared with the matrix method which in other respects is so wonderfully suited to bring out the true correspondence between the quantum theory and the classical ideas. After the discussions with Schrodinger it is very much on my mind to complete a paper dealing with the general principles of the quantum theory such as I spoke of already during your visit. First in these days, however, I have got a chance to settle down to the work.” During the following months the discussions on the foundations of quantum mechanics continued between Bohr and Heisenberg. At that time Heisenberg lived in a little appartment in the attic of the Institute main building, whereas Bohr had moved into the “villa” just built on the Institute premises. Thus there was ample occasion for the two of them to meet also outside regular working hours, and in fact Bohr used to come up to Heisenberg late in the evening to resume the debate on the thought experiments at issue. As Heisenberg has graphically described9: “Like a chemist who tries to concentrate his poison more and more from some kind of solution, we tried to concentrate the poison of the paradox, and the final concentration was such experiments like the electron with the two holes and so on. They were just a kind of quintessence of what was the trouble.” Later in the interview Heisenberg explainslo: “The difficulties in the discussion between Bohr and myself was that I wanted to start entirely from the mathematical scheme of quantum mechanics and use the Schrodinger theory perhaps as a mathematical tool sometimes, but never enter into Schrodinger’s interpretation, which I couldn’t believe. Bohr, however, wanted to take the interpretation in some way very serious and play with both schemes.”

lo

AHQP, interview with Heisenberg, 25 February 1963. Transcript, p. 13. h i d . Transcript, p . 14.

~HQP,

lnteriies w i t h Heirenberg, 25 Feb 63 English

P A R T I : T H E E M E R G E N C E OF T H E C O M P L E M E N T A R I T Y A R G U M E N T

Day after day the discussions went on till late in the night, until by the middle of February 1927 both men had worked themselves into a state of tension and exhaustion. Then Bohr wisely decided to go on a skiing trip to Norway. That he stayed away no less than four weeks testifies to his need for relaxation and clarification of his mind. Also Heisenberg, alone back in Copenhagen, experienced a feeling of relief being able to pursue his own line of thought. When the two friends and adversaries met again each had achieved a decisive breakthrough. 2.

THE UNCERTAINTY RELATION AND THE 7-RAY MICROSCOPE

Bohr left Copenhagen for Norway in mid-February” and already by the 23rd of the month Heisenberg was able to send Pauli a letterI2, fourteen pages long, in which he expounded many of the essential ideas contained in his classic paper on the uncertainty relation^'^. Pauli’s answer has been lost and so has a letter from Bohr in Norway, for which Heisenberg thanked him on March 10. In this letterI4 Heisenberg mentions that he has written up his ideas in the form of a draft of a paper and sent it to Pauli. But he had asked Pauli to return the draft in order that he could discuss the content with Bohr upon the latter’s return. Nevertheless, Bohr arrived back in Copenhagen around March 18 15, only to find that Heisenberg had already sent the paper off for publication. As the printed article bears the date of receipt March 23, 1927, the paper must have been sent off from Copenhagen at a time when Bohr’s return was imminent. One may venture the guess that after the many months of preceding discussions, and knowing as he did Bohr’s thorough method of working, Heisenberg was eager to get the manuscript off before every paragraph of it would be overhauled by Bohr. But things were not to work out quite that way. We know from what Bohr himself told in later life that it was while skiing alone in the mountains around Gudbrandsdalen that he suddenly grasped the general complementarity argument which for so long had been dawning upon him. In fact, the letter to Einstein of April 13, quoted below (p. [21]) already contains the very essence of the argument. Thus he returned to Copenhagen with Cf. the letter from Klein to Saha, 18 February 1927. BSC, microfilm no. 16. Heisenberg to Pauli, 23 February 1927. PWB I, letter [154]. l3 W. Heisenberg, uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z . Phys. 43 (1927) 172-198. Received 23 March 1927. Since this paper is so intimately related to the development of Bohr’s complementarity argument we have found it appropriate to include it in the present volume (p. [I59]) with the kind permission of Heisenberg. l4 Heisenberg to Bohr, 10 March 1927. BSC, microfilm no. 11. l5 Cf. the letter from Bohr to Kronig, 18 March 1927. BSC, microfilm no. 13 l1

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

a rather definite view on the foundations of quantum mechanics, and must have been somewhat taken aback to find how rashly Heisenberg had sent off his paper for publication. That Bohr deeply appreciated the epoch-making character of Heisenberg’s discovery and was enthusiastic about it, is clearly borne out by his letter to Einstein already alluded to. Nevertheless, he was highly critical about many details, and Heisenberg was in for another two months of heated discussions, as he himself relates in his book (ref. 5 ) . On April 4, 1927, Heisenberg writes from Copenhagen in a letter to Pauli: “Otherwise there is here general agreement and thought experiments are constantly discussed. I argue with Bohr over the extent to which the relation p l q l h has its origin in the wave- or the discontinuity-aspect of qu[antum] m[echanics]. Bohr emphasizes that, e.g., in the y-ray microscope the diffraction of the waves is essential; I emphasize that the theory of light quanta and even the Geiger-Bothe experiment are essential. By exaggerating to one side as well as to the other one may argue at length without saying anything new.”

-

Heisenberg 10 Pauli,

,,

4 April 21 PWB

[I611

One of the points to which Bohr had drawn Heisenberg’s attention was the decisive r81e of the finite aperture of the microscope in preventing a precise fixation of the recoil of the electron through a subsequent measurement of the direction of the scattered light quantum. Apparently, Dirac had also raised this point, because in a letter from the end of April Heisenberg wrote to him: Copenhagen, April 27, 1927 Dear Dirac, I am very sorry, that I could not answer your letter before, but I was in Munich for a short vacation-time. - Your difficulties don’t seem to me very serious. In your first example you know first the velocity of the electron. Then you measure the place by the r-ray microscope. The point is, that one can not calculate the velocity afterwards, because the direction of the dispersed light quantum after the collision is not known. If you think of an ordinary type of real microscope, you want an

Heisenberg to Dirac, 27 Aoril 21 AHQP, mf. 59 English

PART I: THE EMERGENCE OF THE COMPLEMENTARITY A R G U M E N T

aperture-angle a, as large as possible, in order to get the accuracy A. If a, is

A

small, your accuracy will not be A but -, the uncertainty in the direction a, hv A h . This of the light quantum will be a, and p1= - *a,, q1= -, plql c v1 example was so discussed by Bohr. I had the same question always discussed in the following way: One imagines the microscope of the simple form:

-

Lc**L

-

I\&-.-_ j

I

.-

4 -

’ /q,

-

I

__L_ 4.-

A\

The holes have the width d (d>A). Every hole gives rise to a diffraction of the I. order - . The accuracy of the place, however, will be d = q l , if the electron d is not too far from the screens. The direction of the light quantum is again A I hv uncertain by the amount - and p1= - -* P141-h. d d c Also your second example is very pretty. In order to measure the velocity by Doppler effect, you want to know the frequency of the dispersed light very accurate. At a certain time to you know, that the electron is on a definite 1 place. The accuracy of the frequency-determination is -, if t l is the time tl

used to the determination (that is just classical wave-theory). The correv1

v1

1 c

sponding accuracy of the velocity is - = - or v1 = - -. The accuracy of c v tl v the place after the determination of v, i.e. after the time t l is apparently

c

v1 tl = - = A V

! You can never get it more accurate! The rest of the calculation

as in the last example. Prof. Bohr says, that one in all those examples sees the very important role, which the wave-theory plays in my theory and, of course, he is quite right. The best wishes to the whole Institute. Yours sincerely, W. Heisenberg

P A R T I: T H E E M E R G E N C E OF T H E C O M P L E M E N T A R I T Y A R G U M E N T

Even as late as in the middle of May, we learn from a letter from Heisenberg to Pauli that he and Bohr were continuing to discuss the p r a y microscope: “Since my return from the Easter vacation - this time particularly pleasant - we have here discussed quantum theory at length. Bohr plans to write a general treatise on the ‘conceptual constitution’ of qu[antum] th[eory] from the point of view: ‘There exist waves and corpuscles’ - if one starts right away with this, one can of course also make everything consistent. In respect to this work, Bohr has drawn my attention to the fact that in my paper yet another important point has been overlooked (Dirac also asked me about it later): With the p r a y microscope one might first imagine that one determines the direction of the incident light quantum as well as the reflected light quantum. Then after the Compton effect both the position and the velocity are known very accurately (more accurately than p l q l h ) . However, this cannot really be done, because of the diffraction of the light (wave theory!). To achieve an accuracy A, the microscope must have an aperture of the order 1. Thus the relation p l q l h is of course obtained, but not quite in the manner I thought. Furthermore, certain points could be better expressed and discussed in every detail, if only one begins a quantitative discussion directly with the waves. Nevertheless, my opinion is of course just as beforehand that in the qu[antum] th[eory] only the discontinuities are interesting and that one can never emphasize them enough. For this reason I am also now, as previously, very happy with this latest work - in spite of the error mentioned - all the results of the paper are correct after all, and I am also in agreement with Bohr concerning these. Otherwise, there is a considerable difference of taste between Bohr and me regarding the word ‘visualizable’.”

-

-

Pauli visited Copenhagen at the beginning of June and he played an important r61e in reconciling Heisenberg with Bohr’s point of view, as we know from a letter from Heisenberg to Bohr written shortly after Pauli’s visit (Bohr Private Correspondence). As a result of the discussions with Bohr, Heisenberg added the following note to his paper (ref. 13): “Note added in proof. After conclusion of the present work, recent investigations by Bohr have led to points of view that permit an essential deepening and refinement of the analysis of the quantum mechanical relationships, attempted in this work. In this connection Bohr has drawn my attention to the fact that in some of the discussions in this work I had overlooked essential points. First and foremost, the uncertainty in the observation does not exclusively depend upon the appearance of discontinuities, but is directly related to the requirement of doing justice at the same time to the different ex-

Heisenberg 16 May 27

t o Pauli,

pwB I , letter [1631 German

P A R T I: T H E E M E R G E N C E O F T H E C O M P L E M E N T A R I T Y A R G U M E N T

perimental facts which find expression in the corpuscle theory on the one hand and the wave theory on the other. E.g., in the use of an imaginary pray microscope one must take into consideration the necessary divergence of the beam of radiation. In the first place, this has the consequence that in the observation of the electron position, the direction of the Compton recoil is only known with an uncertainty which then leads to the relation (1). Furthermore it has not been sufficiently emphasized that the simple theory of the Compton effect can be applied strictly only to free electrons. As Professor Bohr has made clear, the caution which is accordingly necessary in the application of the uncertainty relation is for one thing essential for a comprehensive discussion of the transition from micro- to macro-mechanics. Finally, the considerations about resonance fluorescence are not quite correct, since the relationship between the phase of the light and the phase of the electron motion is not as simple as assumed. Since I was able to become acquainted with and discuss the aforementioned most recent investigations of Bohr, which are soon to appear in a paper on the conceptual constitution of quantum theory, while they were still being worked out, I owe my sincerest gratitude to Professor Bohr.”

Einstein and Bohr during the Solvay Conference 1930 (Courtesy AIP).

PART I : THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

Bohr’s views in the spring of 1927 are beautifully illustrated by the following letter to Einstein (accompanying a proof of Heisenberg’s article). Of special interest is perhaps the fact that Bohr here, just as in the published versions of the Como Lecture, bases the uncertainty relations directly on the “wave-picture” in conjunction with the Einstein-Planck quantum conditions. As alluded to above, the essence of the complementarity argument is already found in this letter: Copenhagen, April 13, 1927 Dear Einstein, Before his holiday trip to the Bavarian mountains Heisenberg asked me to send you a copy of the proofs that he was expecting of a new paper in Zeitschrift fur Physik, which he hoped might interest you. I dare say that this paper, which I am enclosing, represents a very important contribution to the discussion of the general problems of quantum theory. Since the content is closely related to the questions that I have had the great pleasure of discussing with you a number of times - most recently during the unforgettable days in Leiden at the Lorentz celebrations - I should like to avail myself of the opportunity to include some remarks concerning the problem that you discussed recently in the transactions of the Berlin Academy16. It has of course long been recognized how intimately the difficulties of quantum theory are connected with the concepts, or rather with the words that are used in the customary description of nature, and which all have their origin in the classical theories. These concepts leave us only with the choice between Charybdis and Scylla, according to whether we direct our attention towards the continuous or the discontinuous aspect of the description. Yet, at the same time we feel that it is the hopes, conditioned by our own customs, that are here leading us into temptation, inasmuch as it has hitherto been possible to keep ourselves swimming among the realities, as long as we are prepared to sacrifice any accustomed wish. This very circumstance that the limitations of our concepts coincide so closely with the limitations in our possibilities of observation, permits us - as Heisenberg emphasizes - to avoid contradictions. In connection with the question of light quanta it is therefore important to bring the well-known paradoxes into connection with the physical limitations in the concept of a train of monochromatic plane waves. For purely geometrical reasons the description of the wave train is of course connected with a certain indeterminacy in the wavelength that allows a description of the finite extension in the direction of propagation, in the same way as an uncerl6 A. Einstein, ober die Interferenzeigenschaften des durch Kanalstrahlen emittierten Lichtes, Sitmngsber. Preuss. Akad. Wiss., Jg. 1926, pp. 334-340.

Bohr to Einstein, 13 April 27 German text on p. [4181

P A R T I: T H E E M E R G E N C E OF T H E C O M P L E M E N T A R I T Y A R G U M E N ?

tainty in the parallelism of the rays entails a limitation in the cross section of the wave train. All this is a consequence of the well-known laws of optical time determination and the formation of images by optical instruments. If the uncertainty in the vibrational frequency is denoted by Av, then the time that 1

it takes the wave to pass a definite point is at least of the order A t = - .

Av

If further All denotes the uncertainty of the wavelength, then the order of A2

magnitude of the minimal extension of the train is A x = -, and the minimal All ll width A y = - , where E is an angle measuring the divergence of the light &

rays. This uncertainty in the geometrical description of the waves, and therefore in the possibilities of observation of the light quanta, thus stands in a peculiar inverse relationship to the accuracy with which the energy E = hv h and the momentum I = - of the quanta are defined. Thus we have A E A t =

1

.

-

hAll 112 he ll h Av.-=h, and dl,iAx=-- - h , and A I , A y = - = h , all Av A & 112 All in harmony with the general relation between the simultaneous uncertainties of conjugate variables, which according to Heisenberg is a direct consequence of the mathematical laws of quantum mechanics. Through the new formulation we are presented with the possibility of bringing the requirement of energy conservation into harmony with the consequences of the wave theory of light, since according to the character of the description, the different aspects of the problem never appear at the same time. In this connection I also believe that one may avoid the paradox, discussed by you in the Berlin Academyl6, concerning the spectral resolution of the light emitted by a moving atom and observed through a slit perpendicular to the direction of motion17. If we first consider the problem from the point of view of the wave theory, then we find that the uncertainty in the frequency, resulting from the limitation in the time of observation, is 1

V

of the order of magnitude A v = -, where v denotes the velocity and a the a

l7 In the paper cited, Einstein had shown from general arguments that light emitted from a moving atom must be expected to exhibit the same interference effects as radiation from a classical moving oscillator. On this basis he concluded that, in the experiment considered, there would be no effects associated with light quanta, contrary to his earlier expectations. The experiment in question is discussed by Heisenberg in his book: Die physikalischen Prinzipien der Quantentheorie, Hirzel Verlag, Leipzig 1930, p. 59. English translation: The Physical Principles of the Quantum Theory, Chicago University, 1930. Reprinted by Dover Publ. Here p. 79.

PART I: T H E EMERGENCE OF THE COMPLEMENTARITY A R G U M E N T

width of the slit. This result is in harmony with the fact that, owing to the diffraction of the light in the slit, light emitted from the moving atom in a certain A

finite range of directions - will arrive and be observed in a direction perpena

dicular to the motion. Thus one finds again for the frequency interval as a

. c a

consequence of the Doppler effect A v = v - - = - . Considering the V

A

V

a

energy balance, on the other hand, we find that, on account of the spread of frequencies, it is possible to detect sometimes a larger or smaller light quantum by a photoelectric effect. This is connected to the circumstance that kinetic energy may be removed from or supplied to the atom by a radiative recoil that may deviate from the perpendicular direction of observation. That one can observe not merely a statistical, but an individual energy balance, is connected to the fact that, as you indicate in your footnote, no possible ‘light quantum description’ can ever explicitly do justice to the geometrical relations of the ‘ray path’. Heisenberg shows in an exceedingly brilliant manner how his uncertainty relations may be utilized not only in the actual development of quantum theory, but also for the judgment of its visualizable content. In so far as this relation is a direct consequence of the quantum-mechanical formalism, the whole constitutes a very closed system, at least when one limits oneself to mechanical phenomena. As regards such a pedagogically coloured concept as visualizability, however, it seems to me instructive always to keep in mind how indispensable are the concepts of the continuous field theory in the present stage of science. As long as we only talk about particles and quantum jumps, it is difficult to find a simple presentation of the theory, which is based on a reference to the limitation in the possibilities of observation. This is because the uncertainty mentioned is not only connected to the presence of discontinuities but also to the very impossibility of a detailed description in accordance with those properties of material particles and light that find expression in the wave theory. The representation of an electron by a group of de Broglie waves is of course closely analogous to the representation of a light quantum by a group of electromagnetic waves. Hence all the above relations are also valid in this case. Now it follows directly from the uncertainty in the group velocity, corresponding to the uncertainty in the electron momentum, that the group in the course of time spreads out also in the direction of propagation, all this exactly as Heisenberg has worked it out on the basis of quantum mechanics in association with Dirac’s matrix transformation theory. However, I shall not trouble you any longer. One could of course talk for ever about this whole wonderful development. How nice it would be once

P A R T I : T H E EMERGENCE OF T H E COMPLEMENTARITY A R G U M E N T

again to be able to talk to you face to face about all these things. As far as I understand, it is Heisenberg’s intention to try to see you in Berlin on his way back. For a long time I have had the intention of trying to clarify my thoughts on the general questions in a small article, but the development runs so tempestuously, that everything anew becomes quite commonplace. Still I hope soon to finish such an article. With the kindest regards, Yours, N. Bohr With Heisenberg’s departure from Copenhagen in the summer of 1927 to take up a professorship in Leipzig, the close personal collaboration between him and Bohr came to an end. In his New Year’s greeting from 1928, Bohr gives eloquent expression to his feelings in looking back on their companionship during these exciting years: Bohr to Heisenberg, [Dec 281 Danish text on p. (4241

[Hornbak, End of December 19281 Dear Heisenberg, I ought to have answered your nice letter long ago; still I would not let the year come to an end without thanking you for the great pleasure you gave all of us by your visit. Rarely have I felt myself in more sincere harmony with any other human being, and I still rejoice, when I think back on our walks and discussions and not least on the memory of the evening when we listened together to Herffding’s beautiful lecture on Socrates. My wife and I have spent the days between Christmas and New Year out here in Hornbak together with my aunt and Hans and Christian. It has been a lovely time of rest, and I have had much occasion to ponder what the year has brought, and to dream about what the new will bring. I am sitting here, thinking how much we have all learned since that time during your first visit up here when we walked together through Hornbak on our way to Tisvilde and, with interruptions for stone-throwing and jumping across ditches, dreamed about the problems of physics and life. How have you been getting on with your own work in the autumn? Mrs Maar told me the other day that you and Pauli again believe you see brighter vistas ahead. Klein too has worked very much with relativity problems and has some thoughts on which he hopes to build a more satisfactory formulation. I am eager to learn what he has made of it, when he returns from his Christmas vacation. I have occupied myself a good deal with the question of the inobservability of electro[n- ?]magnetism and the Pauli principle; however it is still very unclear. In spite of Pauli’s warnings I am also still prepared for

P A R T I : THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

Werner Heisenberg at his inauguration lecture, Leipzig 1927 (Courtesy Mrs Elisabeth Heisenberg).

further limitations in the applicability of the energy concept. Gamow has of late occupied himself thoroughly with the continuous P-ray spectra; but every search for other solutions has hitherto strengthened my conviction that the difficulties lie very deep. How have you been doing with the superconductivity? Bloch’s beautiful work, which you so kindly sent me, and from which I got much pleasure, taught me of course that the way out which I had indicated was barred. Besides, I have spent most of my time during the autumn philosophizing about the foundations of quantum theory, and I hope now during the Christ-

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

mas vacation to work out an exposition of my ideas, which although they do not bring anything really new might still contribute to a clarification of the concepts. In this connection I have thought about many general problems also outside physics, which I hope we shall be able to talk about together in more detail before too long. For the present I will finish with my most cordial wishes for a happy New Year for yourself and your parents and many kind greetings from home to home, Yours sincerely, N. Bohr 3.

PREPARATION FOR THE COMO LECTURE

Both Bohr himself and Heisenberg (in the addendum to his article) refer to a forthcoming article by Bohr in which he intended to put forward his points of view in a systematic manner. On May 28, there appeared in NatureI8 a short note by Norman Campbell under the title: “Philosophical Foundations of Quantum Theory”. It was followed by an even shorter reply by Jordan. Campbell had conceived the idea that time was “a statistical conception, significant only with regard to large aggregates of atoms; and that it is as meaningless to speak of the time interval between atomic events as of the temperature of an isolated molecule”. In his reply, Jordan referred to Heisenberg’s work on the uncertainty relations (ref. 13, then still in press). Jordan pointed out that the “quantum theoretical conceptions differ from those of Dr. Campbell”, but conceded that “in a certain respect Dr. Campbell’s views are, however, confirmed by the quantum mechanics: for if the atom has specified quantum numbers, the time (and the co-ordinates) are statistically and only statistically, defined”. Not that this exchange of notes is particularly illuminating, but it is mentioned at this point because it incited Bohr to commence work on a note, an answer to Campbell, with the same title, “Philosophical Foundations of Quantum Theory”. Several drafts are preserved in the Niels Bohr Archive, but, as the material expanded, the plan to publish a brief note was abandoned and the substance was included in the manuscript, which was eventually to develop into the final version of the Como Lecture. During the steadily evolving phases of what was to become the Como Lecture, Bohr was assisted by Oskar Klein, and most of the surviving manuscripts are in Klein’s handwriting. The first referenceI9 to the notion of “complementarity” is

’’Nature 119 (1927) 779. l9

Concerning the question of the undated draft for an answer to Campbell and Jordan, cf. p. [28].

P A R T I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

probably that found in a manuscript20 of July 10, [1927]. This extremely interesting document, which is in Bohr’s handwriting (see the facsimile on p. [63]), contains a number of points, alluded to in a single or few sentences in Danish. Many are crucial, like the very first sentence (translated from the Danish original): “All information about atoms expressed in classical concepts”, and the next: “All classical concepts defined through space-time pictures.” These points are rephrased and appear on page 566 of the conference proceedings of the Como Lecture. The notion of complementarity is mentioned on the first page: “[Indeed?], the theory exhibited a duality when one considered on the one hand the superposition principle and on the other hand the conservation of energy and momentum. Complementary aspects of experience that cannot be united into a space-time picture based on the classical theories. ” From time to time there have been discussions as to the extent to which Bohr’s development of the complementarity argument was influenced by sources outside physics. Thus Meyer-Abich21 believes that he can detect influences from “The Principles of Psychology” by William James22, whereas Max Jammer23is convinced that Bohr was influenced not only by James, but even by Kierkegaard and Harald Herffding. To anyone familiar with Bohr’s style of thinking and working these conjectures appear highly unlikely, as has already been indicated in the General I n t r o d ~ c t i o n ~Apart ~ . from the trivial question of the word “complementarity” itself, which is of course by no means uncommon and is not 2o The

Niels Bohr Archive contains a wealth of notes and drafts for the Como Lecture. They are contained in five folders, one of which is labelled Como Lecture I , the remaining four Como Lecture II (Bohr MSS, microfilm no. 11). Besides the quotations in this volume, interesting extracts may be found in ref. 2. The manuscript alluded to in the text above is reproduced on p. [57]. It is found in the second folder labelled Como Lecture II. On the manuscript the year is erroneously given as 1926. K. M. Meyer-Abich, Korrespondenz, Individualitat und Komplementaritat. Eine Studie tur Geistesgeschichte der Quantentheorie in den Beitragen Niels Bohrs, Franz Steiner Verlag, Wiesbaden 1965. Cf. especially pp. 133-140. ” W. James, The Principles of Psychology, H . Holt, New York 1890 (reprinted by Dover, New York 1950). Cf. in particular chapter IX, The Stream of Thought, Vol. 1, pp. 224-290. 23 M. Jammer, The Conceptual Development of Quantum Mechanics, McGraw Hill Book Co., New York 1966, pp. 172-179 and 348-350. 24 There is even doubt as to the time when Bohr actually first read William James. Cf. AHQP, interview with Bohr, 17 November 1962 (p. 5 of the transcript) and interview with Rosenfeld, 22 July 1963 (pp. 8-9 of the transcript).’Cf. also ref. 2, p. 152, and the discussion in Vol. 10.

*’

PART I: THE EMERGENCE O F THE COMPLEMENTARITY ARGUMENT

necessarily derived from any particular source, it should be clear from the foregoing sections that the clarification of Bohr’s mind with regard to the peculiar features of description and observation in quantum mechanics was the result of a long and intense struggle with the physics involved. As Rosenfeld very aptly remarked in his review of Jammer’s bookZS:“Bohr’s conception of complementarity in quantum mechanics is not the expression of a ‘specific philosophical position’, but an inherent part of the theory which has the same validity as its formal aspect and is inseparable from it.” Throughout the summer months the idea of publishing a brief answer to Campbell and Jordan in Nature was maintained. According to KleinZ6a last effort to finish the note was made on the very eve of Bohr’s departure for Como in the middle of September: AHQP,

“

Interview with Klein, 28 Feb 63 English

... Bohr was trying to finish. Then suddenly he said, ‘Now, everything is

ready except the letter to the editor’. We were a little astonished, but he meant a special letter accompanying the ‘letter to the editor’. Finally he signed the letter, and he wrote ‘Niels Bohr’. Then he got very much upset because he should have written ‘N. Bohr’. So his brother said, ‘But your name is “Niels” ’. They were going to take a taxi to the station; their train was going a little after 12:OO. I left for home I think a little before that. The next morning, when I came, I heard immediately that they had just left in the morning, and that was because in the last minute they couldn’t find their passports ... . I heard also that the paper was not sent away, but that he had taken that with him. ’’

As already mentioned, the note was never published, its content being gradually worked into the final version of the Como Lecture. The temporarily completed note mentioned by Klein has not been found, but an uncompleted draft (in English) of an answer to Campbell and Jordan as well as several notes have survived. The draft in question” is called “Philosophical foundations of the quantum theory” and is included in this volume. It is undated, so it may have been written any time after May 28, 1927, when Campbell’s and Jordan’s notes appeared in Nature. Judging from the degree of similarity to the corresponding sections of the published versions of the Como Lecture, one would tend to favour a rather late date for the manuscript, perhaps even shortly before Bohr’s departure for Como. The manuscript deals with the mutually exclusive aspects of light phenomena and the analogous situation for material particles. The notion of complementarity is mentioned on the second page. This

Nuclear Physics A126 (1969) 696. AHQP, interview with Klein, 28 February 1963 (p. 11 of the transcript). Preserved in the Niels Bohr Archive in the second folder labelled Como Lecture II. Reproduced on p. [67].

25

’’ 26

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENI

part of the manuscript agrees largely with pp. 567-569 of the printed version in the conference proceedings, although there are of course differences in detail. Also the wording of the manuscript is in many places evidently provisional. In the later part of the manuscript Bohr comments on the Schrijdinger theory. These comments correspond largely to p. 579 of the proceedings. After mentioning Heisenberg’s uncertainty relations and matrix mechanics the manuscript breaks off.

Bohr’s talk at Como was given on September 16. We do not possess the text of the original lecture28, as the version published in the conference proceedings is of a much later date (although it differs substantially from the final editions in “Nature” and “Naturwissenschaften”). However, the Archive contains a fairly complete (although in places almost illegible) m a n ~ s c r i p t ~ running ~, to 8 pages in Bohr’s handwriting, with the title: “Fundamental problems of the quantum theory”, dated September 13. As this manuscript is the latest we possess before the delivery of the lecture, it is reproduced in full. The material covers essentially $61 and 2 of the version published in the conference proceedings. The manuscript ends by stating a number of points which are also dealt with in the published lecture. However, there is in this manuscript no reference to Campbell and Jordan. The opening sentence of the manuscript is found again only slightly changed at the beginning of 61, p. 566 of the proceedings. There follows in both the manuscript and the lecture a general discussion of the nature of observation and description. However the manuscript lacks substance. Complementarity is not mentioned at this point, nor is the mutually exclusive character of space-time coordination and conservation laws. It is hard to imagine that Bohr in his delivery of the lecture would have restricted himself to the indications given here. The next section is somewhat incomplete here and there, but it is clear that it deals with the nature of light and material particles in much the same way as 52 of the published lecture. Group and phase velocity are discussed and the uncertainty relations derived from the wave picture (as already in the letter to Einstein quoted on p. [21]). The diffraction experiment discussed in the manuscript is left out in the published lecture, where Bohr after a few general remarks proceeds to analyse the pray microscope and the momentum measurement by means of the Doppler effect. These examples are referred to in the manuscript in only a few sentences. Subsequently, some further points are alluded to in single words.

4. THE

COMO CONFERENCE AND CONTINUED W O R K ON THE MANUSCRIPT

The reception of Bohr’s presentation of his new ideas by the distinguished audience was remarkably cool. The discussion (reprinted from the proceedings on p. [137]) hardly touches on the fundamental issues brought forward by Bohr. In later life he used to tell, with humorous comments, how Wigner had aptly summarized the prevailing feelings with the remark: “This lecture will not induce any

28

The stenographic report in the possession of the Archive seems incomprehensible (Como Lecture

Z, [1927]. Bohr MSS, microfilm no. 11).

29 Preserved in the Niels Bohr Archive in the third folder labelled Como Lecture ZZ. Reproduced on P. 1731.

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

one of us to change his own opinion about quantum mechanic^"^^. To do justice to the listeners to this original exposition, and to underline the difficulties confronting them, it may, however, be well to recall that even Rosenfeld at that time was far from appreciating Bohr’s message. This is of course all the more remarkable, since in later years Rosenfeld’s doctrines were identified in wide circles with Bohr’s complementarity point of view - not with complete justice, I should like to add. Rosenfeld relates: AHQP, Interview with Rosenfeld, 1 July 63 English

“In fact, my own view of the Como lecture when I read it was that Bohr was just putting in a rather heavy form things which had been expressed much more simply by Born and which were current in Gottingen at the time. I did not see, I did not feel, any of the subtlety that was in it, and I suppose that this was the general feeling in Gottingen.” In this connection one must also concede that section 5 of the text printed in the conference proceedings is less skilfully woven than the earlier parts, mainly due to the vast number of different topics on which Bohr comments here (in fact this section was changed and subdivided in the Nature version). If this part originally formed the conclusion of the lecture, then one further reason can be given for the bewilderment of the audience. However, the main reason is without doubt to be found in the character of the lecture, its highly non-technical nature, with only a few elementary formulae, but an abundance of subtle comments on various aspects of quantum theory. It was of course not the last time that this characteristic style of Bohr’s created difficulties for the comprehension of his views. Even today his papers present formidable demands on the reader. After the Volta Meeting in Como, Bohr spent a week together with Pauli at Lake Como in order to prepare the publication of an extended version of his lecture. Back in Copenhagen a manuscript entitled: “Das Quantenpostulat und die neuere Entwicklung der Atomistik” was completed and sent to the Naturwissenschaften on October 11, although it was never published in this form. On the same day Bohr wrote to Pauli that he had asked that a copy of the proof be sent to him for critical remarks. This proof has not survived, but fortunately we possess an English version3’ (dated October 12-13) sent to Darwin on October 16. Since this manuscript represents the stage of development of Bohr’s views immediately before the Solvay Meeting, where the first discussion with Einstein took place, it is reproduced in this volume. 30 This remark was also remembered by Rosenfeld; cf. AHQP, interview, 1 July 1963 (p. 19 of the transcript). 31 Manuscript, The Quantum Postulate and the Recent Development of Atomic Theory. Reproduced on p. [89].

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

From the last sentence of the manuscript it is clear that Bohr thought of this article as being different from the Como Lecture: “A more detailed elaboration of this point of view in its application to a number of simple examples was recently given by the author in a lecture at the Volta congress in Como and will soon appear in the transactions of this congress,” However, it is by no means a revival of the idea of writing an answer to Campbell and Jordan. Indeed, these authors are not alluded to at all in this version. In point of fact, the manuscript follows rather closely the first four of the five sections of the lecture published in the transactions (but the manuscript is not divided into sections). In the first section the similarity extends to the detailed formulation of many paragraphs. It deals with the general conditions for observation and description and the nature of light and material corpuscles. A paragraph in the manuscript, where Bohr took a definite stand against de Broglie’s attempt at viewing material particles and light quanta as singularities in the wave-fields, was removed in the printed version. The next section deals with general properties of the wave description. The discussion of group and phase velocities is missing from this manuscript (p. 570 of the proceedings). The uncertainty relations are derived from the wave picture as in the lecture, but the ensuing analysis of the pray microscope and the momentum measurement by the Doppler effect is not included in the manuscript. Section 3 of the published lecture is also clearly prefigured here. It deals with matrix mechanics

Wolfgang Pauli.

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

and Heisenberg’s derivation of the uncertainty relations. At this point the manuscript contains the cautious passage: “It must be borne in mind, however, that we are concerned here with principal difficulties regarding our ordinary concepts much like those already contemplated in connection with the questions of radiation in empty space and isolated material particles. Indeed the shortcomings of our ordinary concepts appear, if possible, even more clearly in the general interaction.” This paragraph was omitted in the printed version. Section 4 of the lecture deals with the Schrodinger description and much of the material is contained in the manuscript. Here there are often important differences in formulation, the printed version representing almost everywhere a decisive improvement. The remark from the lecture about the symbolic nature of the wave description which refers to configuration space (coordinate space) is found in the manuscript but merely as an aside. The elaboration of this point may be due to Pauli’s criticism (vide p. [34]). As regards section 5 of the lecture, the manuscript contains only the paragraph (found on p. 587 of the proceedings) on the “reality of stationary states”. In particular, the short reference to Campbell in this section of the lecture is not found in the manuscript. Also lacking is the discussion of the Stern-Gerlach experiment and the inobservability of the phase, as well as the question of the limitations in the treatment of atoms as isolated systems and several other themes alluded to in section 5 of the lecture.

The comparison above shows that the main parts‘ of Bohr’s views, as they found expression at that time in the Como Lecture, are contained in the present manuscript. Thus we must conclude that the discussion with Einstein at the Solvay Meeting did not have any major impact on the final elaboration of the text in the form in which it appears in the transactions of the conference. As we shall see, this conclusion is probably also valid for the considerably expanded versions published in Nature and Naturwissenschaften. Pauli responded promptly to Bohr’s request for critical remarks. The letter, to which no extract can do justice, runs as follows3*: Pauli to Bohr, 17 Oct 27 German text on p. 14321

Hamburg, October 17, 1927 Dear Professor Bohr, [Sehr verehrter, lieber Herr Professor!] I send you now the proofs ‘with all the critical remarks which occurred to me’, in accordance with your wish. First I should like to say, however, that in general I am very much in agreement with the overall tendency of the note as well as with most of the details. In particular it has become clear to me that the statistical interpretation of the theoretical results always enters at the point when one divides a closed system into two parts, which one interprets as object under observation and measuring instrument, respectively, and then asks 32 Whenever the passages quoted by Pauli from the lost German manuscript could be identified in the English manuscript, the numbering of the pages referred to in Pauli’s letter has been changed so as to apply to the English version. Moreover, the quotations in question have not been translated from Pauli’s letter, but have been replaced by Bohr’s own English version in the manuscript sent to Darwin.

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

what one can say about one part without knowledge of the other. When one interprets the results of the matrix formulation of the quantum theory in this manner, it seems to me that the possibility of relating it to experience is entirely at hand. For this reason it seems to me that your statement (on page 7) ‘the inner consistency of the theory is achieved only at the expense of any immediate interpretation of the results of c a l ~ u l a t i o n is ’ ~non-conclusive ~ or at least could give rise to misunderstandings. Now I shall indicate in succession a couple of places that I think could still be improved. I have written suggestions for purely linguistic improvement in the margin of the proof sheets. Page 4-5 is the only place that I find unclear. However, this may be partly due to the fact that it is not quite idiomatic. (Still, as I read it again now, I already understand it somewhat better.) Further down on page 5 where the ‘proper reduction of the spatial extension of the fields’34is mentioned, one could perhaps still add something for the sake of clarity. This is of course just a point which was not quite satisfactory in the Heisenberg paper; there the ‘reduction of the wavepacket’ seemed a bit mysterious. Now it should of course be emphasized that such reductions first of all are not necessary when all the measuring instruments are included in the system. In order to be able to describe the results of observation theoretically at all, one must ask what can be said about one part of the total system on its own. And then one sees as a matter of course the complete solution - that the omission of the instruments of observation in many cases (not always, of course) may formally be replaced by such discontinuous reductions. About page 6-7 (matrix formulation) I have already spoken above. Page [?I: What does ‘if possible’ mean in this connection? Page ZZ: Of course, I completely agree that the uncertainty relations require closer scrutiny in the case of discrete eigenvalues. However, it seems to me that as far as x and p x are concerned, the relation A X Ap,

-h

remains valid when (independent of the value of the energy) one considers the general solution which at a given time to is confined to a given domain Ax German original: “(auf Seite 6 ) ‘streng genommen ist ihr innerer Zusammenhang auf Kosten jeder unmittelbaren Deutung der Rechenresultate gewonnen’ 34 German original: “auf S. 4 ‘unstetigen Verkleinerung der raumlichen Begrenzung der Felder’ ”. It is strange that in the place of the decisive word “unstetig” (discontinuous) we find in the English version the pallid “proper”.

33

...

”.

P A R T I: T H E E M E R G E N C E OF T H E C O M P L E M E N T A R I T Y A R G U M E N T

and then resolves it into spectral components. Concerning the former relation At A E h , true enough, it seems to me that it no longer makes sense for discrete energy values. Perhaps one should assert only in the case of the relation At A E h (but not for the relation A x Ap, h ) that ‘this relation is based on a continuous sequence of states’35. Page 11-12: Here there is a sentence: ‘The utility of the procedure rests entirely on the circumstance that in interpreting observations it is always possible to reduce the considerations to three of the spatial coordinates This I hold not to be correct and I should raise objections to it because this is a point that I have especially at heart. If we have a system consisting of several particles, then the eigenfunctions ty(ql,. .., qN) in the multidimensional configuration space can, as far as I see, in principle be empirically ascertained by statistical utilization of results of observation. And this applies to the function itself, not only to its mean value integrated over all except three of the coordinates. The square of the absolute value of this function yields the probability that at the same time one of the particles has the coordinates qi’), q:), q:’), another one the coordinates qi2),q?), q:2),... etc. Now, one can in principle reduce the ascertainment of this function to the observation offree particles if one performs collision experiments in which the system under investigation is struck simultaneously by several particles. One may then for example ask, assuming a parallel incidence of all the colliding particles, what the probability is that simultaneously one of the particles flies away after the collision with a given energy in a certain direction, another one with another energy in another direction and so on (these are not independent events). It is true that after all we have here to do with free particles, however with several of them, and it seems to me quite misleading to say that one may here restrict oneself to a total of three (space- or momentum-like) coordinates. (I considered this point very carefully when I wrote that footnote (in my paper on degenerate gases and paramagneti~rn~~) on the ty-function in multidimensional space.)

-

-

-

I have of course here stressed those (few) points with which I did not agree, whereas I hardly spoke at all about the passages which I particularly liked. 35 German original: “ ‘ihre Formulierung wesentlich auf der Annahme einer kontinuierlichen Folge Again, the German formulation is more precise than the English. von Eigenwerten beruht’ 36 German original: “ ‘Die Verwendung der Rechenresultate beruht darauf, da13 man sich bei Vergleich mit der Erfahrung immer auf die Betrachtung von nur drei von den raumartigen Koordinaten beschranken kann’ ”. 37 W. Pauli, Uber Gasenrarrung und Paramagnerismus, Z. Pfiys. 41 (2927’ dU2-623.

”.

P A R T I: T H E E M E R G E N C E O F T H E C O M P L E M E N T A R I T Y A R G U M E N T

This might easily give rise to a false impression. By and large I am very pleased with your paper, and I believe that the passages mentioned above can still easily be improved! I am especially pleased about the final paragraph where you state that the stationary states possess just as much or as little reality as the particles themselves. Indeed, I even hope that the atomicity of the electric charge later may be regarded in analogy with the existence of stationary states, so that the number of elementary charges in a certain domain will appear as a quantum number. But this is all in the distant future! In contrast, I have in the last few days made much progress with the relativistic many-body problem and I definitely believe that there are no longer any fundamental difficulties in the way of its solution. It would be nice if you could already be in Hamburg by Friday. Then I could also still help you with the proof and we might perhaps send it off from here38. Of course, this is only meant as a tentative proposal. In any case I hope to hear from you soon. With the kindest regards to you and also to Klein. Yours, W. Pauli Many regards from Stern! 5.

THE SOLVAY MEETING

1927

With this development the stage was set for the next act which took place at the Solvay Meeting in Brussels, October 24-29, 1927. The theme of the meeting was “Electrons and and during the General Discussion Bohr gave a lecture following some introductory remarks by Lorentz. The version of this lecture printed in the report of the Solvay Meeting is, according to the author’s footnote, a translation of the paper in the Naturwissenschaften and therefore is of a later date. However, the Bohr Archive possesses some rather extensive notes taken by Kramers during Bohr’s lecture together with some notes by the secretary, J. E. Verschaffelt, which may either have been taken independently or worked out on the basis of Kramers’ notes. This part of the notes was found too incomplete to warrant r e p r o d u ~ t i o n ~ ~ . As the reader will learn, many months were still to pass before the completion of the treatise. The proofs in question were superseded by an entirely new manuscript. 39 Vide, dlectrons et photons, Rapports et discussions du cinquieme Conseil de physique tenu a Bruxelles du 24 au 29 octobre 1927 sous les auspices de 1’Institut international de physique Solvay, Gauthier-Villars, Paris 1928. These notes have not yet been microfilmed.

38

1NSTlTUT I N T E R N A T I O N A L DE PHYSIQUE S O L V A Y CINQUIEME CONSElL DE PHYSIWE

-

BHUXELLES

23-29

OCTOERE 1927

P A R T I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

The notes cover the wave-corpuscle aspects of light and matter (corresponding to the first sections of the printed lecture). The pray microscope is analysed, although the notes are somewhat incomplete here (as in many other places), and the rale of the finite wave trains is discussed in connection with the momentum measurement through the Doppler effect (as in the printed versions). After some questions (which are not included in the fair copy) Bohr continues by discussing the significance of the phase and comments on the Stern-Gerlach experiment and the inobservability of the phase in a stationary state (55 of the Como Conference proceedings).

As regards Bohr’s further contributions to the General Discussion, we have included (p.[99]) extracts of his answers to Einstein and Dirac, as they appear in Verschaffelt’s notes. True enough, they are very incomplete, but on the other hand they are of course of great interest, being the verbatim report of Bohr’s answers on this occasion. Moreover we have also included the discussion remarks by Einstein and Dirac as well as by Heisenberg (translated from the printed proceedings), since Bohr explicitly refers to these in his later recollections of the 1927 Solvay debate41. After Compton’s lecture a comment by Bohr is recorded”, in which he referred to the difficulty in defining the functioning of the whole experimental arrangement used by Compton, except in terms of the wave picture of light. Also the concept of frequency is defined within this framework, in contrast to the interpretation of the Compton effect itself on the basis of the photon concept. Bohr also mentions the Bohr-Kramers-Slater paper43. As emphasized by Klaus Stolzenburg” it is interesting that Bohr here speaks about this work as implying a “complete rejection of the idea of the existence of photons”. Klaus Stolzenburg finds that the Bohr-Kramers-Slater paper itself does not take quite SO explicit a stand against the light quantum hypothesis.

Outside the regular sessions of the conference Bohr and Einstein met for private discussions at which Ehrenfest, a close friend of both, was also present. Concerning these discussions we are fortunate to possess Bohr’s own account41, but it cannot compete in dramatic vivacity with the following contemporaneous description in a letter from Ehrenfest (since much of the spontaneity is necessarily lost in the translation, the relevant part of this extraordinary letter is reproduced in German on p. [415]): “Brussels-Solvay was fine! Lorentz, Planck, Einstein, Bohr, Heisenberg, Kramers, Pauli, Dirac, Fowler, Brillouin, Bragg, Compton, Langmuir, Schrodinger, de Broglie, Curie, Wilson, Richardson, Knudsen, Debye and I. 41 N. Bohr, Discussions with Einstein on Epistemological Problems in Atomic Physics, originally in Albert Einstein, Philosopher-Scientist(ed. P.A. Schilpp), The Library of Living Philosophers, Vol. VII, Evanston, Illinois 1949. Reprinted in Atomic Physics and Human Knowledge, John Wiley & Sons, New York 1958. This article is reproduced in Vol. 7. 42 Solvay report (ref. 39), pp. 91-92. Reproduced in Vol. 5, p. [207]. This comment too, is found in Kramers’ notes. Further short discussion remarks by Bohr are found on p. 103 (Compton’s report); p. 184 (Born and Heisenberg’s report); and pp. 207-208 (Schrodinger’s report). 43 N. Bohr, H. A. Kramers and J. C. Slater, The Quantum Theory of Radiation, Phil. Mag. 47 (1924) 785-802. Reprinted in Vol. 5, p. [99]. 44 Ref. 2, p. 186.

Ehrenfest t o Goudsmit, Uhlenbeck 3 N,,” 27 and Dieke,

t,H,.:;mf

no. 61

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

Paul Ehrenfest.

BOHR towering completely over everybody. At first not understood at all (Born was also there), then step by step defeating everybody. Naturally, once again the awful Bohr incantation terminology. Impossible for anybody else to summarize. (Poor Lorentz as interpreter between the British and the French who were absolutely unable to understand each other. Summarizing Bohr. And Bohr responding with polite despair.) (Every night at 1 a.m. Bohr came into my room just to say ONE SINGLE WORD to me, until three a.m.) It was delightful for me to be present during the conversations between Bohr and Einstein. Like a game of chess. Einstein all the time with new examples. In a certain sense a sort of Perpetuum Mobile of the second kind to break the UNCERTAINTY RELATION. Bohr from out of philosophical smoke clouds constantly searching for the tools to crush one example after the other. Einstein like a jack-in-the-box: jumping out fresh every morning. Oh, that was priceless. But I am almost without reservation pro Bohr and contra Einstein. His attitude to Bohr is now exactly like the attitude of the defenders of absolute simultaneity towards him.

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

In one of the very next issues of Naturwissenschaften you will find a paper by Bohr with the main ideas. By amending the error running through the Heisenberg paper [‘der durchlaufenden Fehler von Heisenberg’] he has pushed the uncertainty relations into the foreground, but at the same time in a marvellously simple manner provided them with a quite marvellous universality. Something like this: Consider first solely the questions of LIGHT. Then immediately from pure WAVE KINEMATICS the following uncertainties (for example) d t d v 1. The shorter the time duration of a wave signal, the greater the uncertainty in the definition of its frequency (analogously the wave number and the inverse ~ a v e l e n g t h ~ Further, ~). from this result, on account of the Planck-Einh stein relation E = hv, p = - (momentum), the ‘reciprocal uncertainty relations’

-

A

&*&-h

ax-6p-h

Thus, the reciprocal uncertainty of the space-time data as opposed to the dynamical data emerge in general FIRST OF ALL IN THE DOMAIN OF LIGHT

x y z t contra p q r

&

(In the exponent of the wave function they appear just in the combination 27r i h

- ( X l P l + x2P2 + X3P3 + X4P41.1 So much for light. Now, however, such effects like the Compton effect in particular prove that the CONSERVATION LA W for the energy-momentum vector is valid in the interaction between light and movable matter. THUS, it follows for every such interaction that thanks to the conservation laws (! !!!!!!!!!!) the above reciprocal uncertainty relations are transferred from light to matter (! !!!!!! BRAVO BOHR ! ! !!! !). This might cause you complete despair (witness indeed the desperate attempt of Slater Kramers Bohr) if it were not for the fact that just de Broglie-Schrodinger with the wave calculus and Born-Heisenberg-Dirac with the non-permutative matrix calculus were also ‘coming up with uncertainties’ just from the matter aspect. And not even The meaning is obviously “the wave number (or the inverse wavelength) and the spatial extension of the wave signal”. 45

PART I : THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

somehow with an uncertainty of a different width than the optical, but wonder upon wonder again of width h . Thus Bohr: Downright undeserved magnificent harmony! !!! And by means of the conservation laws you can now with full confidence let the uncertainty relation propagate itself into any arbitrary corner of physics. For instance, above all from very small bodies (electrons) to arbitrarily large ones. For instance to an entire microscope! Thus consider simply the collision between an electron and the moon. The conservation laws see to it that the uncertainty in the dynamical quantities is transferred from the electron to the moon. It is so easy to overlook the fact that the h-uncertainty of x contra p also applies to the large bodies. Because indeed x and the VELOCITY can be so accurately determined at the same time. However, the uncertainty in the velocity should of course be multiplied by the large mass to yield the uncertainty in the momentum. In private discussions with Einstein, Bohr has developed other very nice points. The following, e.g.: Large, massive, rigid reference systems with imperturbable [ ‘unverstoerbaren’] clocks are particularly suited for the fixation of x y z t. But at the same time unable to indicate momentum or energy transfer. This is the way the uncertainty relation shows up in classical mechanics (difficult to notice, but quite unmistakable). By faintly illuminating a small, freely moving body, one may determine its position rather nicely every hour and evaluate the intermediate velocity and momentum with enormous accuracy. Thereby the uncertainty relation APPEARS to be violated. This is, however, merely a misunderstanding. One has here only EVALUATED the momentum for the intermediate time, but not actually measured it. Furthermore, one also notices that these position measurements at 0 1 2 3 o’clock do NOT allow an exact evaluation of the momentum BEFORE 0 o’clock and AFTER 3 o’clock, but only within the uncertainty associated with Compton recoil. Altogether the notion of a ‘conceptual tracking of the particle between the moments of observation’ should be rejected just as the notion of a ‘tracking of a light corpuscle through the wave field between emission and absorption’ (I hope that with this formulation I do not offend against Bohr’s view). In the article in Naturwissenschaften you will see how Bohr constantly returns to the ‘complementary description’ of all experience (on the one hand the mathematically unambiguous MULTIDIMENSIONAL or matrix computation machinery concerning the ever carefully closed stomach of an isolated system (defined, unambiguously sharp, but beyond all observation and all x y z t description). On the other hand the terribly crude (namely with an intrusion in this idyll amounting to at least h with each observation) ‘establishment of the particle in x y z t ’ , however with uncertainty relation).

P A R T I: T H E EMEROENCE O F THE COMPLEMENTARITY ARGUMENT

Bohr says: For the time being we have at our disposal only those words and concepts that yield such a complementary mode of description. But at least we see already that the famous INTERNAL CONTRADICTIONS of quantum theory only arise because we operate with this not yet sufficiently revised language (I know for sure that this last formulation of mine would drive Bohr to COMPLETE DESPAIR). Now read it for yourselves!” Einstein summed up his attitude after the Solvay discussions in the following lines4$ “Concerning the ‘quantum mechanics’ I think that as regards ponderable matter it contains just as much truth as the theory of light without quanta. It might be a correct theory of statistical laws, but an insufficient conception of the individual elementary processes.”

6. THE

Einstein to Sommerfeld. 9 Nov 21 Oerman

FINAL TOUCH

On November 18, 1927, Bohr gave a lecture in Videnskabernes Selskab (The Danish Academy) under the same title as the published version of the Como Lecture. We possess only the brief abstract published in Danish4’ and English48. The further work on the proofs dragged on well into 1928, in spite of Pauli’s assistance. At the beginning of January, Bohr had to confess to Pauli that he had prepared an entirely new manuscript and proposed to visit Hamburg in order that he and Pauli could go through it together. Pauli answered on January 13: Hamburg, January 13, 1928 Dear Bohr, Many thanks for your nice letter. We are all looking forward to your arrival. Of course I and the other physicists (Lenz, Stern, Koch) will be here in Hamburg throughout the month of January (indeed until the end of February). The paper-scissors and glue pots of the Institute are being prepared for you in the best manner. I am really quite happy that you have altered the manuscript. Indeed, after some time I did not particularly like the old one, especially as it seemed to me that ‘the complementarity of causal and space-time description’ requires still further elucidation, and the statistical interpretation of the results of theoretical computation seemed to me to be introduced too abruptly. In any case I already look forward to seeing the new Albert Einstein, Arnold Sommerfeld, Briefwechsel (ed. A. Hermann), Schwabe & Co. Verlag, Basel and Stuttgart 1968, pp. 111-112. 47 Overs. Dan. Vidensk. Selsk. Forh. Juni 1927 - Maj 1928, p. 27. Reproduced on p. [107]. 48 Nature 121 (1928) 78. Reproduced on p. [108]. 46

Pauli to Bohr, 13 Jan 28 German text on p. 14351

P A R T I: T H E EMERGENCE OF T H E COMPLEMENTARITY ARGUMENT

manuscript, and I have the best intentions of exhibiting as critical an attitude towards it as at all possible (the result of this will be that you are going to prolong your sentences still further). All this we are of course going to talk about. Perhaps you will be kind enough to write to us again about the exact hour of your arrival so that if possible we can meet you. Many greetings to your brother (it will be a great pleasure for the mathematicians and me to welcome him here in Hamburg) and cordial greetings to yourself, from your faithful W. Pauli instead of ‘Sie’. P.S. I see that I should perhaps everywhere have written ‘Du’ Please, be patient. As time goes on I shall surely get into the habit. Bohr was much relieved to receive Pauli’s positive reaction and answered promptly: Hotel Pratschli, Arosa, January 15, 1928

Bohr to Pauli, 15 Jan 28 Danish text on p. (4361

Dear Pauli, Your letter gave me great pleasure. Criticism is exactly what I want, and I hope that you will find that the paper has been improved. Or, to express myself more modestly I should say that I believe that I understand various essential points better than when we last met. I shall arrive in Hamburg from Gottingen next Sunday, January 22, at 5.45 p.m. Will you be kind enough to book me a quiet room in the same hotel where I stayed last time*. If it is convenient for you, perhaps we could already the first evening go through the paper in its new form. Then we could discuss the details the following days. I thought a good deal about these matters while skiing here in Arosa, so that I am not coming altogether unprepared. But in particular I look forward to hearing all the latest news. I am very grateful for the offer of paper-scissors and glue pot, and if it is possible I should like to send the proof off from Hamburg, since after all the whole thing must come to an end sometime. Harald is coming to Hamburg on the afternoon of January 29. It would be nice if we could get a small circle of mathematicians and physicists together *at Dammtor where I shall then get off.49 (I see that unconsciously I am already practising for the corrections in the German proof. My address in Gottingen will be Prof. Courant, Wilhelm Weberstrasse.) 49

This sentence is written in German, whereas the rest of the letter is in Danish. This is the point

of the following remark.

P A R T I: T H E E M E R G E N C E OF T H E C O M P L E M E N T A R I T Y A R G U M E N T

that evening to celebrate the beginning of the New Year - even though the event is belated - and discuss hopes and experiences. With many greetings from both of us to all common friends in Hamburg, Yours ever, Niels Bohr

As late as in the middle of March, we find Pauli reluctant to come to Copenhagen, unless he is assured that the final proofs have been returned: Hamburg, March 10, 1928 Dear Bohr, I came back to Hamburg last night and Stern gave me the message that you had passed through here. Now I should like to hear from you what the situation is as regards your paper, precisely formulated: whether the last proofs have now really been sent off so that you have nothing more to do with it. Because, on the one hand 1 think that if you are not quite finished with the paper, neither of us would gain much from my visit to Copenhagen. On the other hand I should very much like to come to chat pleasantly with you (and also with Klein) about various physical and unphysical matters. Thus, would you be kind enough to write to me briefly as soon as possible whether everything is O.K.? - If so, I would come to Copenhagen sometime on the 14th, 15th or 16th (I would then wire the exact date). Many greetings and looking forward to seeing you again soon, I hope. Yours, W. Pauli

Pauli to Bohr, 10 March 28 German text on p. 14371

In a warm and humorous letter, Bohr sought to reassure his friend: Copenhagen, March 13, 1928 Dear Pauli, As agreed I sent the proof off from Holland and I am just expecting to get a last copy any day now to see whether all the corrections have been properly introduced. Even though I thus cannot give an unambiguous answer to your question, I still hope that you are satisfied and that we may look forward to seeing you here very soon. If you, in dread of the experiences of the past, should wish a further guarantee, I may, of course, send you a wire as soon as I have returned the very last proof. However, I think that you ought to show so much confidence in the future that you immediately write or wire me to tell me when you could come. But whatever you do and whenever you come, you know that I shall rejoice at having you here to enjoy together physics as well as other things. I had a nice journey in Holland and much

Bohr to Pauli, 13 March 28 Danish text on p . [437]

P A R T I: T H E E M E R G E N C E O F T H E C O M P L E M E N T A R I T Y A R G U M E N T

pleasure from the visit to Kramers as well as to Ehrenfest. Since my return I have been busy preparing an opposition to Jacobsen’s doctoral thesis, which he defends next Thursday. I have not yet heard how Klein has been getting on in Cambridge. I believe that he will be back in about a week. With many greetings from all of us, Yours, Niels Bohr P.S. By the way, I may tell you that we have now got no less than six boys. But don’t be afraid that they are going to disturb your peace in Copenhagen. Part of the Institute lore is that when Pauli actually came to Copenhagen, it was carefully kept secret from him that the proofs had not yet been sent off50. Finally, on April 13, the paper was published in German in Naturwissenschaften51, and the following day the English version appeared in Natures2. These two versions are essentially identical, but represent a substantial expansion in comparison with the text printed in the transactions from the Como Conference. A list indicating the major additions can be found on p. [l 1I]. Thus, it is sufficient here to state that although several details in the presentation are improved, no basic new features have been added. In particular, there are no additions that warrant the conclusion that they are direct results of the discussions with Einstein at the Solvay Meeting. Besides Pauli, Dirac had also been involved in the work on the proofs, as the following interesting letter shows: [Copenhagen,] March 24, [ 19128 Dear Dirac, Klein, who was very happy for his visit to Cambridge and for all the kindness with which he was received, tells me that we may look forward to a short visit of you in a few weeks’ time on your way to Leiden. It shall be a great pleasure to us all indeed to see you here again and to hear about the latest progress of your work. I hope you will stay in the institute in the rooms which Heisenberg occupied when you were here last and which have been left as guest rooms. As you will have understood already, all expenses connected with your journey and your stay here will be covered by funds which are at

Bohr to Dirac, 24 March 28 English

50

This story seems to originate from Klein. Bohr, Das Quantenpostulat und die neuere Entwicklung der Atomistik, Naturwiss. 16 (1928)

” N.

245-257.

” N . Bohr, The Quantum Postulate and the Recent Development of Atomic Theory, Nature (Suppl.) 121 (1928) 580-590. Reproduced on p. [147].

P A R T I: T H E E M E R G E N C E OF T H E C O M P L E M E N T A R I T Y A R G U M E N T

disposal of the institute for such purpose. I shall be glad soon to hear from you what date we may expect you here, and for how long time you can stay in Copenhagen. I was very thankful for your kind help with the proof of my article. From our discussions in Cambridge and from what Klein told me I do not know, however, whether you are quite in sympathy with the point of view, from which I have tried to represent the paradoxes of the quantum theory. Although of course I realise the tentative character of the formulation, I still believe that the point of view of complementarity is suited to describe the situation. I think, we can not too strongly emphasize the inadequacy of our ordinary perception when dealing with quantum problems. Of course I quite appreciate your remarks that in dealing with observations we always witness through some permanent effects a choice of nature between the different possibilities. However, it appears to me that the permanency of results of measurements is inherent in the very idea of observation; whether we have to do with marks on a photographic plate or with direct sensations the possibility of some kind of remembrance is of course the necessary condition for making

Paul Dirac and Werner Heisenberg.

P A R T I : THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

any use of observational results. It appears to me that the permanency of such results is the very essence of the ordinary causal space-time description. This seems to me so clear that I have not made a special point of it in my article. What has been in my mind above all was the endeavour to represent the statistical quantum theoretical description as a natural generalisation of the ordinary causal description and to analyze the reasons why such phrases like a choice of nature present themselves in the description of the actual situation. In this respect it appears to me that the emphasis on the subjective character of the idea of observation is essential. Indeed I believe that the contrast between this idea and the classical idea of isolated objects is decisive for the limitation which characterises the use of all classical concepts in the quantum theory. Especially in relation with the transformation theory the situation may, I think, be described by saying that any such concepts can be used unaltered if only due regard is taken to the unavoidable feature of complementarity. A point not directly referred to in the article but in which I have been very interested lately is the question of the uni-direction of time. For an isolated system this question has of course no sense. In considering observations, however, it is essential that the light travels towards our eye or the photographic plate. I believe that a closer analysis offers the proper answer to such paradoxes regarding the nature of light as brought forward especially by Lewis. I have been considering to send a short note to Nature concerning these paradoxes, and in this connection I should like also to discuss some of the remarks brought forward at the Solvay conference by Einstein and by yourself. Before a publication of course I should wish to discuss the questions in detail with you, but I hope that your visit here will offer a good opportunity also in this respect. With kindest regards from us all, Yours, [Niels Bohr] On May 14, Bohr sent Pauli a reprint of the published paper together with his sincere thanks for Pauli’s faithful and friendly help53. He had earlier sent a reprint to Schrodinger who answered on May 5 : Schrodinger to Bohr, 5 May 28 German text on p. [4631

Berlin, May 5 , 1928 Dear Professor Bohr, Please excuse my long delay in replying. I had indeed wanted to make sure 53 P W B

I, letter [197].

P A R T I: T H E E M E R G E N C E O P T H E C O M P L E M E N T A R I T Y A R G U M E N T

of the matter by making written representation to the Vice-Chancellor’s of-

fice, and I have only just now received the answer from the Vice-Chancellor. The Vice-Chancellor writes that on account of your recommendation he has of course no longer any hesitation in admitting Mr Merller to the courses. Thus, Mr Merller needs only to register. I believe that the presentation of a study certificate is not required in any case. Many thanks for the reprint of your paper which actually I had already read from the proof sheets kindly lent to me by Mr Planck. I have recently presented the main ideas in our seminar and I should be very curious to know whether I have to some extent hit upon what you mean. It seems to me that there is a very strange relation between Heisenberg’s uncertainty relation and the claim of discrete quantum states. On account of the former the latter can really not be experimentally tested. This is best seen for the action and angle variables. For these

If you now let d w have the magnitude 1, i.e., if you renounce all knowledge whatsoever of the angle variables (since of course everything is periodic in w with period l), then you have d J = h , i.e., just equal to the difference in Jvalues of neighbouring quantum states. One may also verify this fact in a few simple cases, e.g., in the quantization of an ideal gas. If we allow the molecule a latitude in position of the size of the entire gas volume, then the uncertainty in the momentum becomes of the order of magnitude of the momentum difference of neighbouring quantum states.* One further remark: If you want to describe a system, e.g., a mass point by specifying its p and q, then you find that this description is only possible with a limited degree of accuracy. This seems to me very interesting as a limitation in the applicability of the old concepts of experience. But it seems to me imperative to demand the introduction of new concepts, with respect to which this limitation no longer applies. Because what is in principle unobservable should not at all be contained in our conceptual scheme, it should not be possible to represent it within the latter. In the adequate conceptual scheme it should no longer appear as if our possibilities of experience were limited through unfavourable circumstances. However, it will no doubt be very difficult to invent this new conceptual scheme, since - as you em1* 1 1

$ p dq = rnu.21 = nh, mu = nh/21, neighbouring quantum states thus differ in momentum

by the amount h/21. On the other hand, from A p A q = h , one obtains with A q = l just Ap = h / l = 2-h/21.

PART I: THE EMERGENCE O F THE COMPLEMENTARITY ARGUMENT

phasize so impressively - the new-fashioning required touches upon the deepest levels of our experience: space, time and causality. With the most cordial wishes for the welfare of you and your family, I remain with sincere regards, Yours, E. Schrodinger To this letter Bohr replied: Bohr to Schrodinger, 23 May 28 German text on p. [4641

[Copenhagen,] May 23, [19]28

Dear Schrodinger, Many thanks for your kind and substantial letter. Please excuse me that I have not answered you until now. We were of course very grateful for your helpful efforts in connection with Mr Maller’s inquiry. Mr Mlaller is looking forward very much to being able to attend the lectures in Berlin this summer. It was also a special pleasure for me to realize from your remarks that your attitude is not altogether unsympathetic towards considerations of the tendency which finds expression in my article in Naturwissenschaften. Still I am not quite in agreement with your emphasis on the necessity to develop ‘new’ concepts. We have not only, as far as I can see, no basis for such a new-fashioning so far, but the ‘old’ empirical concepts appear to me to be inseparably linked to the foundations of the human means of visualization. True enough, the apparent contrast between the superposition principle and the individuality postulate has revealed the complementary nature of the space-time coordinates and the conservation laws. But I think that we are here concerned with a philosophically consistent and hence satisfying extension of the foundations of our description of nature. In my opinion there is also no question of a more or less arbitrary limitation in the applicability of the classical concepts, but we have to do with the recognition of an inescapable feature of complementarity that emerges in an analysis of the concept of observation and which in many respects recalls the recognition of the general [feature of] relativity. Of course we do not possess in the quantum theory technical equipment that can be compared with that of the theory of relativity. However, I believe that also in this respect the quantum theory is approaching a certain temporary completion. Indeed, I believe that it is already possible to say that any application of classical concepts that permits an unambiguous definition, may also be ascribed a physical interpretation. In this connection I should like to make the following remark concerning your comments on the relationship between the uncertainty relations and the quantum postulate. As I attempted to show in my article, these relations are to be regarded as an inescapable limitation of the possibilities for definition

P A R T I: T H E EMERGENCE OF T H E COMPLEMENTARITY A R G U M E N T

of the space-time vector and the momentum-energy vector of the single individuals, inasmuch as they take into consideration a property of wave groups that expresses a direct consequence of the superposition principle. In the case of interaction between several individuals - and altogether whenever there can be a question of quantization - the uncertainty relations must always be applied with caution. In the very case you mention of an action and angle variable, there exists the possibility of solutions to the wave equation, in which the former is well defined and which one may apply without necessarily having to ask about the simultaneous limitations of the angle variable. Your remark that an angle variable can never exhibit an uncertainty larger than the modulus of periodicity can hardly - as far as I understand - be taken into consideration in this connection. In the interpretation of experiments by means of the concept of stationary states, we are indeed always dealing with such properties of an atomic system as depend on phase relations over a large number of consecutive periods. The definition and applicability of the eigensolutions of the wave equation are of course based on this very circumstance. In the case where ordinary position coordinates are employed, the conjugate variable is not unambiguously determined in an eigensolution, but exhibits a finite range of values in such a way that the product Ap d q is of the order of magnitude nh, where n denotes the number of nodes. Here, in the behaviour of wave groups one also has a close analogy to the uncertainty relation for free particles. Thus, for example the range of values of the variables that may be experimentally ascertained is smaller, compared to the range of values of the individual eigensolutions, the larger the quantum number. Although this circumstance entails a natural transition from micro- to macromechanics, there remains always - as stated in the article - an absolute exclusion between the application of the concept of stationary states and the tracking of the behaviour of the individual particle in the atom. This exclusion provides in my opinion a particularly striking example of the general complementary nature of the description. As I have tried to show in my article, a quite definite meaning may be ascribed to the concept of stationary states as well as to the discrete energy values within their domain of applicability. In this connection the ascertainment that the atom is in a definite state is always associated with a renunciation of the knowledge of the phase of the corresponding eigensolution. As already stated, in this very inobservability of the phase we have again to do with a simple example of the consequences of the superposition principle. In the article I have endeavoured strongly to stress the failure of classical pictures in the quantum theoretical treatment of the interaction problem, .and to emphasize that our entire mode of visualization is based on the abstraction

P A R T I: T H E E M E R G E N C E OF T H E C O M P L E M E N T A R I T Y A R G U M E N T

of free individuals - a point where, in my opinion, the relationship between classical theory and quantum theory is particularly evident. I might add further that just in the case, touched upon in your letter, of the quantization of a gas, this failure shows up so strikingly in the paradoxes of the new statistics. However, I do not quite understand your use of the uncertainty relation in this case, because here of course the momentum variable conjugate to the coordinate does not have an unambiguous value. I have recently been pondering some further questions of a general character, and I hope soon in a short note to be able to show that certain paradoxes in the quantum-theoretical treatment of radiation phenomena may be illuminated by the remark that the fixation of a direction of time is intimately related to the concept of observation. I fear that I have already bored you very much with all these words. However, you must put it down to my enthusiasm. After the many years of struggling in the dark, I feel perhaps especially strongly the fulfilment of the old hopes, brought about by the new discoveries of yourself and others. I often think with great pleasure of our lively discussions in Copenhagen and Brussels and hope that we may soon have an opportunity to resume them. With the kindest greetings also to Planck and Einstein, with whom perhaps you will discuss the content of this letter. Yours sincerely, [Niels Bohr] In accordance with Bohr’s expectations, Schrodinger sent Einstein a copy of his letter as well as of Bohr’s reply, adding the following comments4: “The remark about the uncertainty relation in an ideal gas, further elaborated, runs as follows: Suppose we quantize a molecule that is reflected back and forth through the distance 1. Then we have jpdx = psdx = 21p =nh, nh i.e., p n = The neighbouring quantum values of the momentum differ

Schrodinger to Einstein, 30 May 28 German

merely by the amount

that even with the largest

possible uncertainty in the coordinate (&=I), I still cannot achieve an accuracy in the momentum that allows me to distinguish between neighbouring quantum values. I do not understand at all what Bohr says about this case towards the end of the third page.*”

*

“All this is of course awfully trivial!”

s4 Schrodinger-Planck, Einstein, Lorentz, Briefe zur Wellenrnechanik (ed. K. Przibram), SpringerVerlag, Vienna 1963, pp. 27-28. The dates given in the editorial footnote are in error.

P A R T I : T H E E M E R O E N C E OF T H E C O M P L E M E N T A R I T Y A R O U M E N T

Einstein replied promptlys5 “I think that you have hit the nail on the head. The subterfuge with the arbitrarily large domain for cyclic variables in order to narrow down d p is very clever to be sure. But an uncertainty relation interpreted in this manner appears hardly enlightening. The thing is contrived for free particles and suits only this case in a natural way. Your demand that the concepts p , q should be abandoned, since they can only claim such a ‘wavering sense’ [‘Wackelbedeutung’], seems to me quite justified. The Heisenberg-Bohr soothing-philosophy - or religion? - is so cleverly concocted that for the present it offers the believers a soft pillow of repose from which they are not so easily chased away. Let us therefore let them rest. I am so damned little impressed by this religion that in spite of all I say:

Not:

E and v

but:

E or

v,

and surely: not v, but E (possesses after all reality). But I can devise no mathematical verse out of this. My brain is also already too stale [‘abgeleiert’].” An amusing footnote to the story of the birth of this, Bohr’s first and decisive paper on the foundations of modern quantum theory, is the “introduction” with which the editors of Nature found it appropriate to embellish Bohr’s article56. It is reproduced on the following page.

”Ibid., p. 29. Klaus Stolzenburg (ref. 2) informs us that the author of the “introduction” was one H . S. Allen.

56

Einstein to Schrodinger, 31 May 28 German

Sapplement t o NATURE No. 3050

A P R I L 14, 1928

New Problems in Quantum Theory. IFTEEN years have elapsed since Niels Bohr first published a series of papers which were the beginning of a new epoch in the development of the quantum theory. Adopting the atomic model proposed by Rutherford, in which electrons circle round a massive nucleus under the action of a Coulomb force of electric attraction, Bohr gained immediate success in interpreting the spectrum of hydrogen and of ionised helium. For his purpose he was compelled to assume the existence of ‘stationary states,’ and the emission of monochromatic radiation in the transition between two such states of an atomic system. I n one sense the new method raised as many difficulties as it removed, and to some of the more conservative physicists the account of Bohr’s atom read like a fairy tale. Further progress in the interpretation of line spectra was made through the generalisations of Wilson and Sommerfeld, but in spite of the inclusion of a widening circle of facts and the fulfilment of predictions, it came t o be realised that a more radical procedure was necessary before a consistent and complete theory could be evolved. I n the forward movement few have been more active than Bohr himself. The employment of a spinning electron by Goudsmit and Uhlenbeck removed many discrepancies, and it seems as if some form of magnetic electron is likely t o be accepted as a fundamental constituent of an atomic system. The magneton of S. B. McLaren with its quantum of angular momentum may be regarded as the prototype of all such magnetic electrons. Within the last few years the matrix mechanics of Heisenberg, Born, and Jordan, the quantum algebra of Dirac, and the undulatory mechanics of Schrodinger, have led t o remarkable theoretical developments. The new wave mechanics gave rise to the hope that a n account of atomic phenomena might be obtained which would not differ essentially from that afforded by the classical theories of electricity and magnetism. Unfortunately, Bohr’s statement in the following communication of the principles underlying the description of atomic phenomena gives little, if any, encouragement in this direction. I n classical mechanics it is assumed that the position of a particle (such as an electron) can be determined a t a specified instant of time by means of its co-ordinates. As the time varies it is supposed to be possible t o trace the path of the par-

ticle through space, or to determine its ‘world line ’ in the four-dimensional world. Further, it is assumed that the concept of causality may be applied in considering the effect of the action of external forces. Thus in classical physics we have a causal space-time co-ordination, based on the assumption that the methods or tools of measurement do not affect the phenomena which are observed. I n the new quantum theory the outlook is changed, for any attempt to observe the position or motion of a n electron involves illumination by light, and this implies interaction between the electron and the light employed in making the measurement. The position and the path of an electron become vague. Thus there is introduced in the new quantum mechanics an indefiniteness which contrasts with the clear-cut concepts of classical mechanics. Bohr asserts that in any phenomenon which we may attempt to observe there is a n essential discontinuity, or rather individuality, which may be symbolised by Planck’s constant h. The causal space-time co-ordination of atomic phenomena must on this view be abandoned, and we are left with a somewhat vague statistical description. The strange conflict which has been waged between the wave theory of light and the light quantum hypothesis has resulted in a remarkable dilemma. But now we have a parallel dilemma, for a material particle manifests some of the attributes of wave motion. Can these apparently contradictory views be reconciled ? According to Bohr, the pictures ought to be regarded not as contradictory but as complementary. Radiation in free space is not open to observation, and is a mere abstraction. An isolated material particle likewise can never be observed and is also a n abstraction. It is only through their interaction with other systems that the properties of these abstractions can be defined and observed. It must be confessed that the new quantum mechanics is far from satisfying the requirements of the layman who seeks to clothe his conceptions in figurative language. Indeed, its originators probably hold that such symbolic representation is inherently impossible. It is earnestly to be hoped that this is not their last word on the subject, and that they may yet be successful in expressing the quantum postulate in picturesque form.

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

On this rigmarole Pauli comments tersely and to the point: “AS I now read your letter again, I also think about your paper. Honestly, I have not laughed so much for a long time as when reading the ludicrous comment with which the editors of Nature have prefaced your article. After a superfluous historical survey, in short sentences, of the development of quantum theory in recent years, in which the name of de Broglie is not mentioned, there follows an obscure paraphrase of the following mood: ‘We British physicists would be awfully pleased if in the future the points of view advocated in the following paper should turn out not to be true. Since, however, Mr Bohr is a nice man, such a pleasure would not be kind. Since moreover he is a famous physicist and more often right than wrong, there remains only a slight chance that our hopes will be fulfilled.’ In any case, this is how I read the Nature commentary, and I thought to myself, ‘Sancta simplicitas!’.”

PauIi to Bohr, 16 June 28 German text on p , [4381 On P. [4401

I. ATOMIC THEORY AND WAVE MECHANICS ATOMTEORI OG BOLGEMEKANIK Overs. Dan. Vidensk. Selsk. Forh. Juni 1926 - Maj 1927, pp. 28-29 ATOMIC THEORY AND WAVE MECHANICS Nature 119 (1927) 262 Communication to the Royal Danish Academy on 17 December 1926 ABSTRACT

NIELS BOHR gav en Meddelelse: Afornleori og B d g e mekanik. (I den kvanteteoretiske Beskrivelse af Atomernes Forhold indgaar e t vcesentligt Element af Diskontinuitet, der staar i afgjort Modsaetning ti1 de klassiske mekaniske og elektrodynamiske Teoriers Fordringer. De lovende Resultater, der i den sidste Tid e r opnaaet ved H j d p af den som Belgemekanik betegnede Modifikation af de klassiske Teorier, har imidlertid paany rejst Spcargsmaalet om Muligheden af i Atornbeskrivelsen a t uiidgaa ethvei? Element af Diskontinuitet. Paa Videnskabeiis nuv s r e n d e Standpunkt e r denne Mulighed dog n s p p e ti1 Stede, idet det her drejer sig om Vanskeligheder ved selve de Grundbegreber, hvorpaa saavel Belgemekanikken som de klassiske Teorier hviler.)

Dec. 17.-Niels Bohr: Atomic theory and wave mechanics. The quantum theory of atomic constitution contains an essential element of discontinuity contrasting with the classical theories of mechanics and electrodynamics. I n view of the recent promising results of the modification of classical mechanics known as the wave mechanics, the problem has arisen as to the possibility of avoiding any element of discontinuity in the description of atoms. This possibility, however, would seem excluded in the present state of science, where we meet with difficulties regarding fundamental concepts common for the classical theories and wave mechanics.

11. UNTITLED FRAGMENT FROM FOLDER LABELLED COMO LECTURE II (1927) DANISH TEXT, TRANSLATION AND FACSIMILE

See Introduction to Part I, sect. 3.

P A R T I: T H E E M E R G E N C E O F T H E C O M P L E M E N T A R I T Y A R G U M E N T

“COMO LECTURE 11” The folder “Como Lecture 11”, 1927, contains a large number of drafts and notes, comprising 181 sheets, a few of which have notes on the reverse, making a total of 195 written pages. Most of them are handwritten, in pencil or ink, in Klein’s, Bohr’s, Mrs Margrethe Bohr’s and an unidentified hand. There are a few typed pages and carbon copies. The languages are English, Danish and German. Many sheets are dated, carrying dates between 2 July 1927 and 13 September 1927 (except for 2 pages, dated 1926, apparently by mistake). The main titles are: “Atomteori og Berlgemekanik” (“Atomic Theory and Wave Mechanics”), “Philosophical Foundations of the Quantum Theory”, ‘‘Fundamental Problems of the Quantum Theory”, “Uber die Wellentheorie des Lichts und der Materie” (“On the Wave Theory of Light and Matter”), and “Zur Frage des begrifflichen Aufbaus der Quantentheorie” (“On the Question of the Conceptual Structure of the Quantum Theory”). We have here (as document 11) reproduced 3 pages in Danish written in ink in Bohr’s handwriting with an amendment in pencil in Klein’s handwriting at the end. The first two pages are numbered and are dated 10 July 1926 [1927, as is clear, La., from the reference to Davisson]. As document I11 a manuscript entitled “Philosophical foundations of the quantum theory” is reproduced. It consists of 8 numbered pages written in pencil in Klein’s handwriting with a few corrections by Bohr. It is not dated. As document IV a manuscript entitled “Fundamental problems of the quantum theory” is reproduced. It consists of 8 numbered pages written in ink in Bohr’s handwriting, all dated 13 September 1927. The material is on microfilm Bohr MSS no. 11. As will be apparent from the facsimile reproduction, this manuscript was evidently written in great haste. Hence many words are hardly legible and spelling and punctation are rather careless. In the transcript we have attempted to give as faithful a reproduction as possible, indicating uncertainties by square brackets and question marks. However, some trivial slips of the pen have been corrected. Some of the formulae are hardly legible, but most of the symbols seem to correspond to those used in the printed versions of the Como Lecture.

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

I 10-7-1926 [1927] Enhver Oplysning om Atomer udtrykt i klassiske Begreber Alle klassiske Begreber defineret ved Rum Tids Billeder Derfor Begyndelse ti1 Quantum[theori ?] en stykkevis Brug af Rum Tids Billeder knyttet formelt sammen med Relationer indeholdende Plancks Konstant. og paa Konservation af Energi og Bevagelsesmangde. Forbindelsen af vasentlig diskontinuert og statistisk Art. Bestrabelsen paa at knytte de statistiske Love op ti1 Egenskaberne af Billeder forte saaledes ti1 at de fremtraadte som Almindelig[gorelse] af den klassiske Theory og specielt konvergerer ti1 denne Theoris Forlangender i den Granse [hvor] man i statistisk Anvendelse kan se bort fra det diskontinuerte Element. forte ti1 Erkendelsen af en vidtgaaende Korrespondens mellem Quanteteorien og den klassiske Theory og ti1 det Program at udvikle en [konsistent ?] Kvantitativ Beskrivelse [ved] at s0ge analoge Trak hos den klassiske Theori. Det viste sig dog umuligt med Rum Tids billeder at give denne quantitativt Udtryk. [Ja ?] Theory udviste en Dualitet hvis man saa paa Superpositionsprincippet paa den ene Side og Bevarelsen af Energi og Bevagelsesmasngde paa den anden. Komplementare Sider af Erfaringen der ikke lader sig forene i et Rum Tidsbillede baseret paa de klassiske Theorier. +

I1 10-7-1926 [1927].

Ja som tilmed synes at stride saa starkt mod den klassiske Rum Tidsbeskrivelse at det var w a r t at [se] hvorledes den [kunde] forenes dermed. LysBolger kraver Tid for deres Definition, +

1* L=nA [?I n = &

Partikler maa andre deres Hastighed hurtigt. Lyskvantum kan ikke gmes tilstraekkelig lille Ez = I X = h [?I

-

A E . T- A I * L h .

* [We surmise that these equations are intended to indicate that the extension of the wave group is large compared to the wavelength.]

MS, p. 2

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

Forandring ved Berlgeteorien for Materien. De Broglie. Einstein Bekrzftigelse af Davidson. Ogsaa beskrive en Particle ved Superposition af Berlger. Samme Relation som [for] Lyskvanter. (Relativiteten vasentlig)

MS,P. PI

Mulighed for Rum og Tid Beskrivelse nerjeste knyttet op ti1 “Conservation theorems”. Maaling af Energi eller Impuls med given Nerjagtighed medfarer Tab af Phaserelationer, der medferrer Umulighed af Interferens ved Superposition. Vanskelighed ved Rum og Tid uadskilleligt fra Vanskelighed ved Benyttelse af klassiske Begreber defineret i Rum og Tid. Henvist ti1 statistisk Brug af klassiske Begreber. ferrst forsergt ved at passe Rum Tid Billeder sammen med Bevaring af Energy og Impuls. Korrespondens d.v.s. Forbindelse af de statistiske Love med Billedernes Trak. Muligt i betydeligt Omfang. men ferrte ti1 Formodning om Statistik af Energy. Tydeligt komplementser Side hos Erfaringen “naar den beskrives med klassiske Begreber”. Einstein. Umuligt at udtrykke i Rum Tidsbilleder [hentet ?] fra klassisk Teori. Tydeligt at Modifikation af Tids Rum Beskrivelsen nadvendiggjorde Modifikation af “Erhaltung”

Interpoiation in Klein’s handwriting

Crossed out

Added in Klein’s handwriting

Moderne Udvikling giver Midlerne i Haende. Midlerne symbolske. og Behandling vasentlig statistisk. Heisenberg Undersergelse af Statistikens Sammenhzng med Betingelser for Maalinger. Giver et vasentligt Bidrag ti1 Forstaaelse af Teoriens Modsigelsesfrihed og samtidig ti1 den reffeAnvendelse. Nerje Forbindelse med Rum-Tidsproblemet . Interessant at analysere dette sidste Vekselspil og samtidig [Forholdene for Rum Tidsbeskrivelsen.

?I

Uadskilligheden af Modsigelsesfrihed og Principperne for Teoriens Anvendelse. Problemet om Anskuelighed

P A R T I: T H E E M E R G E N C E O F T H E C O M P L E M E N T A R I T Y A R G U M E N T

TRANSLATION I 10-7-1926 [1927] All information about atoms expressed in classical concepts

All classical concepts defined through space-time pictures Therefore beginning of quantum [theory ?] piecewise use of space-time pictures formally connected by relations containing Planck’s constant. and on conservation of energy and momentum. The connection of essentially discontinuous and statistical kind. The endeavours at connecting the statistical laws with the properties of pictures thus implied that they appeared as generalization of the classical theory, and in particular converge to the demands of this theory in the limit [where] in statistical applications one may disregard the discontinuous element. led to the recognition of a far-reaching correspondence between the quantum theory and the classical theory and to the programme of developing a [consistent ?] quantitative description [by] looking for analogous features in the classical theory. However, it proved impossible to express this quantitatively by space-time pictures. [Indeed ?I, the theory exhibited a duality when one considered on the one hand the superposition principle and on the other hand the conservation of energy and momentum. Complementary aspects of experience that cannot be united into a space-time picture based on the classical theories. --t I1 10-7-1926 [1927]. And which indeed seem to be in such a strong conflict with the classical space-time description that it was difficult to [see], how it [could be] reconciled with it. Light waves require time for their definition, +

L=nA[?]

1 *

n=-

&

* [We surmise that these equations are intended to indicate that the extension of the wave group is large compared to the wavelength.]

MS, p. 2

PART I : THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

Particles must change their velocity rapidly. Light quantum cannot be made sufficiently small E T = I X = h[?] A E . T- A I . L

- h.

Change with wave theory of matter. De Broglie. Einstein Confirmation by Davidson. Also describe a particle by superposition of waves. Same relation as [for] light quanta. (Relativity essential)

MS, P. PI

Possibility for space-time description closely connected with conservation theorems. Measurement of energy or momentum with given accuracy implies loss of phase relations, which implies impossibility of interference by superposition. Difficulty with space and time inseparable from difficulty by application of classical concepts defined in space and time. Compelled to statistical use of classical concepts. first attempted by connecting, space-time pictures with conservation of energy and momentum. Correspondence, i.e. connection of the statistical laws with features of the pictures. Possible to a considerable extent. but led to the hypothesis of statistics of energy. Clearly complementary aspect of experience “when it is described by classical concepts”. Einstein. Impossible to express in space-time pictures [taken over ?] from classical theory. Clear that modification of time-space description necessitated modification of conservation

Interpolation in Klein’s handwriting Crossed out

Added in Klein’s handwriting

Modern development provides the means. The means symbolic. and treatment fundamentally statistical. Heisenberg investigation of the relationship between the statistics and the conditions for measurements. Gives an essential contribution to the understanding of the consistency of the theory and at the same time to its proper application. Close relation to the space-time problem. Interesting to analyse the latter interplay and at the same time the [conditions for ?] the space-time description. Inseparability of consistency and the principles for the application of the theory. The problem of visualizability.

111. PHILOSOPHICAL FOUNDATIONS OF THE QUANTUM THEORY UNPUBLISHED MANUSCRIPT FROM FOLDER LABELLED COMO LECTURE II (1927)

See Introduction to Part I, sect. 3.

P A R T I: T H E EMERGENCE OF T H E COMPLEMENTARITY ARGUMENT

See the editorial note “Como Lecture 11” to document 11. Passages which are crossed out in the manuscript have been omitted, except in one case where the correction could not be deciphered. The punctation is done extremely negligently, and commas have been added by the editors to facilitate the reading.

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

Philosophical foundations of the quantum theory. In connection with the discussion between Drs. Norman R. Campbell and P. Jordan, published under this headline in the issue of Nature May 28, I should be glad to get space in these columns for the following general remarks. The endeavours to develop a consistent description of atomic phenomena based on Planck’s discovery of quantum of action meets as well-known with fundamental difficulties. Due to the contrast between the principles underlying the ordinary description of natural phenomena and the element of discontinuity characteristic for the quantum theory, we must be prepared that every concept used in accounting for the experimental evidence will have only restricted validity when dealing with atomic phenomena. This situation is brought out most clearly through the discussion about the nature of light connected with the light quantum theory of Einstein. On one hand, the concept of corpuscles of light with energy E and momentum I , connected with the period of vibration* 5 and the wavelength I through the relation

ET = ZI = h , might be considered as the rational expression for the conservation of energy and momentum in the photoelectric effect and the Compton effect. On the other hand, the appearance of the quantities 5 and I in these relations reminds us of the vast field of experience which is accounted for so conclusively by the principle of superposition of waves. It seems that we here meet with an unavoidable dilemma, the question being not a choice between two rivalising [competing] concepts, but rather of the description of two complementary sides of the phenomenon. This becomes clear when we try to obtain a closer analysis of the concepts in question. If for instance we ask about the position of a light quantum at a certain moment, we find that, no more than in the case of its energy and momentum, we can define the position of a light quantum at a given time without a consideration of the complementary waves, and that we even here meet that difficulty that a wavetrain [for] the exact definition of its energy and momentum ought to be strictly harmonic and consequently unlimited in space and time**. Only by the superposition of harmonic waves of different wavelengths and directions it is

* [The Greek letters 7 and u are difficult to distinguish in the handwritten documents. We have therefore chosen to use these symbols in accordance with the way they are used in the printed versions of the Como lecture.] ** [The first part of this sentence is crossed out and replaced by a correction in Bohr’s handwriting which is only partly legible.]

MS,p. 2

MS,p. 3

P A R T I: T H E EMERGENCE OF T H E COMPLEMENTARITY A R G U M E N T

MS,p. 4

MS,p.

5

MS,p. 6

possible at a given time to limit the extension in space of the wavefield, introducing at the same time a lack of definition of energy and momentum. On the other hand, a consideration will show that the only way to define the presence of the waves is through an analysis of the very phenomena from which the corpuscular character of light is concluded, i.e., the interaction between light and matter. But just at this point the discussion was until recently most unsatisfactory, because the description of the behaviour of the so-called material particles rested so entirely on the corpuscular idea that no direct connection between the kinematics of light waves and of the exchange of energy and momentum with matter consistent with the relations (1) seemed obtainable. These relations, as well as the postulates of the quantum theory regarding the behaviour of systems composed of material particles, suggest in fact a statistical feature of the whole theory, not only as regards our description of the occurrence of the observable phenomena, but even as regards the validity of the laws of conservation of energy and momentum. This difficulty however is removed through the introduction of an essential wave feature in the description of the behaviour of material particles due to the work of de Broglie and Schrodinger. Notwithstanding the fundamental difference between the electric constituents of matter and a radiation field, de Broglie has pointed [out] how the dilemma of the light quantum leads to the anticipation that essential properties of the electron can only be adequately described by a comparison with a wave whose period and wavelength is [are] related to its energy and momentum just by the above relations, an anticipation [which] is brought out so brilliantly by the experiments on the reflection of electrons in crystals described by Davidson in a recent issue of Nature. The introduction of this idea means that the difficulty of assigning a position to an electron is exactly the same as that in the case of a light quantum, the electronic position being in fact limited only by the superposition of a number of de Broglie waves with different wavelengths and directions, involving an uncertainty in the definition of its energy and momentum. Following up the wave-properties of a free particle, Schrodinger has developed a theory of the interaction between material particles in which the mechanical equations of motion are replaced by a differential equation of a type familiar from wave problems. Through this the quantum theory has entered in a new stage, in which the existence of stationary states does not appear as a separate postulate, but where each such state appears as a possible proper vibration of the wave equation, similar to the harmonic waves in free space representing a component of a radiation field. At first sight this representation might seem at variance with the stability of stationary states involved in the quantum postulates, but it can be shown that in this respect the analogy between a radiation problem and the interaction between material systems is complete in that sense that all the properties, of atoms and free particles may be described by

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

means of the solution of the waveequation, if this is utilized in a way consistent with the conservation of the individual particles, just as we have in the wave description of light to maintain the conservation theorems and to use the wavefields obtained by the superposition of the harmonic components as a measure of the probability for the manifestation of a lightquantum. In fact the use of the Schrodinger waves as a measure of the probability of the “materialisation” of an electron or of an atom in a certain stationary state leads - as shown by Born - to a complete description of the collision phenomena of Franck and Hertz, which may be said to exhibit the stability of the st. states. In this discussion the consideration of the separate proper vibrations plays an essential part, but it is essential to remember that in any linking up of the calculations with an analysis of the measurements, we have to a smaller or larger extent always to do with a superposition of proper vibrations, just as [in] the well-known description of the optical experiments and as in de Broglie’s representation of the motion of an electron as an effect of the propagation of a group of harmonic waves. As Schrodinger has emphasized, this point of view leads to a possibility of elucidating in detail the gradual change of the properties of electrons bound in atoms to those of free particles. This last problem has been studied in detail in the paper of Heisenberg referred to in Dr. Jordan’s article. In this paper Heisenberg analyzes the uncertainty by which the mechanical attributes of particles are described by means of the wavefunctions, and with the intention to show that the apparent paradoxes, involved in the postulates of the quantum theory and in the related symbolic treatment of mechanics based on the matrix representation, are only apparent and involve no real contradiction. I wish to use the opportunity ...

MS,p. 7

MS, p.

s

IV. FUNDAMENTAL PROBLEMS OF THE QUANTUM THEORY UNPUBLISHED MANUSCRIPT DATED 13 SEPTEMBER 1927 FROM FOLDER LABELLED COMO LECTURE ZI (1927) TEXT AND FACSIMILE

See Introduction to Part I, sect. 3 .

P A R T I: T H E E M E R G E N C E OF T H E C O M P L E M E N T A R I T Y A R G U M E N T

See the editorial note “Como Lecture 11” to document 11. As will appear from the facsimile reproduction, this manuscript was evidently written in great haste. Hence many words are hardly legible and the spelling and punctuation are rather careless. In the transcript we have attempted to give as faithful a reproduction as possible, indicating uncertainties by square brackets and question marks. However, some trivial slips of the pen have been corrected.

PART I: THE EMERGENCE O F THE COMPLEMENTARITY ARGUMENT

13-9- 1927.

Fundamental problems of the quantum theory. Characteristic of the quantum theory is the acknowledgement of a fundamental limitation in our classical physical ideas when applied to atomic phenomena. Just on account of this situation however we meet with intricate difficulties when attempting to formulate the contents of the quantum theory in terms of concepts borrowed from the classical theories. Still it would appear that the essence of the theory may be expressed through the postulate that any atomic process open to direct observation involves an essential element of discontinuity or rather individuality completely foreign to the classical ideas and symbolized by Planck’s quantum of action. This postulate at once implies a resignation as regards the causal space time coordination of atomic phenomena. Indeed our usual space time coordination rests entirely on the idea of tools of measurements [the] interaction of which with the phenomena to be observed may be neglected. This appears very clearly in the formulation of the theory of relativity, [so] helpful for the elucidation of the classical theories regarding the space time problem. Indeed as stressed by Einstein any observation or measurement may be said ultimately to rest on the coincidence of two events in the same space time point. Just these coincidences will not be affected by the difference which for the rest may be involved in the space time coordination of different observers. According [to] the quantum postulate, however, every observation of atomic phenomena will just involve an individual process, resulting in an essential interaction. We cannot therefore speak of independent tools of measurements. This point has not escaped attention in the work on the development of the quantum theory especially as regards problems of atomic constitution. Just recently, however, it has been stressed in a very interesting and suggestive way by Heisenberg in connection with a discussion of the physical interpretation of the symbolic method developed in the last years and which have proved themselves [so] wonderfully suited for the elucidation of atomic problems. I shall try however here to present the fundamental problem at issue from a somewhat different point of view, taking the starting point in the analysis of the most elementary features of our experience regarding atomic phenomena. Our usual methods of observation depend on material interactions [either ?] direct or [through?] the medium of radiation, represented by our senses of seeing and touching. Now the modern development of science depends on the applicability of these methods also [to] the atomic phenomena. I think of the fun-

MS, p , 2

P A R T I: T H E EMERGENCE OF T H E COMPLEMENTARITY ARGUMENT

MS,p. 3

damental information regarding the structural units of the atom obtained by Thomson and Rutherford by the study of the passage through atoms of high speed electric particles, and on the elucidation of essential features of the atomic dispersion phenomena, obtained through the application of the electromagnetic theory of light, which has not only allowed to account in such details for the optical activity of material media, but in the hands of Lord Rayleigh has even allowed to count the atoms, and in [the] hands of Thomson to count the number of electrons contained in atoms. Nevertheless just as regards the nature of material particles and radiation we have in the last years in connection with the development of the quantum theory met with paradoxes which strikingly disclosed the limit[ation]s of the classical ideas. These difficulties first disclosed themselves as regards the problem of the nature of light. Notwithstanding the success of the wave theory it has not been possible to account for interchanges of momentum and energy by radiation processes except by Einstein’s idea of individual light quanta, carrying energy and momenta expressed by the well-known quantum relations E=hv

and P = h a

where h is Planck’s constant and v and a the number of vibrations per unit 1

of time and the number of waves per unit length respectively. Also v = - and 7

1

a=- where T and A are the period and wavelength*. Now the very form of

1

the above formulae indicates that we are not dealing with a choice between a wave or corpuscular theory of light. Indeed the wave and corpuscular ideas are able only to account for complementary sides of the phenomena. On one hand the frequency and wave number are only defined by the wave idea, and their measurements depend entirely on an application of the wave theoretical superposition principle. On the other hand the formulas [do?] not only express the [individual?] character of the elementary radiation processes [but in this way?] the definition of energy and momentum may be carried back to the idea of material particles. The recent development however [has now?] disclosed fundamental difficulties in this idea. MS,p. 4

dv do

dE dP

* [Cf. the first footnote on p. [691.1 ** [These formulae are added in pencil on the top of

dE=v dP

**

the page. Here evidently u and v denote the phase velocity and the group velocity, respectively. On the following pages these quantities are denoted by vx and v‘”.]

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

Notwithstanding the very direct way in which the individual character of the electrons is brought out by the evidence regarding the atomic nature of electricity, the discovery of Davidson and Germer of the selective reflection of electrons from metal crystals prove the necessity of applying a wave theoretical superposition principle in order to account for the behaviour of electrons. Indeed the wave character of the electrons is by these experiments shown just as clearly as is the wave character of light by the reflection from an optical grating, as in case of X-rays by reflection from a crystal lattice. As well-known the experiments are in complete accordance with the ideas of de Broglie. Emphasizing the very general validity of the quantum relation between energy and frequency this author was led to the idea of ascribing a frequency to any agency carrying energy. In relation with the principle of relativity these ideas led directly to conjugate a so-called phase-wave to a material particle. Employing for brevity the usual means of expressing harmonics through imaginary exponentials the phase-wave may in its relation to time and space be represented by = ei2n(vt-xux-yuy-zuz)

where ox or a, are the number of wave summits cut by unit length by an axis parallel to the x y and z axis respectively. Now de Broglie found that ax oy a, were just related to the momentum P through the above equation if considered as a vector equation. Also [Thus]* =ei(2x/h)(Ef-P,x-P~y-P,z)

The invariant character of the exponent in the sense of the ox- vt.

2

-_ -. v

v

dv

-. -=v do

dE dP

dE= vdP

theory of relativity is shown thereby that the term within the bracket is nothing else, than the negative value of the scalar product of the space time vector E iQ-yz, ict) and the Impulse Energy vector ~(PxP,,P,[ -1 ) : or 9 = e - i(2n/h) (@).

1c

* [Instead of the coordinates x , y , z , by a slip of the pen Bohr has written a,, or, az.]

** [These formulae are written on the top of

E=rn2.]

the page. Across them is added in pencil P=vrn.

MS,p.

5

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

As emphasized by de Broglie the abstract character of the phase-wave is [already?] indicated by the fact that its velocity of propagation vx is always larger than the velocity of light c . Indeed we have according to relativity E P = v - where v is the velocity of the particle also [thus]

2

As stressed by de Broglie the only way of observing an elementary wave is by interference. A group of waves being necessary to represent a wave front, now he shows that the group velocity v" is just d E =v v==- dv = . do dP

MS, p. 6

d E = v dP.

Indeed the representing of a particle by means of a wave group represents a direct generalization of the light quantum theory, the formula of the latter holding for the special case of particles travelling with the velocity of light. This very point suggests at the same time that also a light wave may be considered as an abstraction, and that reality can only be ascribed to a wave group. Let us now study the properties of such a group somewhat more closely. [?I The limitation of the group in space and time rests on the interference of the elementary waves, by which the field may be said to disappear outside the limitations. Let the limitation of the group in a certain direction say that of the x axis be Ax. This interference claims then composition of the group with a difference in wave number do, so that

A x do,- 1, in the same way we get d y do,-dz

do,-dt d v - 1.

We then see that this limitation is depending on variations of v and 0 or of a limitation in the accuracy with which these quantities are defined. Introducing the quantum relations we have A x dP,-dy

-

dP,-dz dP,-Llt

dE-h

or briefly (df d &) h , We see also [thus] that a limitation of the group in extension in space and time is [depending?] on a Complementarity conjugated to a limitation in accuracy with which energy and momentum can be defined. Indeed we may say that ac-

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

cording to the quantum theory the possibility of a space time coordination is complementary to the possibility of a causal description. Loss of phase of group may [be] considered as allowing consistency.

In order to show the [importance of this fact ?I, we may consider a few simple examples. Let a light beam or a beam of electrons pass out of a hole. Let us assume that a plane monochromatic wave falls on the hole, in order that we may know where they are and let us assume that we close the hole in order to know what time they came out. We see at once that if we open the [hole?] a time t 1 the frequency is only defined by d v = - and the energy of the light quantum

t

and the electron is therefore only known by an accuracy given by A t A E = h . Let us in order to know the energy not care about the time and let the hole be left open, what is then the accuracy with which we know the momentum. If the diameter of the hole is I the outpassing wave will be defracted over an angle of 1 h magnitude a = - . Now the absolute value of the momentum is - = p . The

A

I

component parallel to the hole however will be undetermined to the amount A p = a p also [thus] dp d l - h . Of course these illustrations give nothing new, everything is contained in the above general formula, it shows however clearly how impossible it is in experimental arrangements to go beyond the limitations discussed. The complementary features of the apparent contrary claims of individuality and superposition finds its explanation thereby that such agencies as free material particles and radiation in empty space are abstractions according to the quantum theory. They can only be observed through their interactions.

In order to discuss the possibilities of observation let us consider the simplest case, that of the Compton effect. As well-known just this effect has offered a most successful application of the light quantum theory. For a time, however, it would appear that the application involved a fundamental difficulty. (Description of radiation claims extension in space and time, while change of energy and momentum of electron apparently is punctual [point-like] and instantaneous).

% A/@7\

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Suggestion of statistical character of conservation. Disproved. Solution by wave theory. Opens possibility of uniting space time coordination and conservation principles. [Conjugation uncertainty already implies in latter principles?], QL+ Q;l = Q&+ Q;. Heisenberg’s conclusions but use that it rests on the discontinuity character of process only by a necessity of conjugating individuality with Wave superposition. Examples

(General Compton effect)

[?I

General interaction problem. Simplicity of atomic constitution. (Closed Systems) Possibility of neglecting space time coordination. Use of Postulate. Spectrum Franck [Wood?]. (Proof of stationary states.) Korrespondence Principle. Generalization of classical theory. Kramers. Heisenberg. Symbolic method. Observability

[?]

Superposition Schrodinger Heisenberg. Harmony between [Separation ?] and [individuality?]. Quantum Paradoxes.

13-

e

7

--

V. THE QUANTUM POSTULATE AND THE RECENT DEVELOPMENT OF ATOMIC THEORY [ 11 UNPUBLISHED MANUSCRIPT DATED 12- 13 OCTOBER 1927

See Introduction to Part I, sect. 4.

P A R T I: T H E EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

This manuscript consists of a carbon copy of 12 typewritten pages with a few amendments in ink in Mrs Margrethe Bohr’s handwriting. The formulae and symbols are written in pencil. Except for the first and last page, the pages are dated, carrying the dates 12 and 13 October 1927. The manuscript was found with a letter from Bohr to Darwin of 16 October 1927. It is included in the Darwin Papers, deposited with the American Philosophical Society, Philadelphia, Pennsylvania. The manuscript is on microfilm AHQP no. 36.

PART I : THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

N. Bohr. THE QUANTUM POSTULATE AND THE RECENT DEVELOPMENT OF A TOMIC THEOR Y In connection with recent comments on the principles underlying the quantum theoretical interpretation of atomic phenomena I am glad to get space in these columns for the following general remarks. The quantum theory is characterized by the acknowledgement of a fundamental limitation in the classical physical ideas, when applied to atomic phenomena. The situation thus created is of a peculiar nature, since our interpretation of the experimental material rests extensively upon the classical ideas. Notwithstanding the difficulties which hence are involved in the formulation of the quantum theory, it seems nevertheless that its essence may be expressed in the so-called quantum postulate, which to any atomic process open to direct observation attributes an essential discontinuity or rather individuality, completely foreign to the classical theories and symbolized by Planck’s quantum of action. This postulate implies a resignation as regards the causal space-time coordination of atomic processes. Indeed our usual description of physical phenomena is based entirely on the idea that the phenomena concerned may be observed without disturbing them appreciably. This appears for example clearly in the theory of relativity, which has been so fruitful for the elucidation of the classical theories. As emphasized by Einstein, every observation or measurement ultimately rests on the coincidence of two independent events at the same space-time point. Just these coincidences will not be affected by any differences which the space-time coordination of different observers otherwise may exhibit. Now the quantum postulate implies that no observation of atomic phenomena is possible without their essential disturbance, and that accordingly the idea of means of observation independent of the phenomena or of phenomena independent of means of observation cannot be maintained. This circumstance has far-reaching consequences. On the one hand the exact definition of a system of objects claims the elimination of all external disturbances. But then according to the quantum postulate any possibility of observation will be excluded. On the other hand, if in order to make observation possible we permit certain interactions with suitable means of measurement, not belonging to the system, a rigorous definition of this system is naturally no longer possible, and its description will consequently exhibit a statistical character. The very nature of the quantum theory thus forces us to regard the space-time coordination and the claim of causality, the union of which characterizes the classical theories, as complementary features of the description of experience, symbolizing the idealization of observation and definition respectively.

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This view is clearly brought out by the much discussed question of the nature of light and material particles. As regards light, its propagation in space and time is adequately expressed by the electromagnetic theory. Especially the interference phenomena in empty space and the optical properties of material media are completely governed by the wave-theoretical superposition principle. On the other hand the light quantum idea is the immediate expression of the laws of conservation of energy and momentum for the interaction between radiation and matter. As well known the doubts regarding the validity of the superposition principle on the one hand and of the conservation laws on the other hand, which were suggested by this apparent contradiction, have been definitely disproved through direct experiments. As a consequence there seems to be no other choice than to regard the two views of the radiation phenomena as complementary in the above sense. To an analogous conclusion we are led as regards the nature of material particles. The individuality of the elementary electrical corpuscles is forced upon us by general evidence. Nevertheless recent experiences, before all the discovery of the selective reflection of electrons from metal crystals, claim the use of the wavetheoretical superposition principle in accordance with the original ideas of L. de Broglie. Our point of view is essentially different from that taken by de Broglie in a recent article (Journal de Physique, serie VI, t. 8, p. 225, 1927). This author attempts to reconcile the two apparently contradictory sides of the phenomena by regarding the individual particles or light quanta as singularities in the wave field. It does not seem, however, that any such view resting upon the concepts of classical physics is suited to help us over the fundamental difficulties referred to. On the contrary, the dilemma of the nature of light and material particles seems, as far as classical concepts are used, to be unavoidable and to constitute an adequate summary of the analysis of experiments. In the discussion of these questions it must be kept in mind, that according to the view taken above, radiation in empty space as well as isolated material particles are abstractions, their properties on the quantum theory being observable and definable only through their interaction with other systems. Nevertheless in the present state of science these abstractions are unavoidable for a description of experience. Already in the analysis of the definition of the simplest concepts the statistical character of the description appears clearly. Consider Einstein’s well-known formulae of the light quantum theory

E=hv

P=ha

where E and P denote energy and momentum of a quantum, v and Q frequency and wave number of the corresponding phase wave*. According to de Broglie,

*

[Cf. the first footnote on p. [69].]

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

these formulae are equally valid in the wave representation of material particles and fulfill generally the claims of relativistic invariance. It is now to be remembered, that by limiting the extension of the wave field in space and time, in order to represent individuals, one introduces an uncertainty in the definition of [v] and [a], corresponding to the resolution of the field into a group of elementary waves. Indeed, from the theory of optical instruments the following relations are immediately seen to hold d t dv- 1

MS, p . 4

d x do,- 1

where d t and A x represent the extension of the wave field in time and in a given direction of space, while dv and do, denote the order of magnitude of the corresponding variation in the frequency of the elementary waves and of their wave number in the direction chosen. For the greatest attainable accuracy with which the energy and momentum of the individuals can be defined by the above quantum formulas we get then

A t AE-LIX LIP,-

h.

These reciprocal uncertainty relations were given in a recent paper of Heisenberg (Zs. f. Phys. [43, 172-1981 1927) as the expression of the statistical element which, due to the feature of discontinuity implied in the quantum postulate, characterizes any interpretation of observations by means of classical concepts. It must be remembered, however, that the uncertainty in question is not a simple consequence of a discontinuous change of energy and momentum say during an interaction between radiation and material particles employed in measuring the space-time coordinates of the individuals. According to the above considerations the question is rather that of the impossibility of defining rigorously such a change, when the space-time coordination of the individuals is also considered. In the language of the relativity theory the content of the above uncertainty relations may be summarized in the statement that according to the quantum theory a reciprocal relation exists between the maximum sharpness of definition of the space-time and energy-momentum vectors associated with the individuals. This circumstance may be regarded as a simple symbolical expression for the complementary nature of the space-time description and the claims of causality. At the same time, however, the general character of this relation makes it possible to a certain extent to reconcile the conservation laws with the space-time coordination of observations, the idea of a coincidence of well defined events in a space-time point being replaced by that of unsharply defined individuals within finite space-time regions. Following this analogy to the causal space-time

MS,p . 5

P A R T I: T H E E M E R G E N C E O F T H E C O M P L E M E N T A R I T Y A R G U M E N T

MS, p. 6

description, Heisenberg, in the paper cited, has made an important contribution to the elucidation of the quantum-theoretical methods. Due to the gradual spreading of the wave fields associated with the individuals the statistical character of the description, however, is by no means limited to the inaccuracy expressed by Heisenberg’s relations. Indeed, we are forced to comtemplate a proper reduction of the spatial extension of the fields after every new observation. In this connection Heisenberg points out that in a certain sense we may say that already the ordinary (macroscopic) phenomena are created by observation. Nevertheless a complete account for the principal difficulties of satisfying the claims of causality within a space-time representation of atomic (microscopic) phenomena would seem to be offered only by the view that we are dealing here with complementary features of the description of nature. Below we shall come back to this point in a more general connection. Hitherto we have only regarded certain general features of the quantum problem. The situation implies, however, that the main stress has to be laid on the formulation of the laws governing the interaction between the objects which we symbolize by the abstractions of isolated particles and radiation. Points of attack for this formulation are presented in the first place by the problem of atomic constitution. As well known, it has been possible here by means of an elementary use of classical concepts and in harmony with the quantum postulate to throw light on essential sides of experience. For instance the experiments regarding the excitation of spectra by electronic impacts and radiation are adequately accounted for on the assumption of discrete stationary states and individual transition processes. This is primarily due to the circumstance that in these questions the dependence of the processes on the time may largely be disregarded. It is just this simplification which allowed to trace in the problem of atomic constitution a far-reaching correspondence between the consequences of the classical theory and those of the quantum theory, notwithstanding the fundamental difference between them. In the excitation of spectra this difference appears strikingly in the circumstance that spectral lines which on the classical view are ascribed to the same state of the atom will, according to the quantum postulate, correspond to separate transition processes, between which the excited atom has a choice. A characteristic example of the correspondence referred to is presented by the connection between the classical treatment of dispersion and the formulation given by Einstein of the statistical laws governing the radiative transition processes. For the formulation of the correspondence principle a fragmentary use of classical electrodynamics is, however, in general only sufficient in the limit, where in statistical applications the discontinuities may be disregarded. Although Kramers’ treatment of the dispersion problem gave important hints for the rational development of correspondence considerations, it is only through the

P A R T I : T H E E M E R G E N C E OF T H E C O M P L E M E N T A R I T Y A R G U M E N T

quantum theoretical methods created in the last few years that the general endeavours laid down in this principle have obtained an adequate formulation. As well known, Heisenberg succeeded in taking a decisive step in this direction by showing how in the equations of motion of classical mechanics the ordinary kinematical and mechanical quantities could be replaced by symbols, which refer directly to the individual processes required by the quantum postulate. Through the collaboration of Born, Jordan and Dirac the theory was given a very elegant formulation, the correspondence with the classical theory finding its direct expression in the formal conservation [retention] of the classical laws, while the element characteristic of the quantum theory, Planck’s constant, appears explicitely only in the algorithms to which the symbols, the so-called matrices, are subjected. In fact matrices, which represent canonically conjugated variables in the sense of the Hamiltonian equations, do not obey the commutative law of multiplication, but two such quantities q and p have to fulfill the relation

MS, p. 7

h qp - p q = i 27t

where i is the square root of -1. Indeed this exchange relation expresses strikingly the symbolical character of the matrix formulation of the quantum theory. To begin with, the theory was mainly developed in connection with problems of the mechanics of material particles and is therefore often called quantum mechanics. Recently, however, Dirac has been able to give an analogous treatment of the interaction of radiation and atoms, thus taking a first step towards the development of a rational quantum electrodynamics. It must not be forgotten, however, that it has not yet been possible in the formulation of the matrix problems to do justice to the claims of the relativity theory. This is intimately connected with the aforesaid simplification of the atomic problem appearing in the resignation as to time relations. Although the matrix theory directly takes cognizance of the individual processes, thus placing the question of the possibilities of observation in the foreground, still the inner consistency of the theory is achieved only at the expense of any immediate interpretation of the results of calculation. As pointed out at the beginning, it must be kept in mind that a closed system escapes all possibility of observation and that according to the quantum postulate observations always imply a lack of definition of the system. The applicability of the matrix methods is in the first place due to the circumstance that in atomic problems time averages are to a large extent all that is required. Recently Dirac and Jordan have been able by means of their transformation theory of matrices to extend the use of such averages and to obtain a general statistical formulation

MS, p. 8

P A R T I: T H E EMERGENCE OF T H E COMPLEMENTARITY ARGUMENT

of the quantum theory. On the basis of the methods developed by these authors and in close connection with ideas of Born and Pauli, Heisenberg has in the paper already cited above attempted a closer analysis of the physical content of the quantum theory especially in view of the apparently paradoxical character of the exchange relation. In this connection he has formulated the relation A q A p h as the general expression for the maximum accuracy with which two canonically conjugated variables can simultaneously be observed. In this way Heisenberg has been able to elucidate many paradoxes appearing in the application of the quantum postulate. It must be born[e] in mind, however, that we are concerned here with principal difficulties regarding our ordinary concepts much like those already contemplated in connection with the questions of radiation in empty space and isolated material particles. Indeed the shortcomings of our ordinary concepts appear, if possible, even more clearly in the general interaction problem. A simple description of the situation, however, involving as above the complementary ideas of individuality and superposition is provided by the method of wave mechanics developed by Schrodinger on the basis of the ideas of de Broglie. Already in his first considerations concerning the wave theory of material particles de Broglie pointed out that the stationary states of an atom may be visualized as an interference effect of the phase waves associated with an electron bound. (If) it is true that this point of view at first did not lead beyond the earlier methods of quantum theory as regards quantitative results. Schrodinger, however, succeeded in developing a wave-theoretical method which is equivalent to the matrix method and which has proved to be of decisive importance for the great progress in atomic physics during the last years. Indeed, the proper vibrations of the Schrodinger wave-equation have been found to furnish a representation of the stationary states of an atom meeting all requirements. In particular he was led to associate with each of these proper vibrations a continuous distribution of electricity, suited to represent the electrostatic properties of the atom in a stationary state. Considering further the superposition of two such proper vibrations Schrodinger obtained an instructive illustration of the consequences of the quantum postulate regarding the transition processes between two stationary states. Indeed this superposition was found to result in an oscillating continuous electric distribution which on classical electrodynamics would give rise to a radiation of the proper frequency and of an intensity intimately connected with the consequences of the matrix theory. In view of these results Schrodinger has expressed the hope that the development of the wave theory would eventually remove the irrational element expressed by the quantum postulate and open the way for a complete description of atomic phenomena along the line of the classical theories. As a support for this view Schrodinger in a recent paper (Ann.

-

MS,p. 9

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

d. Phys. 83, p. 956, 1927) emphasizes the fact that the discontinuous exchange of energy between atoms, required by the quantum postulate, from the point of view of the wave theory is replaced by a simple resonance phenomenon. In a discussion of this resonance problem it must be kept in mind, however, that in Schrodinger’s theory we are not concerned with a wave problem directly accessible to observation. Notwithstanding its remarkable adaption to the requirements of quantum theory, Schrodinger’s wave equation theory will appear to constitute a transcription of classical mechanics which for the time being can be interpreted only by the explicit use of the quantum postulate and the concept of material particles. This fact has been emphasized especially by Born, who in connection with his important investigation of collision problems has suggested a simple statistical interpretation of Schrodinger’s wave functions. From a similar point of view Klein has discussed the bearing of Schrodinger’s theory on the radiation problem with special regard to the correspondence principle. Quite apart from its fertility Schrodinger’s attack on the atomic problem constitutes a decided advance particularly because it has served to focus the attention on the general consequence of a superposition of different proper vibrations, as exhibited by the complete solution of his wave equation. This feature is implicitely contained in Dirac’s and Jordan’s generalization of the matrix theory, but its explicit consideration permits a simple elucidation of an essential side of the quantum problem. At first it might seem difficult to attribute a meaning to such a superposition as long as we adhere to the quantum postulate. Indeed, the interpretation of experiments on the excitation of spectra rests essentially on the assumption that an atom is always in some definite stationary state. It must be admitted, however, that a closed atomic system is not accessible to observation and constitutes therefore in a certain sense an abstraction, just as the idea of an isolated particle. Now the superposition of proper vibrations affords an adequate means of taking the external influences into account, which are necessarily connected with observations. The situation here is quite analogous to the question regarding the properties of light and material particles. Although the definition of energy and momentum of the individuals depends upon the conception of an elementary harmonic wave, still any representation of the observed phenomena in space and time rests upon a consideration of the interference within a group of such elementary waves. Just as in our derivation of the reciprocal relations between the accuracies, with which the space-time vector and the momentum-energy vector can be defined, there results in the more general case a limitation in the accuracy of the definition of conjugated variables, which corresponds to Heisenberg’s general uncertainty relation. The case of discrete stationary states requires special considerations, however, in as much as the formulation of this relation is based on a continuous sequence of states. An immediate conse-

MS,p. 10

MS,p . 1 1

P A R T I: T H E E M E R G E N C E OF THE C O M P L E M E N T A R I T Y A R G U M E N T

MS, p. 12

quence of the interference of the elementary waves within a wave group is that one cannot speak of the phase of the group as a whole in the same manner as in the case of individual elementary waves. This ambiguity in the phase, well known from the theory of optical instruments, is essential for the consistency of the representation of individuals by waves. However, the statistical character of the quantum theoretical description in the general case is no more exhausted by the uncertainty relation than in the simple problem of isolated particles. On the whole it appears that only by taking the possibilities of definition and of observation simultaneously into account it is possible to obtain an adequate representation of the correspondence between the quantum theory and the classical theories as well as of their fundamental difference. The formulation of the interaction problem in Schrodinger’s theory seems to be particularly suited to illustrate the view concerning the nature of the quantum theory advanced in this note. Indeed, this view that the space-time description of natural phenomena is to be regarded as an idealization is clearly brought out by the fact that Schrodinger’s formulation depends essentially on the artifice of a multidimensional space. Quite apart from still unsolved difficulties contained in the restriction of the transcribed mechanical problem to the non-relativistic case, this fact prevents a direct comparison with the space-time description of the classical theory. The utility of the procedure rests entirely on the circumstance that in interpreting observations it is always possible to reduce the considerations to three of the spatial coordinates only. This is connected with the fact that in all direct experiences about atomic properties we are concerned with the behaviour of atoms in interaction with radiation or free material particles. We must even admit that the definition of all concepts entering in the description of observational results presupposes these abstractions. This situation must be particularly born[e] in mind when considering the question of the explicit application of the quantum postulate to the interpretation of atomic properties. Indeed, the radical deviation from the causal description which we meet for example in the excitation of spectra appears inseparable from the total insufficiency of the space-time representation of the phenomena. On the whole the concepts of stationary states and individual transitions may be regarded as possessing, within their proper limitations, just the same degree of reality as the concept of individual particles. From the point of view of the complementary nature of observation and definition it appears possible to treat the paradoxes of the quantum theory in a uniform manner in immediate contact with the simplest experiences. A more detailed elaboration of this point of view in its application to a number of simple examples was recently given by the author in a lecture at the Volta congress in Como and will soon appear in the transactions of this congress.

VI. GENERAL DISCUSSION AT THE FIFTH SOLVAY CONFERENCE Electrons et photons, Rapports et discussions du cinquikme Conseil de physique tenu a Bruxelles du 24 au 29 Octobre 1927, Gauthier-Villars, Paris 1928, pp. 253-256, 261-263 and 264-265 and UNPUBLISHED MANUSCRIPT FROM FOLDER LABELLED NOTES FROM SOLVA Y MEETING (1927) EXTRACTS, PARTLY IN TRANSLATION

See Introduction to Part I, sect. 5 .

P A R T I : T H E E M E R G E N C E OF T H E C O M P L E M E N T A R I T Y A R G U M E N T

The discussion remarks by Einstein*, Dirac, and Heisenberg are translated from the French text of the Solvay Report. The folder “Notes from Solvay Meeting”, 1927, contains notes taken during the discussions at the conference. There are 12 pages written in ink in J.E. Verschaffelt’s handwriting, comprising Bohr’s contributions to the discussion of Compton’s report (cf. Vol. 5 ) and to the General Discussion. The languages are English, German and French. The notes are rather incomplete with many gaps. We have here reproduced the most comprehensible parts of Bohr’s answers to Einstein and Dirac. In a few places, indicated by square brackets, we have attempted to fill the gaps, but the reader should be aware of the tentative character of these interpolations. There are 8 further pages written in pencil in Kramers’ handwriting, comprising a remark by Bragg, the first part of the General Discussion, and some scattered notes. The languages are English, French, Danish, and Dutch. Finally there are carbon copies of 20 typed pages, or parts of pages, with additions and corrections in ink in Verschaffelt’s handwriting. They contain reports of various discussions at the meeting, the languages being English, French and German. Also these reports are very incomplete with many gaps. The manuscripts are kept in an envelope labelled (by Kramers): “Solvaykongressen 1927”. This material has not yet been microfilmed.

* Under copyright of the Hebrew University, Jerusalem, Israel, and reproduced with the kind permission of this institution.

P A R T I : T H E E M E R G E N C E OF T H E C O M P L E M E N T A R I T Y A R G U M E N T

GENERAL DISCUSSION OF THE NEW IDEAS PRESENTED.

Translation

Causality, Determinism, Probability.

...

Einstein - I must apologize for not having studied quantum mechanics thoroughly. Nevertheless, I would like to make some general remarks. With respect to this theory, one can take two standpoints regarding its domain of validity which I would like to characterize by means of a simple example. Let S be a screen in which a small opening 0 is made (fig. 2), and let P be a photographic film in the shape of a hemisphere of large radius. Let us suppose that electrons are hitting S in the direction of the arrows. Part of these electrons will pass through 0. Because of the smallness of the opening and of the velocity of the particles, they will distribute themselves uniformly in all directions and react with the film.

Fig.

2.

The following co siderations are common 1 the two ways f perceivin the theory. There are de Broglie waves which fall almost perpendicularly on S and are diffracted at 0. On the other side of 0 one has spherical waves which reach the screen P I and whose intensity at P is a measure of what happens at this place. Now, we can characterize the two points of view as follows: 1. Interpretation I . - The de Broglie-Schrodinger waves do not correspond to a single electron, but to an electron cloud, extended in space. The theory does not give any information about the individual processes, but only about an ensemble of an infinity of elementary processes. 2. Interpretation II. - The theory claims to be a complete theory of the individual processes. Each particle which moves towards the screen as far as one

pp. 253-256

P A R T I: T H E EMERGENCE OF T H E COMPLEMENTARITY A R G U M E N T

can determine from its position and its velocity, is described by a de Broglie-Schrodinger wave packet of small wavelength and small angular aperture. This wave packet is diffracted and, after diffraction, arrives partly at the film P in a resolved state. According to the first point of view which is purely statistical, Iv/l2 expresses the probability that some particle from the cloud is found at the place considered, e.g., at a particular place on the screen. According to the second point of view, /v/12 expresses the probability that at the moment considered the same particle is situated at a particular place (e.g., on the screen). Here, the theory applies to the individual process and claims to give information about everything which is governed by laws. The second interpretation goes further than the first in the sense that all the information resulting from I1 also results from the theory by virtue of I, but the inverse is not true. It is only by virtue of I1 that the theory contains the consequence that the conservation laws are valid for the elementary processes; it is only from 11 that the theory can deduce the result of the Geiger-Bothe experiment and that it can explain the fact that in a Wilson chamber the droplets originating from an a-particle are approximately situated on continuous lines. But, on the other hand, I have objections to make against interpretation 11. The scattered wave moving towards P does not present any preferred direction. If Iv/12 was simply considered as the probability that a definite particle is situated at a certain place at a definite instant, it might happen that one and the same elementary process would act at two or more places of the screen. But the interpretation according to which 1 v/I2 expresses the probability that this particle is situated at a certain place presupposes a very particular mechanism of action at a distance which would prevent the wave continuously distributed in space from acting at two places of the screen. In my opinion one can only counter this objection in the way that one does not only describe the process by the Schrodinger wave, but at the same time one localizes the particle during the propagation. I think that de Broglie is right in searching in this direction. If one works exclusively with the Schrodinger waves, interpretation I1 of I v / l 2 in my opinion implies a contradiction with the relativity postulate. I would still like briefly to indicate two arguments which seem to me to speak against viewpoint 11. One is essentially connected with a multidimensional representation (configuration space) because only this representation makes possible the interpretation of lv/I2 belonging to interpretation 11. Now, it seems to me that there are objections of principle against this multidimensional representation. In fact, in this representation two configurations of a system which only differ by the permutation of two particles of the same kind are represented by two different points (of configuration space), which is not in

PART I : THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

agreement with the new statistical results. Secondly, the peculiarity of the forces of acting only at small spatial distances finds a less natural expression in the configuration space than in the space of three or four dimensions. Einstein having exposed his standpoint, Bohr says:

Manuscript, p. 3

Bohr: I feel myself in a very difficult position because I don’t understand what precisely is the point which Einstein wants to [make]. No doubt it is my fault.

...

MS, p. 4

...

MS, p. 5

Dirac -

Translation

As regards general problem I feel its difficulties. I would put problem in other way. I do not know what quantum mechanics is. I think we are dealing with some mathematical methods which are adequate for description of our experiments. Using a rigorous wave theory we are claiming something which the theory cannot possibly give. [We must realize] that we are away from that state where we could hope of describing things on classical theories. Understand same view is held by Born and Heisenberg. I think that we actually just try to meet, as in all other theories, some requirements of nature, but difficulty is that we must use words which remind of older theories. The whole foundation for causal space-time description is taken away by quantum theory, for it is based on assumption of observations without interference. ... excluding interference means exclusion of experiment and the whole meaning of space and time observation ... because we [have] interaction [between object and measuring instrument] and thereby we put us on a quite different standpoint than we thought we could take in classical theories. If we speak of observations we play with a statistical problem. There are certain features complementary to the wave pictures (existence of individuals). ... The saying that space-time is an abstraction might seem a philosophical triviality but nature reminds us that we are dealing with something of practical interest. Depends on how I consider theory. I may not have understood, but I think the whole thing lies [therein that the] theory is nothing else [but] a tool for meeting our requirements and I think it does.

...

I would now like to state my opinion concerning determinism and the significance of the numbers which occur in the calculus of the quantum theory as they appear to my mind after I have thought about Bohr’s remarks. In the classical theory one starts from certain numbers which completely specify the initial state of the system and one deduces certain numbers which specify the final state. This deterministic theory only applies to an isolated system.

pp. 261-263

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

But, as Bohr remarked, an isolated system is by definition unobservable. One can only observe a system by disturbing it and by studying its reaction to the perturbation. Consequently, since physics only deals with observable quantities the classical deterministic theory cannot be defended. Also in the quantum theory one starts from certain numbers from which one deduces other numbers. Let us try to penetrate into the physical essence of these two sets of numbers. The perturbations which an observer inflicts on a system in order to observe it are directly subject to his control and are acts of his free will. It is exclusively the numbers which describe these acts of free choice which can be taken as initial numbers for a calculation in the quantum theory. Other numbers specifying the initial state of the system are fundamentally unobservable and do not appear in the quantum theoretical treatment. Let us now consider the final numbers obtained as a result of an experiment. It is essential that the result of an experiment must be a permanent registration. The numbers which describe such a result must not only contribute to describing the state of the world at the instant when the experiment is finished, but they must also contribute to describing the world at every subsequent instant. These numbers describe that which is common to all the events in a certain chain of facts linked together by causal links and extending infinitely into the future. We can take as an example an experiment with a Wilson cloud chamber. The causal chain here consists in the formation of droplets of water around ions, the scattering of light by these droplets and the action of this light on a photographic plate where they leave a permanent imprint. The numbers which constitute the result of the experiment describe all the events in this chain equally well and serve to describe the state of the world at any instant subsequent to the one when the chain started. In general one attempts, by theoretical considerations, to extend the chain as far as possible backwards into the past, in order for the numbers obtained as a result of the experiment to apply as directly as possible to the process being examined. In the example which I have just given, one might perhaps attribute the formation of the ions to a /?-particle so that as the result of the experiment one obtains the numbers representing the trajectory of a /?-particle. This conception of the nature of experimental results agrees perfectly with the new quantum theory. This theory describes the state of the world at any instant by a wave function t y which normally develops according to a causal law so that its initial value determines its value at any later instant. It may happen, however, that at a given instant, t l , I,Y can be expanded in a series of the form

w=

c cnwn, n

where the I C / ~are wave functions of such a nature that they cannot interfere with

P A R T I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

each other at an instant later than t l . If this is the case, the state of the world at instants later than tl will be described not by v / , but by one of the cy,. One may say that nature chooses whichever iyn it pleases since the only information which the theory gives is that the probability of choosing any one t+vnis Ic,12. Once the choice is made it is irrevocable and affects any future state of the world. The value of n chosen by nature can be determined by an experiment and the results of all experiments are the numbers describing similar choices by nature. Let us take as an example the case of a simple collision process. The wave packet representing the incident electron is scattered in all directions. One must take as the wave function after the process not any scattered wave, but once again a wave packet moving in a definite direction. From the results of a suitable experiment one might, by reconstructing backwards a chain of events causally linked with one another, determine the direction into which the electron was scattered and thereby conclude that nature chose this direction. If, however, one would place a mirror so that it reflects the electron wave scattered in one direction d l , in such a way that it would interfere with the electron wave scattered in another direction d2 one would not be able to distinquish between the case where the electron is scattered in the direction d2 and the one where it is scattered in the direction dl and reflected again in the direction d2.Hence one would not be able to construct the causal chain sufficiently far and one would not be able to say that nature chose one direction after the collision took place; it is not until later that nature chose the place where the electron would appear. The interference of the v/, forces nature to postpone its choice till later. Dirac having exposed his views:

Manuscript, p . 5

Bohr: Quite see that one must go into details of pictures, if one wants to control or illustrate general statements. I think still that you may simpler put it in my way. Just this distinction between observation and definition allows to let the quantum mechanics appear as generalisation. What does mean: get records which do not allow to work backwards. Even if we took all molecules in photographic plate one would have closed system. If we tell of a record we give up definition of plate. Whole point lies in that by observation we introduce something which does not allow to go on.

...

Heisenberg - I do not agree with Dirac when he says that in the experiment described nature makes a choice. Even if you place yourself very far from your scattering material and if you measure after a very long time, you can obtain interference by taking two mirrors. If nature had made a choice, it would be difficult to imagine how the interferences are produced. Obviously we say that

Translation pp. 264-265

PART I: THE EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

nature’s choice can never be known until the decisive experiment has been done; for this reason we cannot make any real objection to this choice because the expression “nature makes a choice” does not have any physical consequences. I would rather say, as I have done in my latest paper, that the observer himself makes the choice because it is not until the moment when the observation is made that the “choice” has become a physical reality and that the phase relations in the waves, i.e., the ability of interference, is destroyed.

VII. THE QUANTUM POSTULATE AND THE RECENT DEVELOPMENT OF ATOMIC THEORY [2] KVANTEPOSTULATET OG ATOMTEORIENS SENESTE UDVIKLING Overs. Dan. Vidensk. Selsk. Forh. Juni 1927 - Maj 1928, p. 27 THE QUANTUM POSTULATE AND THE RECENT DEVELOPMENT OF ATOMIC THEORY Nature 121 (1928) 78 Communication to the Royal Danish Academy on 18 November 1927 ABSTRACT

NIELS BOHR gav en Meddelelse : Kvantepostulatet 09 Atomteoriens seneste Uduikling. (Den Dualisme, som kendetegner Kvanteteoriens Formulering, er paa laererig Maade blevet belyst gennem de store Fremskridt indenfor Atomteorien i de sidste Aar. Paa Videnskahens nuvierende Trin er denne Dualisme uundgaaelig og ter betragtes som et umiddelbart Udtryk for den ved Kvanteteorien postulerede Svigteii af de elementiereste Principper for vor Naturbeskrivelse.)

Nov. 18.-Niels Bohr : The quantum postulate and the recent development of atomic theory. The dualism which characterises the formulation of the quantum theory has received much illumination through the recent great progress of atomic theory. I n the present state of science, this dualism would seem unavoidable and may be regarded as a direct expression of the fundamental limitation of the ordinary principles of classical physics postulated by the quantum theory.

VIII. THE QUANTUM POSTULATE AND THE RECENT DEVELOPMENT OF ATOMIC THEORY [3] Atti del Congress0 Internazionale dei Fisici

11-20 Settembre 1927, Como-Pavia-Roma, Volume Secondo, Nicola Zanichelli, Bologna 1928, pp. 565-588

Elaboration of the Address given in Como on 16 September 1927 at the International Congress of Physicists on the Occasion of the Centenary of the Death of Alessandro Volta 1st Version and Discussion Remarks, pp. 589-598

See Introduction to Part I, sects. 3 and 4.

P A R T I: T H E EMERGENCE OF THE COMPLEMENTARITY ARGUMENT

THE QUANTUM POSTULATE AND THE RECENT DEVELOPMENT OF ATOMIC THEORY (1928) Versions published in English, German and Danish

English: The Quantum Postulate and the Recent Development of Atomic Theory A Atti del Congress0 Internazionale dei Fisici 11-20 Settembre 1927, Como-Pavia-Roma, Volume Secondo, Nicola Zanichelli, Bologna 1928, pp. 565-588 B Nature (Suppl.) 121 (14 April 1928) 580-590 C “Atomic Theory and the Description of Nature”, Camb. Univ. Press, 1934 (reprinted 196l), pp. 52-91 German: Das Quantenpostulat und die neuere Entwicklung der A tomistik D Naturwiss. 16 (13 April 1928) 245-257 E “Atomtheorie und Naturbeschreibung”, Julius Springer Verlag, Berlin 1931, pp. 34-59 Danish: Kvantepostulatet og Atomteoriens seneste Udvikling F “Atomteori og Naturbeskrivelse” , Festskrift udgivet af K~benhavns Universitet i Anledning af Universitetets Aarsfest November 1929, Bianco Lunos Bogtrykkeri, Copenhagen 1929, pp. 40-68 G “Atomteori og naturbeskrivelse”, J.H. Schultz Forlag, Copenhagen 1958, pp. 47-75 As explained in the Introduction, A and B are very different, B representing a major revision and expansion of the paper. Also, sections have been rearranged and titles added. Both of these versions are therefore included in the present volume (documents VIII and IX). To facilitate a comparison a list of the major changes is given on the following page. C agrees with B , apart from a few minor linguistic improvements. However, the introduction follows A . There are no references. D corresponds to B . E and F correspond to C , apart from a reference to “Atomic Theory and Mechanics” (on p. 53 and p. 61, respectively). In G , the sections are not numbered, the reference to “Atomic Theory and Mechanics” has been removed, and the new Danish orthography introduced. There are no discussion remarks by Bohr reported in the proceedings of the Como Conference.

P A R T I: THE EMERGENCE OF THE COMPLEMENTARITY A R G U M E N I

MAJOR CHANGES FROM VERSION A TO VERSION B The division into sections is changed. Each of the original sections 2 and 5 has been divided into two. Titles are added. A,

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References to paragraphs in B The introduction is shortened. 1st paragraph. The last two sentences, on the complementary aspects of light, are added. 1st paragraph. The reference to Rayleigh is added. 1st paragraph. The last three sentences, concerning the analogy to optics and the r61e of the conservation laws, are added. Last paragraph of section 2. The last three sentences, concerning the r61e of the electric charge, are added. 2nd paragraph. The last two sentences, concerning the irrelevance of the length of the wave train, are added. 4th paragraph. The first two sentences, concerning the irrelevance of the time of observation, are added. Also the fourth sentence, on the concept of velocity, is added. Two whole paragraphs, and the first two sentences of the following paragraph, dealing with the definition of a coordinate system by solid bodies and imperturbable clocks, are added. 3rd paragraph. The last three sentences, on the interaction between object and measuring instrument, are added, whereas a short passage has been moved to another place. The last two paragraphs, on the connection to classical physics and the correspondence principle, have been considerably expanded. The 2nd paragraph, on Heisenberg’s paper, has been considerably expanded. 3rd paragraph. The sentence on the transformation theory has been added. 3rd paragraph. The sentence on the possibilities of definition and observation is added. Also the last sentence of section 4. The 2nd paragraph is considerably expanded. Formula ( 5 ) is added.

P A R T I : THE EMERGENCE OF T H E COMPLEMENTARITY A R G U M E N T

References to paragraphs in B

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2nd paragraph. The first four sentences, concerning the impossibility of defining electron orbits in a stationary state, are added. The last paragraph of section 5 , dealing with the transformation theory, is added. 1st paragraph. One sentence, on collision time and energy conservation, is added. The last two lines are added. 3rd paragraph. The references to Darwin and Kennard are added. The section on the Kaluza-Klein theory is added. This page represents a considerable revision and expansion of the last paragraph in A .

The quantum postulate and the recent development of atomic theory by

N. Bohr - Copenaghen.

Although it is with great pleasure that I follow the kind invitation of the presidency of the congrese to give an account of the present state of the quantum theory in order to open a general discussion on this subject, which takes so central a position in modern physical science, it is with a certain hesitation that I enter on this task. Not only is the venerated originator of the theory present himself, but among the audience there will be several who, due to their participation in the remarkable recent development, will surely be more conversant with details of the highly developed formalism than I am. Still I shall try by making use only of simple consideration and without going into any details of technical mathematical character t o describe to you a certain general point of view which I believe is suited t o give an impression of the general trend of the development of the theory from its very beginning and which I hope will be helpful in order t o harmonize the apparently conflicting views taken by different scientists. No subject indeed may be better suited than the quantum theory to mark the development of physics in the century passed since the death of the great genius, whom we are here assembled t o commemorate. At the same time, just in a field like this where we are wandering on new paths and have t o rely upon our own judgment in order to escape from the pitfalls surrounding us on all sides, we have perhaps more occasion

than ever a t every step t o be remindful of the work of the old masters who have prepared the ground and furnished us with our tools.

S 1. The quantum theory is characterised hy the acknowledgement of a fundamental limitation in t he classical physical ideas, when applied to atomic phenomena. The situation thus created is of a peculiar nature, since our interpretation of the experimental material rests extensively iipon the classical concepts. Notwithstandig the difficulties which hence are involved in the formulation of the quantum theory, it seems, as we shall see, that its essence may be expressed in the so-called qnantum postulate, which to any atomic process attributes an essential discontinuity or rather individuality, completely foreign t o the classical theories and symbolised by Planck’s quantum of action. This postulate implies a renunciation as regards the causal space-time co-ordination of atomic processes. Indeed, Our usual description of physical phenomena is based entirely on the idea that the phenomena concerned may be observed without disturbing them appreciably. This appears, for example, clearly in the theory of relativity, which has been so fruitful for the elucidation of the classical theories. As emphasized by Einstein, every observation or measurement ultimately rests on the coincidence of two independent events at the same space-time point. Jnst these coincidences will not, be affected by any differences which the space-time coordination of different observers otherwise may exhibit. Now the quantum postulate implies that any observation of s t o mic phenomena will involve an interaction with the agency of observation not t o be neglected. Accordingly, an independent reality in the ordinary physical sense csn neither be ascribed t o the phenomena nor to the agencies of observation. This situation has far-reaching consequences. On one hand, the usual definition of the state of a physical system claims the elimination of all external disturbances. But then according to the quantum postulate any possibility of obser-

Tlie q u a n t u m postulate and the recent development

567

vation will be excluded. On the other hand, i f in order to make observation possible we permit certain interactions with suitable means of measurement, not belonging to the system, an unambiguous definition of its state is naturally no longer possible, and there can be no question of causality in the ordinary sense of the word. The very nature of the quantum theory thus forces us to regard the space-time co-ordination and the claim of causality, the union of which characterises the classical theories, as complementary biit exclusive features of the description, syniholising the idealisation of observation and definition respectively. Just as the relativity theory has tgught us that the convenience of sharply distinguishing between space and time solely rests on the smallness of the velocities ordinarily met with compared t o the velocity of light, we learn from the quantum theory that the appropriateness of our usual causal space-time description depends entirely upon the small value of the quantum of action as compared to the actions involved in ordinary sense perceptions. This view is clearly brought out by the much discussed question of the nature of light and the ultimate constituents of matter. As regards light, its propagation in space and time is adequately expressed by the electromagnetic theory. Especially the interference phenomena in vacuo and the optical properties of material media are completely governed by the wave-theoretical superposition principle. Nevertheless the conservation o€ energy and momentum during the interaction between radiation and matter, as evident in the photoelectric and Compton effect, finds its adequate expression just in the light quantum idea put forward by Einstein. As is well known, the doubts regarding the validity of the snperposition principle on one hand mci of the conservation laws on the other, which were suggested by this apparent contradiction, have been definitely disprored through direct experiments. This situation would seem clearly to indicate the impossibility of a causal space-time description of the light phenomena. On one hand, in a t tempting to trace the laws of the time-spatial propagation

5 68

N . Bohr

of light we are according to the quantum postulate confined to statistical considerations. On the other hand, the fulfilment of the claim of causalit'y for the individud light processes characterized by the quantum of action entails a renunciation as regard8 the space time description. As regards the nature of material particles the situation is similar. The individuality of the elementary electrical corpuscles is forced upon us by general evidence. Nevertheless, recent experience, above all the discovery of the selective reflection of electrons from metal crystals, requires the use of the wave theoretical superposition principle in accordance with the original ideas of L. de Broglie. Jiist as in case of light we have consequently in the question of the nature of matter, so far as we adhere to classical concepts, to face an inevitable dilemma, which has to be regarded as the very expression of experimental evidence. I n fact, here again we are not dealing with contradictory but with complementary pictures of the phenomena, which only together offer a natural generalisation of the classical mode of description. I n the discussion of these questions it must be kept in mind, that according to the view taken above, radiation in free space as well as isolated material particles are abstractions, their properties on the quantum theory being observable and definable only through their interaction with other systems. Nevertheless these abstractions are, as we shall see, indispensable for a description of experience in connexion with our ordinary space-time view.

Q 2. The difficulties which a causal space-time descrip-

tion is confronted with in the quantum theory and which repeatedly have been subject of discussions are now placed into the foreground by the recent development of the symbolic methods. An important contribution to the problem of a consistent application of these methods has been made lately by Heisenberg (Zeitsohr. f. P h p . , 43, 172, 1927). In particular he has stressed the peculiar reciprocal uncertainty which affects all observations of atomic quantitties.

T h e qirantum postulate and the recent developnzent

569

Before we enter on his results it will be advantageous t o show, how the complementary features of the description appearing in this uncertainty are unavoidable already in an analysis of the most elementary concepts employed in interpreting experience. The fundamental contrast between the quantum of action and the classical concepts is immediately apparent from the simple formulas which form the common foundation of the theory of light quanta and of the wave theory of material particles. If Planck’s constant be denoted by h, as is wellknown. Et = II = h (1) where E and I are energy and momentum respectively, 7 and I the corresponding period of vibration and wavelength. I n these formulas the two notions of light and matter enter in sharp contrast. While energy and momentum are associated with the concepts of particles and hence may be characterized according to the classical point of view by definite space-time coordinates, the period of vibration and wavelength refer to a plane harmonic wave train of unlimited extent in space and time. Only with the aid of the superposition principle it becomes possible t o attain a connection with the ordinary way of description. Indeed a limitation of the extent of the wave-fields in space and time always can be regarded as resulting from the interference of a group of elementary harmonic waves. As shown by de Eroglie (Thbse, Paris, 1924), the translational velocity of the individual associated with the waves can be represented by just the so-called group-velocity. Let YS denote a plane elementary wave by

A

COS

2% (it’- ZGZ - yGv - X B p

+ 6)

where d and 6 are constants determining respectively the amplitude and the phase. The quantity v = I/. is the frequency, B,, B ~ G* , the wave numbers in the direction of the coordinate axes, which may be regarded as vector compo-

5 70

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nents of the wave number b = 11,~in the direction of propagation. While the wave or phase velocity is given by v , l b , the group velocity is expressed by dv/da. Now according to the relativity theory we have for a particle with the velocity P ) : 2) 1 = --E and vdI = d E , C2

where c denotes the velocity of light. Hence by equation (1)the phase velocity is C*/V and the group velocity v . The circumstance that the former in general is greater than the velocity of light emphasizes the symbolic character of these considerations. At the same time the possibility of identifying the velocity of the particle with the group velocity indicates the field of application of space-time pictures in the quantum theory. Here the complementary character of the description appears since the use of wavegroups is necessarily accompanied by an uncertainty in the definition of period and wavelength and hence also in the definition of the corresponding energy and momentum as given by relation (1). Rigorously speaking, a limited wavefield can only be obtained by the superposition of a manifold of elementary waves corresponding t o all values of v and B,, B,, B ~ . But the order of magnitude of the mean difference between these values for two elementary waves in the group is given in the most favorable case by the condition At Av = AX A0, = Ay A0, = AXAs, = 1

where At, Ax, Ay, Ax denote the extension of the wavefield in time and in the directions of space corresponding to the coordinate axes. These relations well-known from the theory of optical instruments, express the condition that the wave-trains extinguish each other by interference at the space-time boundary of the wave field. They may be regarded also as signifying that the group as a whole has no phase in the same sense as the elementary waves. From equation (1)we find thus At A23 = Ax A I , = d y dI, = AXA I , = h

(2)

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as determining the highest possible accuracy in the definition of the energy and momentum of the individuals associated with the wave field. I n general the conditions for attributing an energy and a momentum value to a wave field by means of formula (1) are much less favorable. Even if the composition of the wavegroup corresponds in the beginning to the relations (2), i t will in the course of time he subject to such changes that it becomes less and less suitable for representing an individual. I t is this very circumstance which gives rise to the paradoxical character of the problem of the nature of light and of material particles. Tn the language of the relativity theory the content of the relations ( 2 ) may be summarized in the statement that according to the quantum theory a general reciprocal relation exists between the maximum sharpness of definition of t,he space-time and energy-momentum 1-ectors associated with the individuals. This circumstance may be regarded as a simple symbolical expression €or the complementary nature of the space-time description and the claims of causality. At the same time, however, the general character of this relation makes it possible to a certain extent t o reconcile the conservation laws with the space-time co-ordination of observations, the idea of a coincidence of well-defined events in space-time points being replaced by t h a t of unnharply defined individuals within finite space-time regions. This circumstance permits to avoid the well-known paradoxes which one encounters in attempting to tlescrihe the scattering of radiation by free electrical particles as well as the collision of two such particles. According to the classical concepts the description of the scattering requires a finite extent of the radiation in space and time, while in the change of the motion of the electron demanded by the quantum postulate one seemingly is dealing with an instantaneous effect taking place at a definite point in space. Just as in the case of radiation, however, it is impossible to define momentum and energy for an electron without considering a finite spacetime region. Furthermore an application of the conserva-

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512 ~

t,ion laws t o the process implies that the accuracy of definition of the energy momentum vector is the same for the radiation and the electron. I n consequence according to relation ( 2 ) the associated space time regions can be given the same size for both individuals in interaction. A similar remark applies to the collision between two material particles, although the significance of the quantum postulate for this phenomenon was disregarded before the necessity of the wave concept was realized. Here this postulate does indeed represent the idea of the individuality of the particles which, transcending the space-time description, meets the claim of causality. The relations ( 3 ) were given by Heisenberg as an expression for the maximum precision with which the space-time coordinates and momentum-energy components of a particle can be measured simultaneously. His view was based on the following consideration: On the one hand the coordinates of a particle can be measured with any desired degree of accuracy, by using for instance an optical instrument, provided radiation of sufficiently short wavelength is used for illumination. According t o the quantum theory, however, the scattering of radiation from the object is always connected with a finite change in momentum, which is the larger the smaller the wavelength of the radiation nsed. The momentum of a partide on the other hand earn be determined with any desired degree of accuracy, by niensuring for instance the Doppler effect of the scattered radiation, provided the wavelength of the radiation is SO large that the effect of recoil can be neglected, but then the determination of the space coordinates of the particle becomes correspondingly less accurate. The essence of this consideration is the inevitability of the quantum postulate in the estimation of the possibilities of measurement. A closer investigation of the possibilities of definition would still seen1 necessary in order t o bring out the general complementary character of the description. Indeed, such a change could not prevent us from ascribing

T h e qriantz~mpostulate and the recent developnient

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accurate values t o the space-time coordinates, as well as to the momentum-energy components before and after the process. The reciprocal uncertainty which always affects the values of these quantities is, as will be clear from the above considerations, rather an outcome of the limited accuracy with which changes in energy and momentum can be defined, provided the wave fields used for the determination of the space-time coordinates of the particle shall be sufficiently limited. I n using an optical instrument for determination of position it is necessary to remember that the formation of the image always requires a convergent beam of light. Denoting by 1 the wavelength of the radiation used, and by E the so-called numerical aperture, the resolving power of a microscope is given by the well-known expression l / 2 , . Even if the object is illuminated by parallel light so that the momentum hit. of the incident light quantum is known both as regards magnitude and direction, the finite value of the aperture will prevent an exact knowledge of the recoil accompanying the scattering. Even in case the momentum of the particle were accurately known before the scattering process, our knowledge of the component of momentum parallel to the focal plane after the observation will consequently be affected by an uncertainty amounting to 2eh/,1. The product of the least inaccuracies with which the positional coordinate and the conjugated component of momentum can be ascertained is therefore just given by formula (2). I n measuring momentum with the aid of the Doppler effect-with due regard to the Compt'on effect-one will employ a parallel wave train. For the accuracy, however, with which the change in wavelength of the scattered radiation can be measured the extent of the wave train in the direction of propagation is essential. If we awume that the directions of the incident and scattered radiation are parallel and opposite respectively to the direction of the podtion coordinate and momentum component t o be mea-

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sured, t8hen ct/21 can be taken as a measure of the accuracy in the determination of the velocity, where I denotes the length of the wave train. For simplicity we here have regarded the velocity of light as large compared to the velocity of the particle. If m represents the mass of the particle, then the uncertainty attached t o the valiie of the momentum after observation is cmIl21. I n this case the magnitude of the recoil 2h/i is sufficiently well defined in order not to give rise to an appreciable uncertainty in the value of the momentum of the particle after observation. Indeed the general theory o€ the Compton effect allows to compute the momentum components in the direction of the radiation before and after the recoil froin the wa've lenghts of the incident and scattered radiation. Even if the positional coordinates of the particle were accurately known in the beginning, onf knowledge of the position after o h s w vation nevertheless will he affected hy an uncertainty. Tndeed, on account of the impossibility to attribute a definite instant to the recoil we know the mean velocity in the direction of observation during the scattering process only with a n accumcy 2 h j m 1 . The uncertainty in the position after ohserT-ation hence is 2 h l i ~ ~ i .Here . too the product of the accuracies in the measurement of position and momentum is thus given by the general formula ( 3 ) . I n the ahore examples we have repeatedly mentioned the velocity of a paxticle. The purpose, however. ~ 7 a sonly to obtain a connection w-ith our ordinary space-time description con>-enient in this case, since, strictly speaking, an unambiguous definition of velocity is excluded by the quantum postulate. This is particularly to be remembered when comparing the results of successive obserITations. We haire seen that the position of an individual at two given moments can be determined to any desired degree of accuracy; but if we from such measurements calculate the 1-elocity of the individual in the ordinary way, it must he clearly realized that we ase dealing with an abstraction, from which no unambiguous information concerning the previous or future

T h e guantitni postulate aiid the recent development

575

behaviour of the individual can be obtained. -4s shown above, this results from the fact that the determination of the position of a particle always involves a complete rupture in the description of its dynamical behaviour, in the same way as the accurate determination of its momentum always involves a gap in the knowledge of its spatial propagation. Just this circumstance reveals in a striking manner the fundamental departure from the classical description of nature. It is true, as emphasized by Heisenberg, that an instructive analogy t o the quantum theoretical point of view is obtained by comparing the uncertainty in the observations of atomi8tic (microscopio) phenomena contained in relation (2) with the uncertainty inherently contained in any observation, due to imperfect measurements, as considered in the ordinary description. He remarlis on that occasion that one even in case of macroscopic phenomena may say, in a certain sense, &hatthey originate in repeated observations. It must not be forgotten, however, that in the classical theories every succeeding observation permits a prediction of future events with ever increasing accuracy, because it improves the knowledge of the initial state of the system.

0 3. Hitherto we have only regarded certain general features of the quantum problem. The situation implies, however, that the main stress has t o be laid on the formulation of the laws governing the intemction between the objects which we symbolise by the abstractions of isolated patrticles and radiation. Points of attack for this formulation are presented in the first place by the problem of atomic constitution. As is well known, it has been possible here by means of an elementary use of classical concepts and in harmony with the quantum postulate t o throw light on essential aspects of experience. For example, the experiments regarding the excitation of spectra by electronic impacts and by radiation are adequately accounted for on t'he assumption of discrete stationary states and individual

576

A’. Bohr

transition processes. This is primarily due to the circumstance that in these questions no closen description of the space time behaviour of the processes is required. It is just this simplification which allowed to trace in the problem of atomic constitution a farreaching correspondence between the consequences of the classical theory and those of the quantum theory, notwithstanding the fundamental difference between them. I n the excitation of spectra this difference appears strikingly in the circumstance that spectral lines which on the classical view are ascribed to the same state of the atom will, according t o the quantum postulate, correspond to separate transition processes, between which the excited atom has a choice. A characteristic example of the correspondence referred to is presented by the connexion traced by Ladenburg and Kramers between the classical treatment of dispersion and t’he formulation given by Einstein of the statistical laws governing the radiative transition processes. For the formulation of the correspondence principle a fragmentary use of classical electro-dynamics is, however, in general only sufficient in the limit, where in statistical applications the discontinuitlies may be disregarded. Although just Kramers’ treatment of the dispersion problem gave important hints for the rational development of correspondence considerations, it is only through the quantum theoretical methods created in the last few years that the general endeavours laid down in the principle mentioned have obtained an adequate formulation. As is well known, Heisenberg succeeded in taking a decisive step as regards the development of a rational quantum mechanics by showing how in the equations of motion of classical mechanics the ordinary kinematical and mechanical quantities could be replaced by symbols, which refer directly to the individual processes required by the quantum postulate. Through the collaboration of Born, Jordan and Dirac the theory was given a very elegant formulation, the correspondence with the classical theory finding its direct expression in the formal conser-

T h e qrrrlntum posliilate arid the recent development

577

vation of the classical laws, while the element characterist,ic of the quantum theory, Planck’s constant, appears explicitly only in the aJgorithms to which the symbols, the so-called matrices, are subjected. In fact, matrices, which represent canonically conjugated variables in the sense of the Hamiltonian equations, do not obey the commutative law of multiplication, but two such quantities q and p have to fulfil the relation

Indeed, this exchange relation expresses strikingly the symbolical character of the matrix formulation of the quantum theory. In cert’ain sense the matrix theory may be described as a calculus with directly observable qnantities. It must be remembered, however, that so far the procedure is limited just to problems, in which in applying the quantum postulate the space-time description may largely be disregarded, and the proper question of observation therefore placed in the background. In pursuing further the correspondence of the quantum laws with classical mechanics, the stress placed on the statistical character of the quantum theoretical description, which is brought in by the quantum postulate, has been of fundamental importance. Here the generalisation of the symbolical method made by Dirac and Jordan represented a great progress by making possible the operation with mafrices, which are not arranged according to the stationary states, but where the possible values of any set of variables may appear as indices of the matrix elements. On the basis of the procedure developed by these authors and in close connexion with ideas of Born and Pauli, Heisenberg has in the paper already cited above attempted a closer analysis of the physical’content of the quantum theory, especially in view of the apparently paradoxial character of the exchange relation (3). In this connection he has formulated the relation LlqLIp-h ~ e s o c o n t oCM

-

Congress0 ad F M C ~ VOI. 11

- 37

(4)

578

N . BOhr

as the general expression for the maximum accuracy with which two canonically conjugated variables can simultaneously be observed. I n this way Heisenberg has been able to elucidate many paradoxes appearing in the application of the quantmm postulate and to a large extent to demonstrate the consistency of the application of the symbolic methods. For the discussion of just these questions the method of wave mechanics developed by Schrodinger has-proved very helpful. I n fact, as we shall see, it permits a general application of the principle of superposition also in the problem of interaction, thus offering an immediate connexion with the aboTre considerations concerning radiation and free particles. $ 4. Already in his first considerations concerning the wave theory of material particles, de Broglie pointed out that the stationary states of an atom may be visualised as an interference effect of the phase wave associated with an electron bound. It is true that this point of view at first did not as regards quantitative results lead beyond the earlier methods of quantum theory, to the development of which Sommerfeld has contributed so essentially. Schrodinger, however, succeeded in developing a wave-theoretical method which is equivalent to the matrix method and which has proved to be of decisive importance for the great progress in atomic physics during the last years. Indeed, the proper vibrations of the Schrodinger wave-equation have been found to furnish a representation of the statimary states of an atom meeting all requirements. The energy of each state is connected with the corresponding period of vibration according to the general quantum relation (1). Furthermore the number of nodes in the various characte ristic vibrations gives a simple interpretation to the concept of quantum number which was already known from the older methods, but at first did not seem t o appear in the matrix formulation. In addition Schrodinger could associate with the solutions of the wave equation a continuous distribution of charge and current, which, if applied t o a characte-

T h e q i i a n ! i ~ mpostulate arid the recent d e v e l o ~ n i e n i

579

risbic vibration, represents the electrostatic and magnetic properties of an atom in the corresponding stationary state. Similarly the superposition of two characteristic solutions corresponds t o a continuous vibrating distribution of electrical charge, which on classical electrodynamics would give rise to an emission of radiation illustrating instructively the consequences of the quantum postulate and the correspondence requirement. regarding the transition process between two stationary states formulated in matrix mechanics. I n view of these results Schrijdinger has expressed the hope that the development of the wave theory would eventually remove the irrational element expressed by the quantum postulate and open the way for a complete description of atomic phenomena along the line of the classical theories. I n support of this view Schrodinger, in a recent paper ( K Ann. d. Phys. )), 83, pag. 956, 1927) emphasises the fact that the discontinuous exchange of energy between atoms required by the quantum postulate, from the point of view of the wave theory, is replaced by a simple resonance phenomenon. It must be kept in mind, however, that just in this resonance problem we are concerned with a closed system which according t o the view here exposed is not accessible to observation. In fact the theory of Schrodinger on this view represents, a8 we shall see, a symbolic transcription of the problem of motion of classical mechanics adapted to the requirements of quantum theory, which can only be interpreted by an explicit use of the quantum postulate. This fact has been emphasised especially by Born ((( Zeitschr. f. Phys. )), 40. 167, 1926) who in connexion with his important investigation of collision problems has suggested a simple statistical interpretation of Schrodinger’s wave functions. In this connexion he has, among other things, formulated the wave theoretical analogy t o the adiabatic principle of Ehrenfest. From the point of the quantum postulate Schrodinger’s considerations regarding the radiation problem have been discussed by Klein ((( Zeitschr. f. Phys. )), 41, 407, 1927) in direct connexion with the correspondence principle.

580

A’. Bohr

Just Schrodinger’s formulation of the problem of interaction would seem t o be particnlarly well suited for the illustration of the nature of quantum theory. As we have seen above, already for a free particle the knowledge of its momentum and energy would exclude any exact information ab.out its space-time coordinates. This entail8 the consequence that in case of interacting particles the classical conception of mutual forces and the derived notion of the potential energy of a system are beset with particular difficulties. It is just these difficulties, which in the Schrodinger wave equation are avoided by replacing the classical expression of the Hamiltonian by a suitable differential operator. That the whole scheme of this theory is essentially formal is already shown by the extensive use of imaginary arithmetical quantities, which it has in common with the matrix theory. But quite apart from this circumstance there can be no question of an immecc diate connection with our ordinary conceptions because the geometrical problem represented by the wave equation is associated with the so-called coordinate space, the number of dimensions of which is equal t o the number of degrees of freedom of the system, such as in general to differ from the three dimensions of ordinary space. Further, Schrodinger’s formulation of the interaction problem, just as the formulation offered by matrix theory, involves a neglect of the finite velocity of propagation of the forces claimed by relativity theory. On the whole it would not seem justifiable, in case of the interaction problem, to demand a visualization by means of ordinary space-time pictures. I n fact, all our knowledge concerning the internal properties of atoms is derived from experiment on their radiation or collision reactions, and the interpretation of experimental facts thus ultimately depends on the abstractions of radiation in free space, and free material particles. Hence, our whole space-time view of physical phenomena as well as the definition of energy and momentum depends entirely upon these abstractions. In judging the applications of ((

((

))

))

T h e q u a n t u m poslulatc and the recent development

581

these auxiliary ideas we should only demand inner consistency, in which connection special regard has t o be paid t,o the possibilities of definition and observation. In the characteristic vibrations of Schrodinger’s wave equation we have, as mentioned, an adequate representation of the stationary states of an atom allowing an unambiguous definition of the energy of the system by means of the general quantum relation (1). This entails, however, that in the interpretation of observations a fundamental renunciation regarding the space-time description is unavoidable. In fact, the consistent application of the concept of stationary states excludes, as we shall see, any specification regarding the behaviour of the separate particles in the atom. I n problems, where a description of this behaviour is essential, we are bound to use the general solution of the wave equation which is obtained by superposition of characteristic s o h tions. We meet here with a complementarity of the possibilities of definition quite analogous t o that, which we have considered earlier in connexion with the properties of light and free material particles. Thus, while the definition of energy and momentum of individuals is attached to the idea of a harmonic elementary wave, every space-time feature of the description of phenomena is, as we have seen, based on a consideration of the interferences taking place inside a group of such elementary waves. Also in the present case the agreement between the possibilities of observation and those of definition can be directly shown. Tn fact, the unambiguous utilization of observations on the behaviour of the particles in the atom depends upon the possibility of neglecting, during the process of observation, the interaction between the particles, thus regarding them as free. This requires, however, that the duration of the process is short compared with the natural periods of the atom, which again means that the uncertainty in the knowledge of the energy transferred in the process is large compared t o the energy differences between neighbouring stationary states.

582

A'. Bohr

In judging the possibilities of observation it must on the whole be kept in mind that the wave mechanical solutions can be visualized only in so far as they can be described with the aid of the concept of free particles. Here the difference between classical mechanics and the quantum theoretical treatment of the problem of interaction appears most strikingly. In the former such a restriction is unnecessary, because the particles are here endowed with an immediate reality D, independently of their being free or bound. This situation is particularly important in connection with the consistent utilization of Schrodinger's electric density as a measure of the probability for electrons being present within given space regions of the atom. Remembering the restriction mentioned this interpretation is seen to be a simple consequence of the assumption that the probability of the presence of a free electron is measured in a similar way by the electric density associated with the wavefield as the probability of the presence of a light qnantum by the energy density of radiation. ((

))

((

9 5 . I n the conception of stationary states we are, as mentioned, concerned with a characteristic application of the quantum postulate. By its very nature this conception means a complete renunciation as regards a time description. From the point of view taken here just this renunciation forms the necessary condition for an unambiguous definition of the energy of the atom. Moreover the conception of a stationary state involves, strictly speaking, the exclusion of all interactions with individuals not belonging to the system. The fact that such a closed system is associated with a particular energy value, may be considered as an immediate expression for the claim of causality contained in the theorem of conservation of energy. This circumstance justifies tlhe assumption of the supramechanical stability of the stationary states, according to which the atom, before as well as after an'external influence, always will be found in a stationary state and which forms the

The quantum postulate and the recenl developmeizt

583

basis for the use of the quantum postulate in problems concerning atomic reactions. In a, judgment of the well-known paradoxes, which this assumption entails for the description of collision and radiation reactions, it is essential to consider the limitations of the possibilities of definition of the reacting free individuals, which i R expressed by relation (2). In fact, if the definition of the energy of the reacting individuals is t o be accurate to such a degree as to entitle us t o speak-of conservation of energy during the reaction, it is necessary, according t o this relation, to coordinate to the transition between two stationary states a time interval long compared to the period associated with this process, aud connected with the energy difference between the stationary states according to relation (1).This is particularly to be remembered when considering the passage of swiftly moving particles through an atom. According to the ordinary kinematics the effective duration of such a passage would be very small as compared with the natural periods of t'he atom, and it seemed impossible to reconcile the principle of conservation of energy with the assumption of the stability of stat'ionary states (cf. Zeitschr. f. Phys. D, 34, 142, 1925). In connexion with a discussion of the paradoxes mentioned above Campbell ((( Phil. Mag. n, I, 1106, 1926) suggested the view that the conception of time itself may be essentially statistical in nature. From the view advanced here, according to which the foundation of space-time description is offered by the abstraction of free individuals, a fundamental distinction between time and space, however, would seem to be excluded by the relativity requirement. The singular position of the time in problems concerned with stationary states is, as we have seen, due to the special nature of such problems. The possibility of a consistent application of the conception of stationary states depends on the fact thatl in any observation, say by means of collision or radiation reactions, permitting a distinction between different stationary states we asre entitled t o disregard the previous history of the ((

584

Y. Bohr

atom. The fact, that the symbolical quantum theory methods ascribe a particular phase t'o each stationary state the value of which depends upon the previous history of the atom, would for the first moment seem t o contradict the very idea of stationary states. As soon as we are really concerned with a time problem, however, the consideration of a strictly closed system is excluded. The use of simply harmonic proper vibrations in the interpretation of observations means therefore only a suitable idealization which in a more rigorous discussion must always be replaced by a group of harmonic vibrations, distributed over a finite frequency interval. Now, as already mentioned, it is a general consequence of the superposition principle that it has no sense t o coordinate a phase value to the group as a whole, in the same manner as may be done for each elementary wave constituting t'he group. This inobservability of the phase, well known from the theory of optical instruments, is brought out in a particularly simple manner in a discussion of the Stern-Gerlach experiment, so important for the inrestigation of the properties of single atoms. As pointed out by Heisenberg, atoms with different orientation in the field may only be separated, if the deviation of the beam is larger than the diffraction a t the slit of the de Broglie waves representing the translational motion of the atoms. This condition means as a simple calculation shows, that the product of the time of passage of the atom through the field, and the uncertainty due t o the finite width of the beam of its energy in the field is a t least equal to the quantum of action. This result was considered by Heisenberg as a support of relation ( 2 ) as regards the reciprocal uncertainties of energy and time values. It would seem, however, that here we are not simply dealing with a measiirement of the energy of the atom at a given time. But since the period of the proper vibrations of the atom in the field is connected with the total energy by relation (1)we realize that the above mentioned condition for separability just means the loss of the phase. This

1 he quantum postulate a n d the

recent d e v e l o p m e n l

-~

585

-

circumstance removes also the apparent contradictions, arising in certain problems concerning the coherence of resonance radiation, which h a r e been discussed frequently, and which were also considered by Heisenberg. To considerer an atom as a closed system, as we have done above, means t o neglect the spontaneous emission of radiation which even in the absence of external influences puts a n upper limit to the life time of the stationary states. The fact that this neglect is justified in many applications is connected with the circiimstance, that the coupling between the atom and the radiation field, which is to be expected on classical electrodynamics, is in general very small compared to the coupling between the particles in the atom. It is, in fact, possible in a description of the state of an atom to a considerable extent to neglect the reaction of radiation thus disregarding the unsharpness in the energy v a l u ~ connected with the life time of the stationary states hccording to relation ( 2 ) (cf. Proc. Camh. Phil. Soc. D, 1924 (Slipplement) or Zeitschr. f . Phys. 13, 7 1 7 , 1923). This is the reason why it is possible to draw conclusions concerning the properties of radiation by using classical electrodpamics. The treatment of the radiation problem by the new quantum theoretical methods meant to begin with just a quantitative formulation of this correspondence consideration. I n the more rigorous form of the theory developed by Djrac ((( Proc. Roy. SOC.n, A. vol. 114, pag. 243, 1927) the radiation field itself is included in the closed system under consideration. Thus it becomes possible in a rational way to take account of the atomistic character of radiation demanded by quantum theory. The renunciation regarding space-time pictures characterizing this treatment may be regarded as a striking illustration of the complementary character of the quantum theory. This is particularly to be borne in mind in judging the radical departure from the causal description of nature met with in radiation phenomena, to which we have referred above in connection with the excitation of spectra. I n view of the asymptotic connection of atomic pro((

((

)),

perties with classical electrodynamics, demanded by the correspondence principle the reciprocal exclusion of the concepttion of stationary states and the description of the behavioiir of individual particles in the atom might be regarded as a difficulty. In fact, the connection in question means that in the limit of large quantum numbers where the relative difference between adjacent stationary states vanishes asymptotically, mechanical pictures of electronic motion may be rationally utilize&. I t must be emphasized, however, that this connection cannot be regarded as a gradual transition towards classical theory in the sense that the quantum postulate would lose its significance for high quantum numbers. On the contrary, the conclusions obtained from the correspondence principle with the aid of classical pictiires depend just upon the assumptions that the conception of stationary states and of individual transition processes are maintained even in this limit. This question offers a very instructive application of the new methods. As shown by Schrodinger ((( Naturwiss. I), 14, 664, 1926), it is possible, in the limit mentioned by superposition of proper vibrations to construct wave groups small in comparison to the ((size of the atom, the propagation of which indefinitely approaches the classicd picture of moving material particles, if the quantum numbers are chosen sufficiently large. I n the particularly simple case of a harmonic vibrator be was able t o show that such wave groupe will keep together even for any length of time, and will oscillate to and fro in a manner corresponding to the classical picture of the motion. This circumstance Schriidinger has regarded-as a support of his hope to construct a pure wave theory without refepring t o the quantum postulate. As emphasized by Heisenberg the simplicity of the caRe of the oscillator, however, is exceptional and intimately connected with the harmonic nature of the corresponding classical motion. Nor is there in this example any possibility for an asymptotical approach towards the problem of free particles. In general the wave group wi!l gradually ))

spread over the whole region of the atom. I n fact, a simple calculation shows that the wave group corresponding to a bound electron can only be followed during a number of periods, which is of the order of magnitude of the quantum numbers associated with the proper vibrations. Here again we meet with the contrast between the wave theory superposition principle and the assumption of' the individuality of particles with which we have been concerned already in the case of free particles. At the same time the asymptotibal connection with the classical theory, t o which a distinction between free and bound particles is unknown, offers the possibility of a particularly simple illustration of the above considerations regarding the consistent utilization of the concept of stationary states. As we have seen, the identification of a stationary state by means of collision or radiation reactions implies e, gap in the time description, which ip a t least of the order of magnitude of the periods associated with transitions between stationary states. In the limit of high quantum numbers these periods. however, may be interpreted as periods of revolution. Thus we see that no causal connection can be obtained between observations leading t o the fixation of a stationary state and earlier observations on the behaviour of the separate particles in the atom. Summarising, i t might be said that the concepts of stationary states and individual transition processes within their proper field of application possess just as much or as little reality as the very idea of individual particles. I n both cases we are concerned with a demand of causality complementary to the space-time description, the adequate application of which is limited only by the restricted possibilities of definition and observation. I t seems, in fact, possible, when due regard is taken of the complementary feature required by the quantum postulate, with the aid of the symbolic methods to build up a consistent theory of atomic phenomena, which may be considered as a rational generalization of the ordinary space((

))

588

Y. Bohr

time description. This view does not mean, however, that the classical electron theory can be regarded simply as the limiting case of a vanishing quantum of action. Indeed, the connection of the latter theory with experience is based on assumptions which can scarcely be separated from the group of problems of the quantum theory. A hint in this direction was already given by the well known difficulties met with in the attempts of accounting for the individuality of ultimate electrical particles on general mechanical and electrodynamical principles. I n this respect also the general field theory formulated in relativity theory has not fulfilled the expectation it had given rise to. Notwithstanding this situation the classical electron theory has quite recently been the guide for a further development in connection with the idea, first put forward by Compton, that the ultimate electrical particles besides their mass and charge are endowed with a magnetic moment. This assumption, introduced with striking success in the discussion of the anomalous Zeeman effect by Goudsmit and Uhlenbeck, has essentially advanced the correspondence interpretation of the spectral laws and the periodic system. One might say, indeed, that the hypothesis of the magnetic electron together with the resonance problem elucidated by Heisenberg, which occurs in the quantum theoretical description of the behaviour of atoms with several electrons, have brought this interpretation to a certain completion. Just the intimate connection, however, obtained between the exclusion principle of Pauli, so important for the problem of atomic constit,ution, and the correspondence treatment excludes the hope of elucidating by means of the methods hitherto used the difference expressed through this principle in the behaviour of ultimate electrical particles and of the individuals known as light quanta. Such an elucidation would seem to be possible only by means of a rational quantum theoretical transcription of the general field theory, in which the ultimate electrical particles have found their natural position.

Discussioiie su lla coinuiiicazione Bohr

BORN: Herr Prof. Bohr hat die Anschauungen, die wir

uns iiber die Grundbegriffe der Quantentheorie gebildet haben, in so treffender Weise dargestellt, dass mir nur iibrig bleibt,

einige Bemerkungen hinzuzuf ugen. Zuerst mochte ich betonen, dass die Quantentheorie heute ein einheitliches Denkgebaude darstellt, in dem die urspriinglichen Formalismen vereinigt sind, die auf Heisenberg’s Ideen aufgebaute Matrizentheorie und die von de Broglie und Schrodinger entwickelte Schwingungstheorie. Sodann scheint es mir wichtig, hervorzuheben, dass die neue Quantentheorie den Determinismus, der die ganze Naturforschung bisher beherrscht hat, aufgibt. Aber die Aufgabe der Kausalitiit im strengsten Sinne ist nur ein scheinbarer Verzicht. Denn die mechanistische Naturauffassung, wie sie bisher in Geltung war, musste zur Vorausberechnung zukiinftiger Ereignisse die Annahme machen, dass der Zustand der Welt in einem. Augenblick vollstiindig in allen Einzelheiten bekannt sei. Aber diese Annahme ist eine Illusion. Die eigentliche Erkenntnis der Quan tentheorie besteht darin, dass die Naturgesetze selbst die vollstandige Fixierung des Zustandes eines abgeschlossenen Systems verbieten. J e genauer eine Koordinate gemessen wird, um so ungenauer ist der zugehorige Impuls bestimmt. Das liegt an der Wellennatur der Materie und wird in der von Herrn Bohr angegebenen Ungenauigkeitsrelation von Heisenberg formuliert . Wird z. 33, ein breites Biindel von parallel fliegenden Lichtquanten durch eine enge Blende geschickt, so breitet es sich hinter dieser umsomehr aus, je enger die Oeffnung ist, d. h. die

zur Blendenebene parallelen Impulskomponenten werden unbestimmt. Der Grund hierfiir ist der Umstand, dass das Licht nicht nur korpuskulare, sondern auch wellemrtige Eigenschaften hat j letztere bewirken eine Beugung an der Blende, ein fBcherartiges Ausbreiten der Lichterregung : Man kann diese Zusammenhang von Korpuskeln und Wellen in vielen FB’len so beschreiben, dass man sagt : Jeder Vorgang besteht aus Wellen, deren IntensitLt die Wahrscheinlichkeit f iir das Aufreten von individuellen Korpuskeln bestimmt. Eine solche Buffassung hat sich bei den Stosserscheinungen bewLhrt, iiber die ich eigentlich hier berichten wollte. Ich mochte jetzt nur einige Resultate mitteilen, die bei der quantitativen Berechnung gewonnen worden sind. Herr Wenzel und Herr Oppenheimer haben gezeigt, dass beim Stoss einer Alfa-Partikel gegen einen Kern das beriihmte Gesetz von Rutherford herauskommt, wenigstens fur hohe Geschwindigkeiten. Ob f iir kleinere Geschindigkeiten merkliche Abweichungen von der klassischen Formel zu erwarten sind, ist noch nicht ganz sicher gestellt. Die wellenmechanische Stosstheorie liefert, wie ich zeigen konnte, die allgemeinen Gesetze, die zuerst von Franck und Hertz beobachtet worden sind. Quantitativ ist der Stoss eines Teilchens gegen ein Wasserstoffatom durchgerechnet worden. Neuerdings hat mein Schuler Herr Elsasser nicht nur die diskreten Anregungsstufen, sondern auch das kontinuierliche Spektrum, d. h. die Stossjonisation in Betracht gezogen. Die Resultate sind behiedigend, wenn man beriicksichtigt, dass die Rechnungen nur angenLhert f iir posse Geschwindigkeiten gefiihrt werden konnten. Eine besonders hiibsche Anwendung der Stosstheorie hat Herr Dirac gemacht, indem er die Dispersion des Lichts als Stoss von Lichtquanten gegen ein Atom auffasste, wobei es vorkommen kann, dass das Quant vom Atom absorbiert wird. E r e r u l t so eine Dispersionsformel mit DBmpfungsglied, also eine Theorie der Breite von Spektrallinien und der Lebensdauer angeregter ZustLnde. Zum Schluss mochte ich noch sagen, dass die versohiedenen mathematischen Formalismen jetzt einheitlich als SonderfBlle einer allgemeinen Operatortheorie angesehen werden konnen. Anschliessend an einen Ansatz von Herrn Wiener und mir haben die Herren Jordan und Dirac diesen Kalkul entwickelt und neuerdings hat ihn Herr v. Neumann in eine mathematisch strenge Form gebracht. Der Gedanke ist der: In ((

))

der alten Theorie hat die Angabe des Wertes einer Koordinate q einen Sinn; in der neuen dagegen lautet die entsprechende Angabe: Es besteht eine gewisse Wahrscheinlichkeit dafiir, dass eine Koordinate q einen bestimmten Wert hat. Jeder Koordinate q entspricht also ein Verteilungsgesetz, eine Funktion. Alle Probleme der Quantenmechanik lassen sich darauf zuruckfuhren, das Verteilungsgesetz einer Grosse Q aus dem einer andern q zu berechnen. Aber wie beim Licht nicht die IntensitLten, welche die Haufigkeit der Lichtquanten bestimmen, einfachen Gesetzen folgen, sondern die Amplituden, so ist es auch hier; man muss Wahrscheinlichkeitsamplituden einf uhren, deren Quadrate die Wahrscheinlichkeiten und damit das beobachtbare Verteilungsgesetz bestimme. Diese Amplituden lassen sich dann nach relativ einfechen Regeln berechnen. Es lauft dies heraus auf die Bestimmung der Hauptachsen einer Flache zweiten Grades im Raume von unendlich vielen Dimensionen. Es scheint, dass diese Theorie auf alle Fragen der Quantenmechanik die richtige Antwort gibt. ((

))

Mr. KRAMERS: I shall not be able to add anything fundamental to Professor Bohr’s exposition of the physical principles underlying the news quantum mechanics. I should, however, like to draw your attention to some examples which illustrate, how the difficulties and paradoxes, which one earlier met in the quantum theory, are come over by the modern views. I am thinking especially of the principle of conservation of energy and momentum, which seemed to contradict the wave-theory of light. On account of this difficulty, Bohr, Kramers & Slater, a few years ago, proposed the view that the conservation laws must be considered as statistical laws, which only are valid if a great number of elementary processes are considered. AS you all know, this view had to be abandoned when Bothe & Geiger, and Compton & Simon published their fundamental experimental researches on the Compton-effect. The difficulty, that the results of these experiments were a t variance with the wave theory of light, disappears definitely, if the Broglie wave theory of matter is taken into account; a t the same time, however, certain fundamental features of the Bohr-Kramers-Slater theory remain valid, viz. as regards the statistical character of the occurrence of the elementary processes, which plays an important part also in the modern theory. ((

))

592

H. A . Kramers

The main point in the modern explanation of the Comptoneffect is, that the conservation laws in these experiments only have a meaning to a certain approximation, due to the fact that on the wave mechanics, the occurrence of a Compton-effect in a definite region in space, and within a definite region of time can only be described by means of light waves and matter-waves, which are not exactly monochromatic waves with a definite direction .of propagation in space, but which form beams limited in time and space. Due to this cirgumstance the amounts of energy and momentum and their changes involved in t h e Compton effect are not exactly defined. We must conclude, following the beautiful arguments of Heisenberg, that the validity of the conservation-laws is limited by an uncertainty which Ls fundamentally involved in the measurements which one might try to apply in order to test these laws as exactly as possible. Similar considerations hold as regards the famous question of the one-to-one correspondence between the emission processep in a light source, and the absoiption processes to which the light of the source may give rise. Also here, the assumption of an exact validity of the conservation-laws suggested by the light-quantum paint in view, Reemed to contradict the principles of the wave theory. Again wave mechanics show the way out of the difficulty, since they te!l us what to expect if measurements were made in order to test the mentioned one-ko-one correspondence. If these measurements show that a transition takes place within a certain time interval, an uncertainty is necessarily involved in the frequency of the corresponding radiation. An exact correspondence between the emission of a light quantum in the source and the absorption of the same 1) light-quantum i n the absorbing matter is therefore out of question, in agreement with the Bob-Kramers-Slater theory. On the other hand, an approximate correspondence, the limits of which are directly derived from Heisenberg’s considerations, can under suitable conditions always be found; as an example of such a correspondence we may consider the Geiger-Bothe experiment; on the theory the times a t which the scattered light quantum and the recoil electrons are observed, are not exactly defined, but this uncertainty is much smaller than the accuracy with which the time is measured in the experiment. ((

HEISENBERG : Ich mochte zuerst uber eine Frage sprechen, auf die Herr Prof. Bohr selbst in seinem Vortrag hingewiesen hat. Die physikalische Bedeutung der Ungenauigkeitsrelation A p Aq h und ihr Zusamnienhang mit den allgemeinen von Bohr hervorgehobenen Gesichtspunkten ist ja erst durch die Untersuchungen von Bohr ganz klargestellt worden. Es sei deshalb hier nur auf eine Analogie hingewiesen, welche diese Relation mit einer Beziehung in den Grundlagen der Relativitatstheorie hat: I n der Relativithtstheorie ist von fundamentaler Wichtigkeit die Definition des Beg-riffes der Gleichzeitigkeit. I n jede lIessung, die zur Definition dieses Reg-riffes dient, geht die Lichtgeschwindigkeit notwendig ein. Gabe es Experimente, die eine genauere Definition ermoglichten, z. B. Signale, die sich mit t‘eberlichtgeschwindigkeit fortpflanzen, so ware die Relativitatstheorie unmoglich. Eine solche genauere Definition ist aber unmoglich und dadurch ist Platz geschaffen f u r das Postulht der k o ~ s t a n t e nLichtgeschwindigkeit. Aehnlich ist es mit der Relation A p Aq h. Gabe es eine genauere Definition von Ort und Impuls, so ware die Quantenmechanik unmoglich. Erst die allgemeine Gultigkeit der Relation P p A q 2 h schafft Raum f u r die Interferenzerscheinungen, die f iir die Quantenmechanik typisch sind. Auf eine weitere Analogie mochte ich Sie aufmerksam machen. I n der Relativitiitstheorie muss zu jeder Beobachtung das Koordinatensystem angegeben werden, von dem aus sie gemacht wird. Erst die Wahl des Koordinatensystems teilt die Welt ein in Rauni und Zeit. I n der Quantenmechanik spielt, wie Herr Prof. Bohr dargelegt hat, die Beobachtung eine ganz merkwdrdige Rolle. Na n konnte die ganze Welt als e i n mechanisches System behandeln, aber dann bleibt nur noch ein mathematisches Problem ubrig, der Zugang zu den Beobachtungen ist dann versperrt. 1-m zur Beobachtung zu gelangen, muss man also irgendwo ein Teilsystem aus der Welt ausschneiden und uber dieses Teilsystem eben Aussagen oder Beobachtungen 11 machen. Dadurch zerstort man dort den feinen Zusammenhaag der Erscheinungen und a n der Stelle, wo wir den Schnitt zwischen dem zu beobachtenden System einerseits, dem Beobachter und seinen Apparaten andererseits machen , mussen wir Schwierigkeiten fur unsere Anschauung erwarten. Aus der Gleichung A p Aq h wissen wir, dass wir nicht gleichzeitig Ort und Impuls eines Teilchens beliebig genau beobachten konnen. Jede Beobachtung teilt

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Resoconto del Congress0 dei Fisici

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Vol. I1 - 3 8

594

W . H e i s e j i b e r g , E. Fermi

in gewisser Weise die Welt ein in bekannte und unbekannte oder besser: mehr oder weniger genau bekannte Grossen. mochte ich ein Beispiel geben. Denken Sie a n die kriiftefreie Bewegung eines Elektrons. Aus dem Vortrag von Prof. Bohr haben Sie gehort, wie mit einem Mikroskop der Platz des Elektrons etwa rnit der Genauigkeit Ap gemessen werden kann, wobei dann die Geschwindigkeit mit der Genauigkeit Ap bekannt ist (Ap A q h ) . Wenn man a n der Born’schen statistischen Deutung der de Broglie-Wellen festhalt, so muss man also das Resultat der Mikroskopbeobachtung dahin interpretieren, dass die Beobachtung ein Wellen-oder Wahrscheinlichkeitspaket konstituiert, das von Wellen des Frequenzbereiches,

-

welcher A p entspricht ( p

=

i),

durch Superposition in dem von

Bohr hervorgehobenen Sinne zusammengesetzt ist. Dieses Wellenpaket bewegt sich nicht nur geradlinig im Raum fort, sondern es breitet sich auch im Lauf der Zeit aus. Fur eine neue Reobachtung gibt das Wellenpaket die Wahrscheinlichkeit, das Elektron a n einer bestimmten Stelle zu finden. Die neue Beobachtung selbst aber reduziert wieder das Paket auf die urspriingliche Grosse Aq, sie trifft eine Auswahl aus einer Fulle von Moglichkeiten und reduziert dadurch f iir die Zukunft die Moglichkeiten. Diese unstetige Aenderung des Wellenbildes bei einer Beobachtusg scheint mir ein wesentlicher Zug der Quantenmechanik. Man muss eben Ernst machen mit der Vorstellung der Wahrscheinlichkeitswellen 1). Die Wellen haben niclit die unmittelbare RealitBt, die wir friiher den Wellen der Maxwell’schen Theorie zugeschrieben haben. Mann muss sie als Wahrscheinlichkeitswellen deuten und daher plotzliche Aenderungen bei jeder neuen Beobachtung erwarten. ((

FERMI: La nuova teoria dei quanti, di cui il prof. Bohr ci ha illustrate le basi concettuali, getta nuova luce sopra il problema dell’ irradiazione. & mio desiderio richiamare l’attenzione sopra alcuni aspetti di questo problema. La questione della reazione della radiazione meccanica quantistica pub esser trat tata con il seguente metodo che permette di ottenere facilmente dei risultati concreti, bench& esso sia in buona parte ipotetico. Consideriamo per semplicit& un sistema nel quale siano ad un

Ijiscicssione sulla conzziiiicu;ici,ie llolij

595

certo istante eccitati due stati quantici nel senso di Schrodinger; rlecondo 1’ interpretazione statistica cib vorrh dire che se si facesse una esperienza per decidere in quale di questi due stati l’atomo si trova effettivamente, esisterebbe un rapport0 definito per le probabilith di trovarlo nell’uno o nell’altro di essi. Si pub allora facilmente calcolare il momento elettrico del sistema il quale viene ad essere una funzione del tempo e dh, secondo la teoria classica dell’ irradiazione, origine a una irradiazione con frequenza determinata secondo la relazione di Bohr dalle energie dei due stati, e a una reazione di radiazione che nella teoria classica i! rappresentata da una forza avente in prima approssimazione la grandezza

Questa forza pub formalmente rappesentarsi come derivante d a un potenziale variabile col tempo

Si pub fare I’ ipotesi che per tener conto della reazione della radiazione basti aggiungere a1 potenziale del sistema il potenziale (1).Effettivamente si trova clie 1’ ipotesi conduce a sisultati plausibili; si trova infatti che, come conseguenza di tale aggiunta fil potenziale del sistema, esso passa dallo stato di maggiore energia a quello di energia minore in un tempo dell’ordine di grandezza delle vite medie atomiche. Si pub anche facilmente seguire il mod0 in cui avviene tale passaggio tra i due stati quantici e calcolare quindi p. e. la larghezza e la forma della riga spettrale che viene emessa nel salto quantioo (l). Circa i nuovi metodi statistici nella meccanica dei qusnti, vorrei fare qualche considerazione. f3 noto che la teoria dei quanti permette di determinare in mod0 del tutto naturale le dimensioni delle celle in cui B neceseario dividere lo spazio delle fasi secondo la statistica di Boltzmann e Maxwell; se perb in base a tale determinazione si vuole costruire la statistica di un gas perfetto ci si accorge che esea non B sufficiente all0 scopo, (4)

E. FERMI,(I Rend. Liiicei I ) , 5, 705, 1927.

596

1;. F e i m i , 11”. Heisonberg

poioh6 quando le dimensioni del recipiente in cui il gas 15 racchiuso diventano grandi gli stati quantici diventano piu fitti per modo che alla fine viene a cessare ogni effetto della loro discontinuit&. Per superare queste difficolth sono stati recentemente fatti due tentativi, uno per parte di Einstein e uno da me ; nel caso di Einstein si ammette tra le molecole del gas una dipendenza statistica del tip0 di quella proposta da Bose per il caso dei quanti di luce; nel caso mio si B invece applicato a tutto il gas, considerato come un unico sistema costituito da tutte le molecole (che sono eguali tra di loro) il principio di esclusione di Pauli. I rapporti tra queste due statistiche sono stati chiariti per mezzo della nuova meccanica dai lavori di Heisenberg, Dirac e Winter; essi han dimostrato che se si ha un sistema che contiene delle particelle identiche tra di loro i suoi termini si dividono in gruppi per mod0 che non i! in alcun mod0 possibile ottenere dei passaggi tra due termini che appartengano a gruppi differenti. Uno di questi gruppi soddisfa alla statistica di Bose-Einstein e l’altro a1 principio di esclusione di Pauli e quindi alla statistica proposta dall’autore. L’esperienza ha fino ad ora dimostrato che gli elettroni di un atomo e anche i corpuscoli positivi soddisfano sempre a1 principio di esclusione. Applicando a1 gas di elettroni nell’ interno di un metallo tale statistica, Pauli ha potuto spiegare il fatto che il paramagnetismo dei metalli alcalini solidi 6 considerevolmente minore di quanto corrisponderebbe a1 valore del momento magnetic0 dell’elettrone, e il Prof. Sommerfeld ci ha mostrato come si possa in base ad essa render conto anche di molte altre proprietA della conduzione metallica. Si pub anche cercare in base alle stesse ipotesi di costruire una teoria dei metalli capace di render conto delle forze che tengono insieme la compagine del metallo. Basta per cib c o d derare gli ioni positivi del metallo disposti ai vertici del reticolato cristallino del metallo e calcolare poi la distribuzione degli elettroni di valenza sotto l’azione delle forze elettrostatiche con metodo simile a quello applicato da Debye e Huckel nella loro teorh degli elettroliti forti, ed applicando naturalmente la nuova statistica a1 posto di quella classica. I calcoli numerici necessari per questa teoria sono perb assai lunghi e non sono ancora completi.

HEISENBERG : Es hat sich in der letzten Zeit herausgestellt, dass auch f u r die positiven Elektronen oder Protonen die Fermi-Statistik gilt. Der Gang dor Untersuchung war etwa lolgender : Die Gleichheit der negativen Elektronen und ihIe statistischen Eigenschaften aussern sich z.B. im Heliumspektrum in der Einteilung aller Terme in ein N Ortho- und Parasystem D. Ueberghnge vom Ortho- zum Parasystem sind sehr selten. Aehnlich lLsst sich das Bandenspektrum z.B. des Wasserstoffmolekuls einteilen, der Gleichheit der beiden Kerne entsprechend, in zwei Systeme, die man auch Ortho- und Paraaystem nennen kann. Von einer Rotationsbande gehort immer abwechselnd eine Linie zum Ortho- eine zum Paraspektrum; daraus folgt ein charakteristischer Intensitatswechsel in den Rotationsbanden, der von Hund und mir naher studiert wurde. Hund rechnete die spezifische Wiirme des Wasserstoffmolekuls aus, wobei er in gewohnlicher Weise thermisches Gleichgewicht zwischen allen moglichen Zustanden des Molekuls annahm. Bei Annahme der Fermi-statistik ergab sich aber so keine Uebereinstimmung mit der Erfahrung. Die Schwierigkeit wurde von Denniaon aufgeklart. Die Uebergange zwischen Ortho- und Parasystem sind ausserordentlich selten, die Rechnung zeigt, dass sie bei tiefen Temperaturen etwa alle paar Monate vorkonimen. Also hkngt die apezifische Warme des Wasserstoffgases wesentlich davon ab, ob die Messung in kurzen oder langen Zeiten ausgefuhrt wird.. F u r kurze Zeiten verhtilt sich Wasserstoff wie ein Gemisch von zwei Gasen, einem Ortho- und einem Paragas. Jetzt ergibt die Annahme der Fermi-Statistik Uebereinstimmung mit der Erfahrung. Das Resultat der Untersuchung war also: Die Protonen gehorchen der Fermi-Statistik, ferner besitzen sie, 1 h wie die negativen Elektronen, ein Drehmoment der Grosse -. 2 2t

Das Studium der Bandenspektren durch Hund gab iibrigens noch einige interessante Resultate hinsichtlich der komplizierteren Atomkerne: Heliumkerri und Sauerstoffkern haben k e i n Drehmoment, vide andere Kerne z.B. Stickstoff haben ein Drehmoment. Dies stimmt gut zu den neuesten Beobachtungen von Aston uber die Bindungsenergie der Kerne, nach denen auch Helium, Kohle und Sauerstoffkern als besonders stabil anzusehen sind.

598

1v. Paz!zi

PAULI: Es muss wohl als ein Schonheitsfehler der bisher erreichten Fassung der Quantenmechanik angesehen werden, dass das Aequivalenz-Verbot D eine zusiitzliche Annahme notwendig rnacht, welche besagt, dass von den verschiedenen Losungen der quastenmechanischen Gleichungen nur eine, die schiefsymetrische, in der Natur realisiert ist. Vielleicht gibt die von Dirac herriihrende Fassung der Quantenelektrodynamik einen Fingerzeig, in welcher Richtung die Reantwortung dieses noch ungelosten Problems gesucht werden muss. I n der Dirac’schen Theorie wird nLmlich die Einf iihrung der Orts-und Zeitkoordinaten der Lichtquapten in einem weniger engem Anschluss an die Nethoden der Punktmechanik vollzogen als die Einf uhrung der Koordinaten der materiellen Teilchen. Jedenfalls scheint das Problem einer niiheren Begrundung des Aequivalenz-Verbotes mit dem der Struktur der Elektronen und Protonen zusammenzuhangen. ((

IX. THE QUANTUM POSTULATE AND THE RECENT DEVELOPMENT OF ATOMIC THEORY [4] Nature (Suppl.) 121 (1928) 580-590 Elaboration of the Address given in Como on 16 September 1927 at the International Congress of Physicists on the Occasion of the Centenary of the Death of Alessandro Volta 2nd Version

See Introduction to Part I, sects. 4-6, as well as the editorial note to document VIII.

Supplement to ‘‘ Nature,” April 14, 1928

580

The Quantum Postulate and the Recent Development of Atomic Theory.’ By Prof. N. BOHR,For. Mem. R.S. connexion with the discussion of the physical a t what point the concept of observation ininterpretation of the quantum theoretical volving the quantum postulate with its inherent methods developed during recent years, I should ‘ irrationality ’ is brought in. This situation has far-reaching consequences. like to make the following general remarks regardIn one hand, the definition of the state of ing the principles underlying the description of physical system, as ordinarily understood, atomic phenomena, which I hope may help t o harmonise the different views, apparently so diver- shims the elimination of all external disturbances. 3ut in that case, according to the quantum gent, concerning this subject. )ostulate, any observation will be impossible, 1. QUANTUM POSTULATE AND CAUSALITY. knd, above all, the concepts of space and time The quantum theory is characterised by the ose their immediate sense. On the other hand, acknowledgment of a fundamental limitation in f in order to make observation possible we perthe classical physical ideas when applied to atomic nit certain interactions with suitable agencies )f measurement, not belonging to the system, phenomena. The situation thus created is of a peculiar nature, since our interpretation of the tn unambiguous definition of the state of the experimental material rests essentially upon the iystem is naturally no longer possible, and classical concepts. Notwithstanding the diffi- bere can be no question of causality in the culties which hence are involved in the formulation rdinary sense of the word. The very nature of the of the quantum theory, it seems, as we shall see, pantum theory thus forces us to regard the spacethat its essence may be expressed in the so-called ime co-ordination and the claim of causality, the quantum postulate, which attributes to any atomic inion of which characterises the classical theories, process a n essential discontinuity, or rather in- LS complementary but exclusive features of the dividuality, completely foreign t o the classical iescription, symbolising the idealisation of observa;ion and definition respectively. Just as the relatheories and symbolised by Planck’s quantum of action. ivity theory has taught us that the convenience This postulate implies a renunciation as regards )f distinguishing sharply between space and time the causal space-time co-ordination of atomic pro- *ests solely on the smallness of the velocities cesses. Indeed, our usual description of physical lrdinarily met with compared to the velocity of phenomena is bmed entirely on the idea that the ight, we learn from the quantum theory that the phenomena concerned may be observed without tppropriateness of our usual causal space-time disturbing them appreciably. This appears, for lescription depends entirely upon the small value example, clearly in the theory of relativity, which )f the quantum of action as compared to the has been so fruitful for the elucidation of the tctions involved in ordinary sense perceptions. classical theories. As emphasised by Einstein, [ndeed, in the description of atomic phenoevery observation or measurement ultimately rests mena, the quantum postulate presents us with on the coincidence of two independent events a t ihe taak of developing a ‘ complementarity ’ theory the same space-time point. Just these coincid- the consistency of which can be judged only by ences will not be affected by any differences which weighing the possibilities of definition and obserthe space-time co-ordination of different observers vation. This view is already clearly brought out by the otherwise may exhibit. Now the quantum postulate implies that any observation of atomic much-discussed question of the nature of light and phenomena will involve a n interaction with the the ultimate constituents of matter. As regards agency of observation not to be neglected. Accord- light, its propagation in space and time is adeingly, a n independent reality in the ordinary quately expressed by the electromagnetic theory. physical sense can neither be ascribed to the Especially the interference phenomena in vacw phenomena nor to the agencies of observation. snd the optical properties of material media are After all, the concept of observation is in so far completely governed by the wave theory superarbitrary as it depends upon which objects are position principle. Nevertheless, the conservation included in the system to be observed. Ultimately of energy and momentum during the interaction every observation can of course be reduced to our between radiation and matter, as evident in the sense perceptions. The circumstance, however, photoelectric and Compton effect, finds its adequate that in interpreting observations use has always expression just in the light quantum idea put to be made of theoretical notions, entails that for forward by Einstein. As is well known, the every particular case it is a question of convenience doubts regarding the validity of the superposition * The content of this paper is essentially the same as th8t of a lecture principle on one hand and of the conservation laws on the other, which were suggested by this apparent on the present state of the uantum theory delivered on Sept. 16 1927, at the Volta celebration% Como. For a summary of the theor; contradiction, have been definitely disproved just previous to the development ?! the new methods the reader :1 referred to a lecture of the author Atomic Theory and Mechanics through direct experiments. This situation would published in this periodical (NA$UxE I18 SOD. 1925). The ra id seem clearly to indicate the impossibility of a development which has taken place sdce hks giv& rise to a consi&r. able number of publications. The present paper is conflned to a causal space-time description of the light phenofew references to recent articles which have a special bearing on the subject now under discmdon. mena. On one hand, in attempting to trace

IN

I

Supplement to

‘‘ Nature,” April 14, 1928

the laws of the time-spatial propagation of light according to the quantum postulate, we are confined to statistical considerations. On the other hand, the fulfilment of the claim of causality for the individual light processes, characterised by the quantum of action, entails a renunciation as regards the space-time description. Of course, there can be no question of a quite independent application of the ideas of space and time and of causality. The two views of the nature of light are rather t o be considered as different attempts a t an interpretation of experimental evidence in which the limitation of the classical concepts is expressed in complementary ways. The problem of the nature of the constituents of matter presents us with an analogous situation. The individuality of the elementary electrical corpuscles is forced upon us by general evidence. Nevertheless, recent experience, above all the discovery of the selective reflection of electrons from metal crystals, requires the use of the wave theory superposition principle in accordance with the original ideas of L. de Broglie. Just as in the case of light, we have consequently in the question of the nature of matter, so far as we adhere to classical concepts, t o face an inevitable dilemma, which has to be regarded as the very expression of experimental evidence. In’fact, here again we are not dealing with contradictory but with complementary pictures of the phenomena, which only together offer a natural generalisation of the classical mode of description. I n the discussion of these questions, it must be kept in mind that, according to the view taken above, radiation in free space as well as isolated material particles are abstractions, their properties on the quantum theory being definable and observable only through their interaction with other systems. Nevertheless, these abstractions are, as we shall see, indispensable for a description of experience in connexion with our ordinary space-time view. The difficulties with which a causal space-time description is confronted in the quantum theory, and which have been the subject of repeated discussions, are now placed into the foreground by the recent development of the symbolic methods. An important contribution to the problem of a consistent application of these methods has been made lately by Heisenberg (Zeitschr. f. Phys., 43, 172 ; 1927). I n particular, he has stressed the peculiar reciprocal uncertainty which affects all measurements of atomic quantities. Before we enter upon his results it will be advantageous t o show how the complementary nature of the description appearing in this uncertainty is unavoidable already in an analysis of the most elementary concepts employed in interpreting experience. 2. QUANTUMOF ACTIONAND KINEMATICS. The fundamental contrast between the quantum of action and the classical concepts is immediately apparent from the simple formula which form the common foundation of the theory of light quanta and of the wave theory of material particles. If

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Planck’s constant be denoted by h, as is well known, Er=Ih=h, . . * (1) where E and I are energy and momentum respectively, r and h the corresponding period of vibration and wave-length. I n these formula the two notions of light and also of matter enter in sharp contrast. While energy and momentum are associated with the concept of particles, and hence may be characterised according to the classical point of view by definite space-time co-ordinates, the period of vibration and wave-length refer to a plane harmonic wave train of unlimited extent in space and time. Only with the aid of the superposition principle does it become possible to attain a connexion with the ordinary mode of description. Indeed, a limitation of the extent of the wavefields in space and time can always be regarded as resulting from the interference of a group of elementary harmonic waves. As shown by de Broglie (ThBse, Paris, 1924), the translational velocity of the individuals associated with the waves can be represented by just the so-called group-velocity. Let us denote a plane elementary wave by A cos 2 ~ (- X~U ,t - y ~ -y ZU, + S ) , where A and 6 are constants determining respectively the amplitude and the phase. The quantity v = l / r is the frequency, ox,u,,,u, the wave numbers in the direction of the co-ordinate axes, which may be regarded as vector components of the wave number a = l / X in the direction of propagation. While the wave or phase velocity is given by v/ir, the group-velocity is defined by dvldir. Now according to the relativity theory we have for a particle with the velocity v :

I

V

= -E C2

and vdI = dE,

where c denotes the velocity of light. Hence by equation (1) the phase velocity is c2/vand the groupvelocity v. The circumstance that the former is in general greater than the velocity of light emphasises the symbolic character of these considerztions. At the same time, the possibility of identifying the velocity of the particle with the group-velocity indicates the field of application of space-time pictures in the quantum theory. Here the complementary character of the description appears, since the use of wave-groups is necessarily accompanied by a lack of sharpness in the definition of period and wave-length, and hence also in the definition of the corresponding energy and momentum as given by relation (1). Rigorously speaking, a limited wave-field can only be obtained by the superposition of a manifold of elementary waves corresponding to all values of v and us, uu,uZ. But the order of magnitude of the mean difference between these values for two elementary waves in the group is given in the most favourable case by the condition AtAv = AXAU, = AyAuY = AzAu, = 1, where At, Ax, Ay, Az denote the extension of the wave-field in time and in the directions of space corresponding t o the co-ordinate axes. These

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relations-well known from the theory of optical instruments, especially from ,Rayleigh’s investigation of the resolving power of spectral apparatus -express the condition that the wave-trains extinguish each other by interference a t the space-time boundary of the wave-field. They may be regarded also as signifying that the group as a whole has no phase in the same sense as the elementary waves. From equation (1) we find thus : AtAE = AXAI, = AyAI, = A z A I ~= h , (2) as determining the highest possible accuracy in the definition of the energy and momentum of the individuals associated with the wave-field. I n general, the conditions for attributing an energy and a momentum value t o a wave-field by means of formula (1) are much less favourable. Even if the composition of the wave-group corresponds in the beginning to the relations (2)) it will in the course of time be subject to such changes that it becomes less and Iess suitable for representing an individual. It is this very circumstance which gives rise t o the paradoxical character of the problem of the nature of light and of material particles. The limitation in the classical concepts expressed through relation (2) is, besides, closely connected with the limited va,lidity of classical mechanics, which in the wave theory of matter corresponds to the geometrical optics, in which the propagation of waves is depicted through ‘rays.’ Only in this limit can energy and momentum be unambiguously defined on the basis of space-time pictures. For a general definition of these concepts we are confined to the conservation laws, the rational formulation of which has been a fundamental problem for the symbolical methods to be mentioned below. I n the language of the relativity theory, the content of the relations (2) may be summarised in the statement that according to the quantum theory a general reciprocal relation exists between the maximum sharpness of definition of the spacetime and energy-momentum vectors associated with the individuals. This circumstance may be regarded as a siinple symbolical expression for the complementary nature of the space-time description and the claims of causality. At the same time, however, the general character of this relation makes it possible to a certain extent to reconcile the conservation laws with the space-time coordination of observations, the idea of a coincidence of well-dehed eventa in a space-time point being replaced by that of unsharply defined individuals within finite space-time regions. This circumstance permits us to avoid the well-known paradoxes which are encountered in attempting to describe the scattering of radiation by free electrical particles as well as the collision of two such particles. According to the classical concepts, the description of the scattering requires a finite extent of the radiation in space and time, while in the change of the motion of the electron demanded by the quantum postulate one seemingly is dealing with an instantaneous effect taking place a t a definite

point in space. Just as in the case of radiation, however, it is impossible to define momentum and energy for a n electron without considering a finite space-time region. Furthermore, an application of the conservation laws to the process imp!ies that the accuracy of definition of the .energy momentum vector is the same for the radiation and the electron. I n consequence, according to relation (2))the associated space-time regions can be given the same size for both individuals in interaction. A similar remark applies to the collision between two material particles, although the significance of the quantum postulate for this phenomenon was disregarded before the necessity of the wave concept was realised. Here this postulate does indeed represent the idea of the individuality of the particles which, transcending the space-time description, meets the claim of causality. While the physical content of the light quantum idea is wholly connected with the conservation theorems for energy and momentum, in the case of the electrical part,icles tho electric charge has to be taken into account in this connexion. It is scarcely necessary t o mention that for a more detailed description of the interaction between individuals we cannot restrict ourselves to the facts expressed by formula (1) and (2), but must resort t o a procedure which allows us to take into account the coupling of the individuals, characterising the interaction in question, where just the importance of the electric charge appears. As we shall see, such a procedure necessitates a further departure from visualisation in the usual sense.

3. MEASUREMENTS IN THE QUANTUM THEORY. I n his investigations already mentioned on the consistency of the quantum theoretica1 methods, Heisenberg has given the relation (2) as an expression for the maximum precision with which the space-time co-ordinates and momentumenergy components of a particle can be measured simultaneously. His view was based on the following consideration: On one hand, the coordinates of a particle can be measured with any desired degree of accuracy by using, for example, an optical instrument, provided radiation of sufficiently short wave-length is used for illumination. According t o the quantum theory, however, the scattering of radiation from the object is always connected with a finite change in momentum, which is the larger the smaller the wave-length of the radiation used. The momentum of a particle, on the other hand, can be determined with any desired degree of accuracy by measuring, for example, the Doppler effect of the scattered radiation, provided the wave-length of the radiation is so large that the effect of recoil can be neglected, but then the determination of the space co-ordinates of the particle becomes correspondingly less accurate. The essence of this consideration is the inevitability of the quantum postulate in the estimation of the possibilities of measurement. A closer investigation of the possibilities of definition would

Supplement to '' Nature," April 14, 1928 still seem necessary in order t o bring out the general complementary character of the description. I n deed, a discontinuous change of energy and momentum during observation could not prevent us from ascribing accurate values to the space-time co-ordinates, as well as to the momentum-energy components before and after the process. The reciprocal uncertainty which always affects the values of these quantities is, as will be clear from the preceding analysis, essentially an outcome of the limited accuracy with which changes in energy and momentum can be defined, when the wavefields used for the determination of the space-time co-ordinates of the particle are sufficiently small. I n using a n optical instrument for determinations of position, it is necessary to remember t h a t the formation of the image always requires a convergent beam of light. Denoting by h the wave-length of the radiation used, and by E the so-called numerical aperture, t h a t is, the sine of half the angle of convergence. the resolving power of a microscope is given by the well-known expression X j Z e . Even if the object is illuminated by parallel light, so that the momentum hlh of the incident light quantum is known both as regards magnitude and direction, the finite value of the aperture will prevent an exact knowledge of the recoil accompanying the scattering. Also, even if the momentum of the particle were accurately known before the scattering process, our knowledge of the component of momentum parallel to the focal plane after the observation would be affected by an uncertainty amounting to 2ehIX. The product of the least inaccuracies with which the positional co-ordinate and the component of momentum in a definite direction can be ascertained is therefore just given by formula (2). One might perhaps expect t h a t in estimating the accuracy of determining the position, not only the convergence but also the length of the wave-train has t o be taken into account, because the particle could change its place during the finite time of illumination. Due t o the fact, however, t h a t the exact knowledge of the wave-length is immaterial for the above estimate, it will be realised t h a t for any value of the aperture the wave-train can always be taken so short t h a t a change of position of the particle during the time of observation may be neglected in comparison t o the lack of sharpness inherent in the determination of position due t o the finite resolving power of the microscope. I n measuring momentum with the aid of the Doppler e f f e c t w i t h due regard to the Compton e f f e c t o n e will employ a parallel wave-train. For the accuracy, however, with which the change in wave-length of the scattered radiation can be measured the extent of the wave-train in the direction of propagation is essential. If we assume that the directions of the incident and scattered radiation are parallel and opposite respectively t o the direction of the position co-ordinate and momentum component t o be measured, then cX/21 can be taken as a measure of the accuracy in the determination of the velocity, where 1 denotes the length of the wave-train. For sim-

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plicity, we here have regarded the velocity of light as large compared to the velocity of the particle. If m represents the mass of the particle, then the uncertainty attached to the value of the momentum after observation is cmhl21. I n this case the magnitude of the recoil, 2h/h, is sufficiently well defined in order not to give rise to an appreciable uncertainty in the value of the momentum of the particle after observation. Indeed, the general theory of the Compton effect allo-s us to compute the momentum components in the direction of the radiation before and after the recoil from the wavelengths of the incident and scattered radiation. Even if the positional co-ordinates of the particle were accurately knonn in the beginning, our knowledge of the position after observation nevertheless will be affected by an uncertainty. Indeed, on account of the impossibility of attributing a definite instant t o the reroil, we know the mean velocity in the direction of observation during the scattering process only with an accuracy 2h,mh. The uncertainty in the position after observation hence is Shllmch. Here, too, the product of the inaccuracies in the measurement of position and momentum is thus given by the general formula (2). J u s t as in the case of the determination of position, the time of the process of' observation for the determination of momentum may be made as short as is desired if only the wavelength of the radiation used is sufficiently small. The fact t h a t the recoil then gets larger does not, as we have seen, affect the accuracy of measurement. It should further be mentioned, t h a t in referring to the velocity of a particle as we have here done repeatedly, the purpose has only been t o obtain a connexion with the ordinary space-time description convenient in this case. As it appears already from the considerations of de Broglie mentioned above, the concept of velocity must always in the quantum theory be handled with caution. It will also be seen that an unambiguous definition of this concept is excluded b y the quantum postulate. This is particularly t o be remembered when comparing the results of successive observations. Indeed, the position of a n individual a t two given moments can be measured with any desired degree of accuracy ; but if, from such measurements, we would calculate the velocity of the individual in the ordinary way, it must be clearly realised that we are dealing with an abstraction, from which no unambiguous information concerning the previous or future behaviour of the individual can be obtained. According to the above considerations regarding the possibilities of definition of the properties of individuals, it will obviously make no difference in the discussion of the accuracy of measurements of position and momentum of a particle if collisions with other material particles are considered instead of scattering of radiation. I n both cases we see t h a t the uncertainty in question equally affects the description of the agency of measurement and of the object. I n fact, this uncertainty cannot be avoided in a description of the behaviour of individuals with respect to a co-ordinate system

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‘‘ Nature,’’ April 14, 1928

fixed in the ordinary way by means of solid bodies and unperturbable clocks. The experimental devices-opening and closing of apertures, etc.are seen to permit only conclusions regarding the space-time extension of the associated wave-fields. I n tracing observations back to our sensations, once more regard has t o be taken t o the quantum postulate in connexion with the perception of the agency of observation, be it through its direct action upon the eye or by- means of suitable auxiliaries such as photographic plates, Wilson clouds, etc. It is easily seen, however, t h a t the resulting additional statistical element will not influence the uncertainty in the description of the object. It might even be conjectured t h a t the arbitrariness in what is regarded as object and what as agency of observation would open u p a possibility of avoiding this uncertainty altogether. I n connexion u i t h the measurement of the position of a particle. one might, for example, ask whether the momentum transmitted by the scattering could not be determined by means of the conservation theorem from a measurement of the change of light momentum of the microscope-including source and photographic plate-during the process of observation. A closer investigation shows, however, that such a measurement is impossible, if a t the same time one wants to know the position of the microscope with sufficient accuracy. I n fact, it follows from the experiences which have found expression in the wave theory of matter, that the position of the centre of gravity of a body and its total momentum can only be defined within the limits of reciprocal accuracy given by relation ( 2 ) . Strictly speaking, the idea of observation belongs to the causal space-time way of description. Due to the general character of relation ( 2 ) , however, this idea can be consistently utilised also in the quantum theory, if only the uncertainty expressed through this relation is taken into account. As remarked by Heisenberg, one may even obtain an instructive illustration t o the quantum theoretical description of atomic (microscopic) phenomena b y comparing this uncertainty with the uncertainty, due t o imperfect measurements, inherently contained in any observation as considered in the ordinary description of natural phenomena. He remarks on t h a t occasion t h a t even in the case of macroscopic phenomena we may say, in a certain sense, that they are created b y repeated observations. It must not be forgotten, however, t h a t in the classical theories any succeeding observation permits a prediction of future events with everincreasing accuracy, because it improves our knowledge of the initial state of t h e system. According to the quantum theory, just the impossibility of neglecting the interaction with the agency of measurement means t h a t every observation introduces a new uncontrollable element. Indeed, it follows from the above consideratione t h a t the measurement of the positional coordinates of a particle is accompanied not only by a finite change in the dynamical variables, but also the fixation of its position means a complete rupture

n the causal description of its dynamical be-

Laviour, while the determination of its momentum h a y s implies a gap in the knowledge of its patial propagation. J u s t this situation brings but most strikingly the complementary character )f the description of atomic phenomena Fhich “ppearsas a n inevitable consequence of the contrast letween the quantum postulate and the distincion between object and agency of measurement, nherent in our very idea of observation.

4. CORRESPOXDESCE PRINCIPLE AXD XATRIX THEORY. Hitherto we have only regarded certain general eatures of the quantum problem. The situation mplies, however, t h a t the main stress has to be aid on the formulation of the laws governing the nteraction between the objects which me symiolise by the abstractions of isolated particles and .adiation. Points of attack for this formulation me presented in the first place by the problem of ttomic constitution. As is 1% ell known, it has been 3ossible here, by means of an elementary use of dassical concepts and in harmony with the quantum postulate, t o throw light on essential aspects of 3xperience. For example, the experiments regarding the excitation of spectra by electronic impacts md by radiation are adequately accounted for on the assumption of discrete stationary states and individual transition processes. This is primarily due to the circumstance that in these questions no closer description of the space-time behaviour of the processes is required. Here the contrast with the ordinary way of description appears strikingly in the circumstance that spectral lines, which on the classical view would be ascribed to the same state of the atom, will, according t o the quantum postulate, correspond t o separate transition processes, between which the excited atom has a choice. Notaithstanding this contrast, however, a formal connexion with the classical ideas could be obtained in the limit, where the relative difference in the properties of neighbouring stationary states vanishes asymptotically and where in statistical applications the discontinuities may be disregarded. Through this connexion it was possible to a large extent to interpret the regularities of spectra on the basis of our ideas about the structure of the atom. The aim of regarding the quantum theory as a rational generalisation of the classical theories led to the formulation of the so-called correspondence principle. The utilisation of this principle for the interpretation of spectroscopic results was based on a symbolical application of classical electrodynamics, in which the individual transition processes were each associated with a harmonic in the motion of the atomic particles to be expected according to ordinary mechanics. Except in the limit mentioned, where the relative difference between adjacent stationary states may be neglected, such a fragmentary application of the classical theories could only in certain cases lead to a strictly quantitative description of the phenomena. Especially the connexion developed by

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Supplement to “Nature,” April 14, 1928 Ladenburg and Kramers between the classical treatment of dispersion and the statistical laws governing the radiative transition processes formulated by Einstein should be mentioned here. Although i t was just Kramers’ treatment of dispersion that gave important hints for the rational development of correspondence considerations, it is only through the quantum theoretical methods created in the last few years that the general aims laid down in the principle mentioned have obtained an adequate formulation. As is known, the new development was commenced in a fundamental paper by Heisenberg, where he succeeded in emancipating himself completely from the classical concept of motion by replacing from the very start the ordinary kinematical and mechanical quantities by symbols, which refer directly t o the individual processes demanded by the quantum postulate. This was accomplished by substituting for the Fourier development of a classical mechanical quantity a matrix scheme, the elements of which symbolise purely harmonic vibrations and are associated with the possible transitions between stationar :t states. By requiring t h a t the frequencies ascribt i to the elements must always obey the combination principle for spectral lines, Heisenberg could introduce simple rules of calculation for the symbols, which permit a direct quantum theoretical transcription of the fundamental equations of classical mechanics. This ingenious attack on the dynamical problem of atomic theory proved itself from the beginning to be an exceedingly powerful and fertile method for interpreting quantitatively the experimental results. Through the work of Born and Jordan as well as of Dirac, the theory was given a formulation which can compete with classical mechanics as regards generality and consistency. Especially the element characteristic of the quantum theory, Planck’s constant, appears explicitly only in the algorithms t o which the symbols, the so-called matrices, are subjected. In fact, matrices, which represent canonically conjugated variables in the sense of the Hamiltonian equations, do not obey the commutative law of multiplication, but two such quantities, q and p , have to fulfil the exchange rule

h Pq-4P=d-lIg,.

*

Indeed, this exchange relation expresses strikingly the symbolical character of the matrix formulation of the quantum theory. The matrix theory has often been called a calculus with directly observable quantities. It must be remembered, however, t h a t the procedure described is limited just to those problems, in which in applying the quantum postulate the space-time description may largely be disregarded, and the question of observation in the proper sense therefore placed in the background. I n pursuing further the correspondence of the quantum laws with classical mechanics, the stress placed on the statistical character of the quantum theoretical description, which is brought in by the

quantum postulate, has been of fundamental importance. Here the generalisation of the symbolical method made by nirac and Jordan represented a great progress by making possible the operation with matrices, which are not arranged according to the stationary states, but where the possible values of any set of variables may appear as indices of the matrix elements. I n analogy to the interpretation considered in the original form of the theory of the ‘ diagonal elen-nts ’ connected only with a single stationary state. ah time averages of the quantity to be represented, the general transformation theory of niatrices permits the representation of such averages of a mwhanical quantity, in the calculation of which any set of variables characterising the ‘ state ’ of the system have given values, v hile the canonically conjugated variables are allo\ved t o take all possible values. On the basis of the procedure (leveloped by these authors and in close connexion n i t h itleas of Born and Pauli, Heisenberg has in the paper already cited above attempted a closer analysis of the physical content of the quantum theory, especially in view of the apparently paradoxical character of the exchange relation ( 3 ) . I n this connexion he has formulated the relation

.

Aq1p-h . . * (4) as the general expression for the maximum accuracy with which two canonically conjugated variables can simultaneously be observed. I n this way Heisenberg has been able to elucidate many paradoxes appearing in the application of the quantum postulate, and t o a large extent to demonstrate the consistency of the symbolic method. I n connexion with the complementary nature of the quantum theoretical description. we must, as already mentioned, constantly keep the possibilities of definition as well as of observation before the mind. For the discussion of just this question the method of wave mechanics developed by Schrodinger has, as we shall see, proved of great help. It permits a general application of the principle of superposition also in the problem of interaction, thus offering an immediate connexion with the above considerations concerning radiation and free particles. Below we shall return t o the relation of wave mechanics t o the general formulation of the quantum laws by means of the transformation theory of matrices.

5 . WAVEMECHANICS AND QUANTUM POSTULATE. Already in his first considerations concerning the wave theory of material particles, de Broglie pointed out t h a t the stationary states of an atom may be visualised as a n interference effect of the phase wave associated with a bound electron. It is true t h a t this point of view a t first did not, as regards quantitative results, lead beyond the earlier methods of quantum theory, to the development of which Sommerfeld has contributed so essentially. Schrodinger, however, succeeded in developing a wave - theoretical method which has opened up new aspects, and has proved to be of decisive

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importance for the great progress in atomic physics during the last years. Indeed, the proper vibrations of the Schrodinger wave equation have been found t o furnish a representation of the stationary states of an atom meeting all requirements. The energy of each state is connected with the corresponding period of vibration according t o the general quantum relation (1). Furthermore, the number of nodes in the various characteristic vibrations gives a simple interpretation to the concept of quantum number which was already known from the older methods, but a t first did not seem t o appear in the matrix formulation. I n addition, Schrodinger could associate with the solutions of the wave equation a continuous distributicn of charge and current, which, if applied t o a characteristic vibration, represents the electrostatic and magnetic properties of a n atom in the corresponding stationary state. Similarly, the superposition of two characteristic solutions corresponds to a continuous vibrating distribution of electrical charge, which on classical electrod-ynamicswould give rise t o a n emission of radiation, illustrating instructively the consequences of the quantum postulate and the correspondence requirement regarding the transition process between two stationary states formulated in matrix mechanics. Another application of the method of Schrodinger, important for the further development, has been made by Born in his investigation of the problem of collisions between atoms and free electric particles. I n this connexion he succeeded in obtaining a statistical interpretation of the wave functions, allowing a calculation of the probability of the individual transition processes required by the quantum postulate. This includes a wavemechanical formulation of the adiabatic principle of Ehrenfest, the fertility of which appears strikingly in the promising investigations of Hund on the problem of formation of molecules. I n view of these results, Schrodinger has expressed the hope t h a t the development of the wave theory will eventually remove the irrational element expressed by the quantum postulate and open the way for a complete description of atomic phenomena along the line of the classical theories. I n support of this view, Schrodinger, in a recent paper (Ann.d. Phys., 83, p. 956 ; 1927), emphasises the fact that the discontinuous exchange of energy between atoms required b y the quantum postulate, from the point of view of the wave theory, is replaced by a simple resonance phenomenon. I n particular, the idea of ipdividual stationary states would be a n illusion and its applicability only an illustration of the resonance mentioned. It must be kept in mind, however, t h a t just in the resonance problem mentioned we are concerned with a closed system which, according to the view presented here, is not accessible t o observation. I n fact, wave mechanics just as the matrix theory on this view represents a symbolic transcription of the problem of motion of classical mechanics adapted t o the requirements of quantum theory and only to be interpreted by an explicit use of the quantum postulate. Indeed, the two formulations of the

interaction problem might be said to be complementary in the same sense as the wave and particle idea in the description of the free individuals. The apparent contrast in the utilisation of the energy concept in the two theories is just connected with this difference in the startingpoint. The fundamental difficulties opposing a spacetime description of a system of particles in interaction appear a t once from the inevitability of the superposition principle in the description of the behaviour of individual particles. Already for a free particle the knowledge of energy and 1110mentum excludes, as we have seen, the exact knowledge of its space-time co-ordinates. This implies that an immediate utilisation of the concept of energy in connexion x i t h the classical idea of the potential energy of the system is excluded. I n the Schrodinger wave equation these difficulties are avoided by replacing the classical expression of the Hamiltonian by a differential operator by means of the relation

where p denotes a generalised component of momentum and q the canonically conjugated variable. Hereby the negative value of the energy is regarded as conjugated to the time. So far, in the wave equation, time and space as well as energy and momentum are utilised in a purely formal way. The symbolical character of Schrodinger’s method appears not only from the circumstance t h a t its simplicity, similarly to t h a t of the matrix theory, depends essentially upon the use of imaginary arithmetic quantities. But above all there can be no question of an immediate connexion with our ordinary conceptions because the ‘ geometrical ’ problem represented by the wave equation is associated with the so-called co-ordinate space, the number of dimensions of which is equal to the number of degrees of freedom of the system, and hence in general greater than the number of dimensions of ordinary space. Further, Schrodinger’s formulation of the interaction problem, just as the formulation offered by matrix theory, involves a neglect of the finite velocity of propagation of the forces claimed by relativity theory. On the whole, it would scarcely seem justifiable, in the case of the interaction problem, to demand a visualisation by means of ordinary space-time pictures. I n fact, all our knowledge concerning the internal properties of atoms is derived from experiments on their radiation or collision reactions, such t h a t the interpretation of experimental facts ultimately depends on the abstractions of radiation in free space, and free material particles. Hence, our whole space-time view of physical phenomena, as well as the definition of energy and momentum, depends ultimately upon these abstractions. In judging the applications of these auxiliary ideas we should only demand inner consistency, in which connexion special regard has to be paid to the possibilities of definition and observation.

Supplement to “Nature,” April 14, 1928 In the characteristic vibrations of Schrodinger’s wave equation we have, as mentioned, a n adequate representation of the stationary states of an atom allowing an unambiguous definition of the energy of the system by means of the general quantum relation (1). This entails, however, t h a t in the interpretation of observations, a fundamental renunciation regarding the space-time description is unavoidable. I n fact, the consistent application of the concept of stationary states excludes, as we shall see, any specification regarding the behaviour of the separate particles in the atom. I n problems where a description of this behaviour is essential, we are bound t o use the general solution of the wave equation which is obtained by superposition of characteristic solutions. We meet here with a complementarity of the possibilities of definition quite analogous t o t h a t which we have considered earlier in connexion with the properties of light and free material particles. Thus, while the definition of energy and momentum of individuals is attached to the idea of a harmonic elementary wave, every space-time feature of the description of phenomena is, as we have seen, based on a consideration of the interferences taking place inside a group of such elementary waves. Also in the present case the agreement between the possibilities of observation and those of definition can be directly shown. According t o the quantum postulate any observation regarding the behaviour of the electron in the atom will be accompanied by a change in the state of the atom. As stressed by Heisenberg, this change will, in the case of atoms in stationary states of low quantum number, consist in general in the ejection of the electron from the atom. A description of the ‘ orbit ’ of the electron in the atom with the aid of subsequent observations is hence impossible in such a case. This is connected with the circumstance that from characteristic vibrations with only a few nodes no wave packages can be built up which would even approximately represent the ‘ motion ’ of a particle. The complementary nature of the description, however, appears particularly in t h a t the use of observations concerning the behaviour of particles in the atom rests on the possibility of neglecting, during the process of observation, the interaction between the particles, thus regarding them as free. This requires, however, that the duration of the process is short compared with the natural periods of the atom, which again means t h a t the uncertainty in the knowledge of the energy transferred in the process is large compared to the energy differences between neighbouring stationary states. I n judging the possibilities of observation it must, on the whole, be kept in mind t h a t the wave mechanical solutions can be visualised only in so far as they can be described with the aid of the concept of free particles. Here the dserence between classical mechanics and the quantum theoretical treatment of the problem of interaction appears most strikingly. I n the former such a restriction is unnecessary, because the ‘ particles ’ are here endowed with a n immediate ‘ reality,’

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independently of their being free or bound. This situation is particularly important in conriesion with the consistent utilisation of Schrodinper’s electric density as a measure of the probability for electrons being present within given space regions of the atom. Remembering the restriction mentioned, this interpretation il: seen t o be a simple consequence of the assumption that the probability of the presence of a free electron is expressed by the electric density associated 1% ith the wave-field in a similar way to that by a hich the probability of the presence of a light quantum is given by the energy density of the radiation. As already mentioned, the means for a general consistent utilisation of the classical concepts in the quantum theory have been created through the transformation theory of Dirac and Jordan, by the aid of which Heisenberg has formulated his general uncertainty relation (4). I n this theory also the Schrodinger wave equation has obtained an instructive application. I n fact, the characteristic solutions of this equation appear as auxiliary functions which define a transformation from matrices with indices representing the energy values of the system to other matrices, the indices of which are the possible values of the space coordinates. It is also of interest in this connexion to mention t h a t Jordan and Klein (Zeitsch. f. Phys., 45, 751 ; 1927) have recently arrived at the formulation of the problem of interaction expressed by the Schrodinger wave equation, taking as starting-point the wave representation of individual particles and applying a symbolic method closely related to the deep-going treatment of the radiation problem developed by Dirac from the point of T-iew of the matrix theory, to which we shall return below. 6. REALITYOF STATIOVARY STATES. I n the conception of stationary states me are, as mentioned, concerned u ith a characteristic application of the quantum postulate. By its very nature this conception means a complete renunciation as regards a time description. From the point of view taken here, just this renunciation forms the necessary condition for an unambiguous definition of the energy of the atom. Moreover, the conception of a stationary state involves, strictly speaking, the exclusion of all interactions with individuals not belonging to the system. The fact t h a t such a closed system is associated with a particular energy value may be considered as an immediate expression for the claim of causality contained in the theorem of conservation of energy. This circumstance justifies the assumption of the supra-mechanical stability of the stationary states, according to which the atom, before as well as after a n external influence, always will be found in a well-defined state, and which forms the basis for the use of the quantum postulate in problems concerning atomic structure. I n a judgment of the well-known paradoxes which this assumption entails for the description of collision and radiation reactions, it is essential t o consider the limitations of the possibilities of

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definition of the reacting free individuals, which is expressed by relation (2). I n fact, if the definition of the energy of the reacting individuals is t o be accurate t o such a degree as t o entitle us t o speak of conservation of energy during the reaction, it is necessary, according to this relation, t o coordinate to the reaction a time interval long compared to the vibration period associated with the transition process, and connected with the energy difference between the stationary states according to relation ( 1 ) . This is particularly t o be remembered when considering the passage of swiftly moving particles through a n atom. According to the ordinary kinematics, the effective duration of such a passage would be very small as compared with the natural periods of the atom, and it seemed impossible to reconcile the principle of conservation of energy with the assumption of the stability of stationary states (cf. Zeits. f. Phys., 34, 142 ; 1926). I n the wave representation, however, the time of reaction is immediately connected with the accuracy of the knowledge of the energy of the colliding particle, and hence there can never be the possibility of a contradiction nith the law of conservation. I n connexion with the discussion of paradoxes of the kind mentioned, Campbell (Phil. Afug., i. 1106 ; 1026) suggested the view that the conception of time itself may be essentially statistical in nature. From the view advanced here, according t o which the foundation of space-time description is offered by the abstraction of free individuals, a fundamental distinction between time and space, however, would seem t o be excluded by the relativity requirement. The singular position of the time in problems concerned with stationary states is, as we have seen, due t o the special nature of such problems. The application of the conception of stationary states demands tha t in any observation, say by means of collision or radiation reactions, permitting a distinction between different stationary states, we are entitled to disregard the previous history of the atom. The fact that the symbolical quantum theory methods ascribe a particular phase to each stationary state the value of which depends upon the previous history of the atom, would for the first moment seem to contradict the very idea of stationary states. As soon as we are really concerned with a time problem, however, the consideration of a strictly closed system is excluded. The use of simply harmonic proper vibrations in the interpretation of observations means, therefore, only a suitable idealisation which in a more rigorous discussion must always be replaced by a group of harmonic vibrations, distributed over a finite frequency interval. Now, as already mentioned, it is a general consequence of the superposition principle that it has no sense t o co-ordinate a phase value t o the group as a whole, in the same manner as may be done for each elementary wave constituting the group. This inobservability of the phase, well known from the theory of optical instruments, is brought out in a particularly simple manner in a discussion of the Stern-Gerlach experiment, so important for

the investigation of the properties of single atoms. As pointed out by Heisenberg, atoms with different orientation in the field may only be separated if the deviation of the beam is larger than the diffraction a t the slit of the de Broglie waves representing the translational motion of the atoms. This condition means, as a simple calculation shows, that the product of the time of passage of the atom through the field, and the uncertainty due to the finite width of the beam of its energy in the field, is a t least equal to the quantum of action. This result was considered by Heisenberg as a support of relation (2) as regards the reciprocal uncertainties of energy and time values. It would seem, however, t h a t here we are not simply dealing with a measurement of the energy of the atom a t a given time. But since the period of the proper vibrations of the atom in the field is connected with the total energy by relation ( I ) , we realise that the condition for separability mentioned just means the loss of the phase. This circumstance removes also the apparent contradictions, arising in certain problems concerning the coherence of resonance radiation, which have been discussed frequently, and were also considered by Heisenberg. To consider an atom as a closed system, as we have done above, means to neglect the spontaneous emission of radiation which even in the absence of external influences puts an upper limit to the lifetime of the stationary states. The fact t h a t this neglect is justified in many applications is connected with the circumstance that the coupling between the atom and the radiation field, which is t o be expected on classical electrodynamics, is in general very small compared to the coupling between the particles in the atom. It is, in fact, possible in a description of the state of a n atom to a considerable extent t o neglect the reaction of radiation, thus disregarding the unsharpness in the energy values connected with the lifetime of the stationary states according to relation (2) (cf. Proc. Camb. Phil. SOC., 1924 (Supplement), or Zeits. f . Phys., 13, 117 : 1923). This is the reason why it is possible to draw conclusions concerning the properties of radiation by using classical electrodynamics. The treatment of the radiation problem by the new quantum theoretical methods meant to begin with just a quantitative formulation of this correspondence consideration. This was the very startingpoint of the original considerations of Heisenberg. It may also be mentioned that an instructive analysis of Schrodinger’s treatment of the radiation phenomena from the point of view of the correspondence principle has been recently given by Klein (Zeits. f . Phys., 41,707 ; 1927). I n the more rigorous form of the theory developed by Dirac (Proc. Roy. Soc., 9,vol. 114, p. 243 ; 1927) the radiation field itself is included in the closed system under consideration. Thus it became possible in a rational way to take account of the individual character of radiation demanded by the quantum theory and to build u p a dispersion theory, in which the final width of the spectral lines is taken into consideration.

Supplement to “Nature,” A p r i l 14, 1928 The renunciation regarding space-time pictures characterising this treatment would seem to offer a striking indication of the complementary character of the quantum theory. This is particularly to be borne in mind in judging the radical departure from the causal description of Nature met with in radiation phenomena, t o which we have referred above in connexion with the excitation of spectra. In view of the asymptotic connexion of atomic properties with classical electrodynamics, demanded by the correspondence principle, the reciprocal exclusion of the conception of stationary states and the description of the behaviour of individual particles in the atom might be regarded as a difficulty. I n fact, the connexion in question means that in the limit of large quantum numbers where the relative difference between adjacent stationary states vanishes asymptotically, mechanical pictures of electronic motion may be rationally utilised. It must be emphasised, however, that this connexion cannot be regarded as a gradual transition towards classical theory in the sense that the quantum postulate would lose its signscance for high quantum numbers. On the contrary, the conclusions obtained from the correspondence principle with the aid of classical pictures depend just upon the assumptions t h a t the conception of stationary states and of individual transition processes are maintained even in this limit. This question offers a particularly instructive example for the application of the new methods. As shown by Schrodinger (Naturwiss., 14, 6 6 4 ; 1926), it is possible, in the limit mentioned, by superposition of proper vibrations t o construct wave groups small in comparison to the ‘ size ’ of the atom, the propagation of which indefinitely approaches the classical picture of moving material particles, if the quantum numbers are chosen sufficiently large. I n the special case of a simple harmonic vibrator, he was able t o show t h a t such wave groups will keep together even for any length of time, and will oscillate t o and fro in a manner corresponding to the classical picture of the motion. This circumstance Schrodinger has regarded as a support of his hope of constructing a pure wave theory without referring to the quantum postulate. As emphasised by Heisenberg, the simplicity of the case of the oscillator, however, is exceptional and intimately connected with the harmonic nature of the corresponding classical motion. Nor is there in this example any possibility for a n asymptotical approach towards the problem of free particles. I n general, the wave group will gradually spread over the whole region of the atom, and the ‘motion’ of a bound electron can only be followed during a number of periods, which is of the order of magnitude of the quantum numbers associated with the proper vibrations. This question has been more closely investigated in a recent paper by Darwin (Proc. Roy. SOC.,A, vol. 117, 258 ; 1927), which contains a number of instructive examples of the behaviour of wave groups. From the viewpoint of the matrix theory a treatment of analogous problems has been carried out by Kennard ( Z e i t s .f. Phys., 47, 326 ; 1927).

689

Here again we meet with the contrast between the wave theory superposition principle and the assumption of the individuality of particles with which we have been concerned already in the case of free particles. At the same time the asymptotical connexion with the classical theory, to which a distinction between free and bound particles is unknown, offers the possibility of a particularly simple illustration of the above considerations regarding the consistent utilisation of the concept of stationary states. As we have seen, the identification of a stationary state by means of collision or radiation reactions implies a gap in the time description, which is a t least of the order of magnitude of the periods associated with transitions between stationary states. Now, in the limit of high quantum numbers these periods may be interpreted as periods of revolution. Thus we see at once that no causal connexion can be obtained between observations leading to the fixation of a stationary state and earlier observations on the behaviour of the separate particles in the atom. Summarising, it might be said that the concepts of stationary states and individual transition processes within their proper field of application possess just as much or as little ‘ reality ’ as the very idea of individual particles. I n both cases we are concerned with a demand of causality complementary t o the space-time description, the adequate application of which is limited only by the restricted possibilities of definition and of observation. 7 . THE PROBLEM OF THE ELEMENTARY

PARTICLES.

When due regard is taken of the complementary feature required by the quantum postulate, it seems, in fact, possible with the aid of the symbolic methods t o build u p a consistent theory of atomic phenomena, which may be considered as a rational generalisation of the causal space-time description of classical physics. This view does not mean, however, t h a t classical electron theory may be regarded simply as the limiting case of a vanishing quantum of action. Indeed, the connexion of the latter theory with experience is based on assumptions which can scarcely be separated from the group of problems of the quantum theory. A hint in this direction was already given by the well-known difficulties met with in the attempts to account for the individuality of ultimate electrical particles on general mechanical and electrodynamical principles. I n this respect also the general relativity theory of gravitation has not fulfilled expectations. A satisfactory solution of the problems touched upon would seem to be possible only by means of a rational quantum-theoretical transcription of the general field theory, in which the ultimate quantum of electricity has found its natural position as a n expression of the feature of individuality characterising the quantum theory. Recently Klein (Zeits. f. Phys., 40, 188 ; 1927) has directed attention t o the possibility of connecting this problem with the five-dimensional unified

590

Xupplement to ‘‘ Nature,’’ April .74,1928

representation of electromagnetism and gravitation proposed by Kaluza. I n fact, the conservation of electricity appears in this theory as an analogue to the conservation theorems for energy and momentum. J u s t as these concepts are complementary to the space-time description, the appropriateness of the ordinary four-dimensional description as well as its symbolical utilisation in the quantum theory would, as Klein emphasises, seem t o depend essentially on the circumstance that in this description electricity always appears in well-defined units, the conjugated fifth d‘imension being as a consequence not open t o observation. Quite apart from these unsolved deep-going problems, the classical electron theory u p to the present time has been the guide for a further development of the correspondence description in connexion with the idea first advanced by Compton that the ultimate electrical particles, besides their mass and charge, are endowed with a magnetic moment due t o an angular momentum determined by the quantum of action. This assumption, introduced with striking success by Goudsmit and Uhlenbeck into the discussion of the origin of the anomalous Zeeman effect, has proved most fruitful in connexion with the new methods, as shown especially by Heisenberg and Jordan. One might say, indeed, that the hypothesis of the magnetic electron, together with the resonance problem elucidated by Heisenberg (Zeita.f. Phys., 41, 239; 1927), which occurs in the quantumtheoretical description of the behaviour of atoms with several electrons, have brought the correspondence interpretation of the spectral laws and the periodic system to a certain degree of completion. The principles underlying this attack have even made it possible t o draw conclusions regarding the properties of atomic nuclei. Thus Dennison (Proc. Roy. SOC.,A, vol. 115, 483; 1927), in connexion with ideas of Heisenberg and Hund, has succeeded recently in a very interesting way in showing how the explanation of the specific heat of hydrogen, hitherto beset with difficulties, can be harmonised with the assumption that the proton is endowed with a moment of momentum of the same magnitude as t h a t of the electron. Due to its larger mass, however, a magnetic moment much smaller than t h a t of the electron must be associated with the proton. The insufficiency of the methods hitherto developed as concerns the problem of the elementary particles appears in the questions just mentioned from the fact that they do not allow of an unambiguous explanation of the difference in the

Prin:eu

i x

Grcnt 1:rituin t y H. S;

behaviour of the electric elementary particles and the ‘ individuals ’ symbolised through the conception of light quanta expressed in the so-called exclusion principle formulated by Pauli. I n fact, we meet in this principle, so important for the problem of atomic structure as well as for the recent development of statistical theories, with one among several possibilities, each of which fulfils the correspondence requirement. Moreover, the difficulty of satisfying the relativity requirement in quantum theory appears in a particularly striking light in connexion with the pyoblem of the magnetic electron. Indeed, it seemed not possible to bring the promising attempts made by Darwin and Pauli in generalising the new methods t o cover this problem naturally, in connexion with the relativity kinematical consideration of Thomas so fundamental for the interpretation of experimental results. Quite recently, however, Dirac (Proc. of the Roy. SOC., A, 117, 6 1 0 ; 1928) has been able successfully to attack the problem of the magnetic electron through a new ingenious extension of the symbolical method and so t o satisfy the relativity requirement without abandoning the agreement with spectral evidence. I n this attack not only the imaginary complex quantities appearing in the earlier procedures are involved, but his fundamental equations themselves contain quantities of a still higher degree of complexity, that are represented by matrices. Already the formulation of the relativity argument implies essentially the union of the spacetime co-ordination and the demand of causality characterising the classical theories. I n the adaptation of the relativity requirement to the quantum postulate we must therefore be prepared t o meet with a renunciation as to visualisation in the ordinary sense going still further than in the formulation of the quantum laws considered here. Indeed, we find ourselves here on the very path taken by Einstein of adapting our modes of perception borrowed from the sensations to the gradually deepening knowledge of the laws of Nature. The hindrances met with on this path originate above all in the fact that, so to say, every word in the language refers t o our ordinary perception. In the quantum theory we meet this difficulty a t once in the question of the inevitability of the feature of irrationality characterising the quantum postulate. I hope, however, that the idea of complementarity is suited to characterise the situation, which bears a deep-going analogy to the general difficulty in the formation of human ideas, inherent in the distinction between subject and object.

K. C L A R K LIMITED, , &u&nLurph.

APPENDIX W. HEISENBERG

UBER DEN ANSCHAULICHEN INHALT DER QUANTENTHEORETISCHEN KINEMATIK UND MECHANIK

Z. Phys. 43 (1927) 172-198

See Introduction to Part I, sects. 1 and 2. For an English translation of this paper, see J.A. Wheeler and W.H. Zurek (eds.), Quantum Theory and Measurement, Princeton Univ. Press 1983, p . 6 2 .

Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Von

W. Heisenberg

Xit 2 Abbildungen.

in Kopenhagen.

(Eingegangen am 23. Marz 1927.)

I n der vorliegenden Arbeit werden zunachst exakte Definitionen der \Vorte : Ort, Geschwindigkeit, Energie usw. (z. B. des Elektrons) aufgestpllt. die auch in d e r

Quantenrnechanik Giiltigkeit behalten, und es wird gezeigt, daO knnonisch konjugierte GriiDen sirnultan nur rnit einer charakteristischen T'ngenauigkeit bestirnmt werden konneu ( 5 1). Diese Ungenauigkeit ist iler eigentliche G r u n d fur das Auftreten statistischer Zusaminenhaoge in. der Quantenmechanik. Ihre rnsthematische Foriiiulierung gelingt mittels d e r D i r a c - J o r d a n s c h e n Theorie ($ 2). Yon den so gewonnenen Grundsatzen ausgehend wird gezeigt, wie die rnnkroskopischen Vorgange aus der Quantenmechanik heraus verstauden werden kijnneu ($ 3). Zur Erlauterung der Theorie werden einige besondere Gedankenexperirnente tliskutiert i d 4 ) .

Eine physikalische Theorie glauben wir dann anschaulicli zu verstehen, wenn wir uns in allen einfachen Fallen die experimentellen Konsequenzen dieser Theorie qualitativ denken Itonnen, und wenn wir gleiclizeitig erkannt haben, daW die ,inwendung der Theorie niemals innere Widerspruche enthalt. Zum Beispiel glauben wir die E i n s t e i n s c h e Vorstellung Tom geschlossenen dreidimensionalen Raum anschaulicli zu verstehen, weil fur uns die experimentellen Konsequenzen dieser Vorstellung widerspruchsfrei denkbar sind. Freilich widersprechen diese Konsequenzen unseren gewohnten anschaulichen Rauni-Zeitbegriffen. W i r konnen uns aber davon uberzeugen, da13 die Mtiglichkeit der Anwendung dieser gewohnten Raum-Zeithegriffe auf sehr groDe Raume weder aus unseren Denkgesetzen noch aus der Erfahrung gefolgert werden kann. Die anschauliche Deutung der Quantenmechanik ist bisher nocli voll innerer lviderspruche, die sich im Kampf der Mehungen um Diskontinuums- und Kontinuumstheorie, Korpuskeln und \Yellen auswirken. Schon daraus mijchte man schliellen, dall eine Deutung der Quantenmechanik mit den gewohnten kinematischen und mechanischen Begriffen jedenfalls nicht moglich ist. Die Quantenmechanik war ja gerade aus dern Versuch entstanden, mit jenen gewohnten kinematischen Begriffen zu brechen und an ihre Stelle Beziehungen zwischen konkreten experimentell gegebenen Zahlen zu setzen. Da dies gelungen scheint, wird andererseits das mathematische Schema der Quantenmechanik auch keiner Revision bedurfen. Ebensowrnig wird eine Revision der Raum-Zeitgeometrie fur kleine Raume und Zeiten notwendig sein, da wir durch Wahl hinreichend schwerer Massen die quantenrnechanischen Gesetze den

W. Heisenberg, Uber den anschaul. Inhalt d. quantentheoret. Kinematik usw.

173

klassischen beliebig annahern konnen, auch wenn es sich um noch so kleine Rlume und Zeiten handelt. Aber daO eine Revision der kinematischen und mechanischen Begriffe notwendig ist, scheint aus den Grundgleichungen der Quantenmechanik unmittelbar zu folgen. Wenn eine bestimmte Masse m gegeben ist, hat es in unserer gewohnten Anschauung einen einfach verstandlichen Sinn, vom Ort und der Geschwindigkeit des Schwerpunkts dieser Masse m zu sprechen. I n der Quantenmechanik aber sol1 eine Relation p 9 - q p =

.-

h

zwischen Masse, Ort 2 xa und Geschwindigkeit bestehen. Wir haben also guten Grund, gegen die kritiklose Anwendung jener Worte Ort und Geschwindigkeit Verdacht zu schijpfen. Wenn man zugibt, daO fur Vorgnnge in sehr kleinen Raumen und Zeiten Diskontinuitaten irgendwie typisch sind, SO ist ein Versagen eben der Begriffe ,,Ort und , GeschwindigkeitUsogar unmittelbar

I-t

I Fig. 1.

I -t

i Fig. 2.

plausibel: Denkt man z. B. an die eindimensionale Bewegung eines Massenpunktes, so wird man in einer Kontinuumstheorie eine Bahnkurve z ( t ) fur die Bahn des Teilchens (genauer: dessen Schwerpunktes) zeichnen konnen (Fig. l), die Tangente gibt jeweils die Geschwindigkeit. In einer Diskontinuumstheorie dagegen wird etwa an Stelle dieser Kurve eine Reihe von Punkten endlichen Abst.andes treten (Fig. 2). I n diesem Falle ist es offenbnr sinnlos, von der Geschwindigkeit an einem bestimmten Orte zu sprechen, v e i l ja die Geschwindigkeit erst durch zwei Orte definiert werden kann und wed folglich utugekehrt zu jedem Punkt je zwei verschiedene Geschwindigkeiten gehoren. Es entsteht daher die Frage, ob es nicht durch eine genauere Analyse jener kinematischen und mechanischen Begriffe mijglich sei, die bis jetzt in der anschaulichen Deutung der Quantenmechanik bestehenden Widerspriiche aufzuklken und zu einem anschaulichen V e r s t i d n i s der quantenmechanischen Relationen zu kommen I). 1) Die vorliegende Arbeit ist aus Bestrebungen und Wiinschen entstanden, denen schon vie1 friiher, vor dem Entstehen der Quantenmechanik, andere Forscher deutlichen Ansdruck gegeben haben. Ich erinnere hier besonders an B o h r s Ar12 *

1i 4

\V. Heisenherg,

5

1. D i e B e g r i f f e : O r t , B a h n , G e s c h w i n d i g k e i t , E n e r g i e . Urn das quantenmechanische Verhalten irgend eines Gegenstandes verfolgen zu konnen, muD man die Xasse dieses Gegenstandes und die \I echselwirkungskrafte mit irgendwelcheu Feldern und anderen Gegenstanden kennen. S u r dann kann die H a m i l t o n sche Funktion des quantenmechanischen Systems aufgestellt werden. [Die folgenden dberlegungen sollen sich im allgemeinen auf die nichtrelativistische Quantenmechanik beziehen, da die Gesetze der quantentheoretischen E l e k t r o d p a m i k noch -ehr un\ ollstandig bekannt sind] '). Uber die ,,Gestalt des Gegenstandes ist irgendwelche weitere Aussage unnotig, am zweckmaoigsten bezeichnet man die Gesamtheit jener Wechselwirkungskrafte mit dem Worte Gestalt. W e n n man sich daruber klar werden will, was unter dem Worte ,.Ort des Gegenstandes', z. €3. des Elektrons (relativ zu einem gegebenen Bezugssystem), zu verstehen sei, so mu0 man bestimmte Experirnente angeben, mit deren Hilfe man den .,Ort des Elektrons" zu messen gedenkt; anders h a t dieses W o r t keinen Sinn. An solchen Experimenten, die im Prinzip den ,,Ort des Elektrons" sogar beliebig genau zu bestimmen gestatten, ist kein Mangel, z. B.: Alan beleuchte das Elektron und betrachte es unter einem Mikroskop. Die hochste erreichbare Genauigkeit der Ortsbestimmung ist hier im aesentlichen durch die Wellenlange des benutzten Lichtes gegeben. Man wird aber im Prinzip etwa ein ~ - S t r a h l - J l i k r o s k o p bauen und mit diesem die Ortsbestimmung so genau durchfuhren konnen, wie man will. Es ist indessen bei dieser Restimmung ein Kebenumstand wesentlich : der Comptoneffekt. Jede Beobachtung des vom Elektron kommenden Streulichtes setzt einen lichtelektrischen Effekt (im Auge, auf der photographischen Platte, in der Photozelle) voraus, kann also auch so gedeutet werden, daD ein Lichtquant das Elektron trifft, an diesem reflektiert oder abgebeugt wird und dann durch die Idinsen des Mikro([

beiten iiber die Grundpostulate d e r Quantentheorie (z. €3. ZY. f . Phgs. 13. 117. 1923) und E i n s t e i n s Diskussionen iiber d c n Zusammenhang zwiachen Wellenfeld und Lichtquanten. Am klarsten sind i n neuester Zeit die hier besprochenen Probleme diskutiert und die auftretenden F r a g e n teilweise b e a n t n o r t e t worden von 11'. P a u l i (Quantentheorie, Handb. d. Phys., Bd. XXIII, weiterhin als I . c. z i t i e r t ) : d u r c h die Quantenmechanik h a t sich an der Formulierung dieser Problenie durch P a u l i nur wenig geandert. Es ist mir auch eine besondere Freude, an dieser Stelle H e r r n W. P a u l i fur die vielfache Anregung zu danken, die ich aus gemeinsamen miindlichen und schriftlichen Diskussionen ernpfangpn habe, und die zu d e r vorliegenden Arbeit wesentlich beigetragen h a t . I) In jiingster Zeit siud jedoch auf dieseni (+ebiet groOe Fortschritte erzielt worden durch Arbeiten von P . D i r a c [Proc. Roy. SOC.( A ) 114, 24:3. 1927 und s p a t e r erscheinende T:ntersuchungen].

n b e r den anfichauljchen Inhalt d e r quantentheoretiscben Kinematik usw.

175

skops nochmal abgelenkt den Photoeffekt auslirst. Im Augenblick der Ortsbestimmung, also dem Sugenblick, in dem das L i c h t q u a t vom Elektron abgebeugt wird, verandert das Elektron seinen Impuls unstetig. Diese iinderung ist urn SO grober, je kleiner die Wellenlange des benutzten Lichtes, d. h. je genauer die Ortsbestimmung ist. I n dem Moment, in dem der Ort des Elektrons bekannt ist, 'kann daher sein h p u l s nur bis auf GrBDen, die jener unstetigen h d e r u n g entsprechen, b e k m t sein ; also je genauer der Ort bestimrnt ist, desto ungenauer ist der Imp& bekannt und umgekehrt ; hierjn erblicken wir eine direkte anschauliche Er1L lauterung der Relation P Q - QP = .~-: Sei yI die Genauigkeit, mit

L xa

-

der der Wert y bekannt ist (ql ist etwa der mittlere Fehler von q), also hier die Wellenlange des Lichtes, y1 die Genauigkeit, mit der der Wert p bestimmbar ist, also hier die unstetige Anderung von 1) beim Comptoneffekt, so stehen nach elementaren Formeln des ('omptoneffekts y1 und g1 in der Beziehung P1'11 h. (1)

-

DaO diese Beziehung (1) in direkter mathematischer Verbindung mit der Vertauschungsrela,tion P Q - q p =

1.

steht, wird spater ge2 ni zeigt werden. Hier sei darauf hingewiesen, daD Gleichung (1) der prazise Ausdruck fur die Tatsachen ist, die man friiher durch Einteilung des Yhasenraumes in Zellen der GroDe 11 zu beschreiben suchte. Zur Bestimmung des Elektronenortes kann man auch andere Experimente, z. B. StoDversuche vornehmen. Eine genaue Messung des Ortes erfordert St6De mit sehr schnellen Partikeln, da bei langsamen Elektronen die Beugungserscheinungen, die nach E i n s t e i n eine Folge der de Broglie\Vellen sind (siehe z. B. Ramsauereffekt) eine genaue Hestimrnung des Ortes verhindern. Bei einer genauen Ortsmessung andert sich der Impuls des Elektrons also wieder unstetig und eine einfache Abschatzung der Genauigkeiten mit den Formeln der d e Broglieschen il'ellen gibt wieder die Relation (1). Durch diese Diskussion scheint der Begriff ,Ort des Elektrons" klar genug definiert und es sei nur noch ein Wort uber die ,,Crr6De' des Elektrons hinzugefiigt. W enn zwei sehr schnelle Teilchen im sehr kurzen Zeitintervall dt hintereinander das Elektron treffen. so liegen die durc,h die beiden Teilchen definierten Orte des Elektrons einander sehr nahe in einem Abstand 47. Aus den Gesetzen, die bei u-Strahlen beobachtet sind, schlieDen wir, dab sich dl bis auf (;roOen der Ordnung 10-1:'cm F-

176

W. Heisenberg,

herabdrucken lafit, wenn nur At hinreichend klein und die Teilchen hinreichend schnell gewahlt werden. Diesen Sinn hat es, wenn wir sagen, das Elektron sei eine Korpuskel, deren Radius nicht groDer als 10-12cm ist. Gehen wir nun uber zum Begriff ,Bahn des Elektrons". Unter Bahn verstehen wir eine Reihe von Raumpunkten (in einem gegebenen Bezugssystem), die das Elektron als ,Orteu nacheinander annimmt. Da wir schon wissen, was unter ,Ort zu einer bestimmten Zeit" zu verstehen sei, treten hier keine neuen Schwierigkeiten auf. Trotzdem ist leicht einzusehen, daD z. B. der oft gebrauchte Ausdruck: die , 1 S- Bahn des Elektrons im Wasserstoffatom von unserem Gesichtspunkt aus keinen Sinn hat. Urn diese 1 S-,,Balm" z u messen, mii€ite namlich das Atom mit Licht beleuchtet werden, dessen Wellenlhge jedenfalls erheblich kiirzer als 10-Bcm ist. Von solchem Licht aber genugt, ein einziges Lichtquant, urn das Elektron d l i g aus seiner .,Bahn*'zu werfen (weshalb von einer solchen Bahn immer nur ein einziger Raumpunkt definiert werden kann), das Wort ,,Bahn" hat hier also keinen vernunftigen Sinn. Dies kann ohne Kenntnis der neueren Theorien schon einfach aus den experimentellen Moglichkeiten gefolgert werden. Dagegen kann die gedachte Ortsmessung an vielen Xtomen im 1 SZustand ausgefuhrt werden. (Atome in einem gegebenen st ationaren' Zustand lassen sich z. B. durch den Stern-Gerlachversuch im Prinzip isolieren.) E s mu0 also fur einen bestimmten Zustand z. B. 1 S des Atoms eine Wahrschejnlichkeitsfunktion fur die Orte des Elektrons geben, die dem Mittelwert der klassischen Bahn uber alle Phasen entspricht und die durch Nessungen beliebig genau feststellbar ist. Xach B o r n ') ist diese Funktion durch +ls(q)Gls(q) gegeben, wenn die zum Zustand 1 S gehorige S c h r t i d i n g e r s c h e Wellenfunktion bedeutet. Mit D i r a c ' ) und J o r d a n ' ) mochte ich im Hinblick auf spatere Verji

1) Die statistische Bedeutung der d e B r o g l i e - Wellen wurde zuerst forrnuliert von A. E i n s t e i n (Sitxungsber. d. p r e d . Akad. d. Wiss. 1925, S. 3). Dieses statistische Element in der Quantenmechanik spielt dann eine wesentliche Rolle bei M. B o r n , W. H e i s e n b e r g und P. J o r d a n , Quantenmechanik I1 (2s. f . Phys. 86, 557, 1926), bes. Kap. 4, § 3, und P. J o r d a n (ZS. f. Phys. 37, 376, 1926); es wird in einer grundlegenden Arbeit von 112. B o r n (ZS. f. Phys. 38, 803, 1926) mathematisch analysiert und zur Deutung der StoDphanomene benutzt. Die Begrundung des Wahrscheinlichkeitsansatzes aus der Transformationstheorie der Matrizen findet sich in den Arbeiten: W. H e i s e n b e r g (ZS. f. Phys. 40,501,1926). P. J o r d a n (ebenda 40, 661, 1926), W. P a u l i (Anm. in ZS. f . Phys. 41, 81, 1927), P. D i r a c (Proc. Roy. SOC.(A) 113, 621, 1926), P. J o r d a n (ZS. f . Phys. 40, 809, 1926). Allgemein ist die statistische Seite der Quantenmechanik diskutiert bei P. J o r d a n (Naturwiss. 15, 105, 1927) und 11. B o r n (Naturwiss. 16, 238, 1927).

fiber den anschaulichen Inhalt der quantentheoretischen Kinematik nsw.

177

allgemeinerungen sagen : Die Wahrscheinlichkeit ist gegeben durch s (1 s,9) 8 q ) ) W O s (1 s, 4) diejenige Kolonne der Transformationsmatrix S(E,q) von E nach 4 bedeutet, die zu E = E,s gehort (E = Energie). Darin, daO in der Quantentheorie zu einem bestimmten Zustnnd, z. B. 1 S, nur die Wahrscheinlichkeitsfunktion des Elektronenortes angegeben werden kann, mag man mit B o r n und J o r d a n einen charakteristisch statistischen Zug der Quantentheorie im Gegensatz zur klassischen Theorie erblicken. Nan kann aber, wenn man will, mit D i r a c auch sagen, daB die Statistik durch unRere Experimente hereingebracht sei. Denn offenbar wgre a u c h i n d e r k l a s s i s c h e n T h e o r i e nnr die Wahrscheinlichkeit eines bestimmten Elektronenortes angebbar, solange wir die Phasen des Atoms nich t kennen. Der Unterschied zwischen klassischer und Quantenmechanik besteht vielmehr darin: Klassisch konnen wir uns durch vorausgehende Experimente immer die Phase bestimmt denken. In Wirklichkeit ist dies aber unmtiglich, weil jedes Experiment zur Bestimmung der Phase das Atom zerstort bzw. verandert. In einern bestimmten stationaren ,,ZustandY des Atoms sind die Phasen prinzipiell unbestimmt. was man als direkte Erlauterung der bekannten Gleichungen

s(1

Ef-ttE=m

I&

oder

Jw-wJ=-

k

2 ni

ansehen kann. ( J = Wirkungsvariable, w = Winkelvariable.) Das W o r t Gescbwindigkeit" eines Gegenstandes ltiOt sich durch Messungen leicht definieren, wenn es sich um krtiftefreie Bewegungen handelt. Man kann z. B. den Gegenstand mit rotem Licht beleuchten und durch den Dopplereffekt des gestreuten Lichtes die Geschwindigkeit des Teilchens ermitteln. Die Bestimmung der Geschwindigkeit wird um SO genauer, ;le langwelliger das benutzte Licht ist, da dann die Geschwindigkeitsanderung des Teilchens pro Lichtquant durch Comptoneffekt um SO geringer w i d . Die Ortsbestimmung wird entsprechend ungenau, wie es der Gleichung (1) entspricht. Wenn die Geschwindigkeit des Elektrons im Atom in einem bestimmten Xugenblick gemessen werden SOU, SO wird man etwa in diesem Xugenblick die Kernladung und die Krafte von den ubrigen Elektronen plotzlich verschwinden lassen, SO daO die Bewegung von da ab kraftefrei erfolgt, und wird dann die oben angegebene Bestimmung durchfiihren. Wieder kann man sich, wie oben, leicht iiberzeugen, dao cine Funktion p (t) fiir einen gegebenen Zustand eines Atoms, z. B. 1 S , nicht definiert werden kann. Dagegen gibt e8

178

W. Heisenberg,

wieder eine FVabrscheinlichkeitsfunktion von p in diesem Zustand, die nach D i r a c und J o r d a n den Wert S ( l S , p ) s(1 S , p ) hat. S ( 1 S,p) bedeutet wieder diejenige Kolonne der Transformationsmatrix S (J,'.1)) von E nach p, die zu E = Els gehort. SchlieDlich sei noch auf die Experimente hingewiesen, welche p e statten, die Energie oder die W e r t e der Wirkungsvariablen ,7 zii messell: solche Experimente sind besonders wichtig, da wir nur mit ihrer Hilfe definieren konnen, was wir meinen, wenn wir von der diskontinuierliclieii Anderung der Energie und der J sprechen. Die F r a n c k - H e r t z scheii S t o h e r s u c h e gestatten, die Energiemessung der Atome wegen der Gultigkeit des Energiesatzes i n der Quantentheorie zuruckzufuhren auf die Energiemessung geradlinig sich bewegender Elektronen. Iliese Slessung IaDt sich im Prinzip beliebig genau durchfuhren, wenn man nur auf d i e gleichzeitige Bestimmung des Elektronenortes, d. h. der Phase verzichtet I/ put(vgl. oben die Bestimmung von p ) , der Helation E t - t E = Zni sprechend. Der Stern - Gerlachversuch gestattet die Bestimlnung d w magnetischen oder eines mittleren elektrischen Moments des Atoms, also die Messung von GrijDen, die allein von den Ll'irkungsvariablen ,I ahhangen. Die Phasen bleiben prinzipiell unbestimmt,. Ebensowenip \vie es sinnvoll ist, von der Frequenz einer Lichtwelle in einem bestimniten Augenhlick zu sprechen, kann yon der Energie eines ;\toms in einem hestimmten Moment gesprochen werden. Dem entspricht im Stern-Gerlachversuch der Umstand, daD die Genauigkeit der Energiemessung urn so geringer wird, je kiirzer die Zeitspanne ist, i n der die Xtome unter deni EinfluD der ablenkenden Kraft stehen l). Eine obere Grenze fur die ablenkende Kraft ist namlich dadurch gegeben, daD die potentielle Energie jener ablenkenden Kraft innerhalb des Strahlenbundels nur um Betrkge variieren darf, die erheblich kleiner sind als die Energiedifferenzen d r r stationaren Zustbnde, wenn eine Bestimmung der Energie der stationaren ZustPnde moglich sein soll. Sei El ein Energiebetrag, der dieser Bedingung genugt (El gibt zugleich die Genauigkeit jener Energiemessung an). so ist also El/d der Hochstwert der ablenkenden Kraft, wenn d die Breite des Strahlenbundels (meDbar durch die Weite der benutzten Blende) ~

Die Winkelablenkung des Atomstrahls ist dann El -, 4 t, dp die Zeitspanne bezeichnet, in der die Atome unter EinfluD der ablenkendrn bedeutet.

1)

Vgl. hierzu W. P a u l i , 1. c .

-70

S.61.

Ober den anschaulichen Inhalt der quantentheoretischen Kinematik usw.

179

Kraft stehen, p den Impuls der Atome in der Strahlrichtung. Diese Ablenkung mu9 mindestens gleicher GroBenordiiung sein wie die natiirliche durch Beugung an der Blende hervorgebrachte Verbreiterung des Strahls, damit eine Messung moglich sei. Die Winkelablenkung durch Beugung ist etwa A / d , wenn A die de B r o g l i e s c h e Wellenlange bezeichnet, also

El t , N /A. (2) Diese Gleichung entspricht . der Gleichung (1) und zeigt, wie eine genaue Energiebestimmung nur durch eine entsprechende Ungenauigkeit in der Zeit erreicht werden kann. 5 2. D i e D i r a c - J o r d a n s c h e T h e o r i e . Die Resultate des vorhergehenden Abschnitts milchte man zusammenfassen und verallgemeinern in dieser Behauptung: A l l e B e g r i f f e , d i e i n d e r k l a s s i s c h e n T h e o r i e z u r Beschreibung eines mechanischen S y s t e m s v e r w e n d e t w e r d e n , l a s s e n s i c h a u c h f u r a t o m a r e V o r g a n g e analog den k l a s s i s c h e n B e g r i f f e n e x a k t d e f i n i e r e n . Die Experimente, die solcher Definition dienen, tragen aber rein erfahrungsgemao eine Unbestimmtheit in sich, wenn wir von ihnen die simultane Bestimmung zweier kanonisch konjugierten GroOen verlangen. Der Grad dieser Unbestimmtheit ist durch die (auf irgendwelche kanonisch konjugierten GroDen erweiterte) Relation (1) gegeben. Es liegt nahe, hier die Quantentheorie mit der speziellen Relativitatstheorie zu vergleichen. Nach der Relativitatstheorie laOt sich das Wort ,,gleichzeitig’ nicht anders definieren, als durch Experimente, in welche die Ausbreitungsgeschwindigkeit des 1,ichts wesentlich eingeht. Gabe es eine , scharfere Definition der Gleichzeitigkeit, also z. B. Signale, die sich unendlich schnell fortpflanzen, so ware die Relativitatstheorie unmaglich. Weil es solche Signale aber nicht gibt, weil vielmehr in der Definition der Gleichzeitigkeit schon die Lichtgeschwindigkeit vorkommt, ist Raum geschaffen fiir das Postulat der k o n s t a n t e n Lichtgeschwindigkeit, deshalb steht dieses Postulat mit dem sinngernlflen Gebrauch der Worter “Ort, Geschwindigkeit, Zeit nicht in Widerspruch. Ahnlich steht es mit der Definition der Begriffe: ,,Elektronenort, Geschwindigkeit in der Quantentheorie. -411e Experimente, die wir zur Definition dieser Worte verwenden konnen, enthalten notwendig die durch Gleichung (1) angegebene Ungenauigkeit, wenn sie auch den einzelnen Begriff p , q exakt zu definieren gestatten. Gabe es Experimente, die gleichzeitig eine .sch#rfereY Bestimmung von p und (1

1130

W. Eeisenberg,

ermoglichen, als es der Gleichung (1) entspricht, so ware die Quantenmechanik unmirglich. Diese Ungenauigkeit, die durch Gleichung (1) festgelegt ist, schafft also erst Raum fur die Giiltigkeit der Beziehunqen, die in den quantenmechanischen Vertauschungsrelationen h P q - q P = .zi ihren pragnanten Ausdruck finden ; sie ermoglicht diese Gleichung, ohne daD der physikalische Sinn der GroDen p und y geandert werden muDte. F u r diejenigen physikalischen Phanomene, deren quantentheoretisclie Formulierung noch unbekannt ist (z. B. die Elektrodynamik), bedeutet Gleichung ( 1 ) eine Forderung, die zum Auffinden der neuen Gesetze nutzlich sein mag. F u r die Quantenmechanik laDt sich Gleichung (1) durch eine geringfugige Verallgemeinerung aus der D i r a c - J o r d a n schen Formulierung herleiten. Wenn wir fur den bestimmten X e r t 17 irgend eines Parameters den O r t q des Elektrons z u q' bestimmen mit einer Genauigkeit ql, so konnen w i r dieses Faktum durch eine \\.ahrscheinlichkeitsamplitude S (7, q ) zum Ausdruck bringen, die nur in einem Gehiet der ungefahren GroDe y, urn q' von Null merklich verschieden ist. Insbesondere kann man z. B. setzen

Dann gilt fur die z u p gehorige IVahrscheinlichkeitsamplitude

S(17,P) = JS(17, Y ) s ( a , P ) d ( r .

(1)

F u r S ( q , p ) kann nach J o r d a n gesetzt werden

Dann wird nach (4)S ( 7 , p ) nur fur Werte von p , fur welche

~~(2-'-Py')Y1 I1

nicht wesentlich groDer als 1 ist, merklich von Null rerschieden sein. Insbesondere gilt im Falle (3):

"0

ober den anschaulichen Inhalt der quantentheoretischen Kinematik nsw.

18 1

Die Annahme (3) fur S ( q , p) entspricht also dem experimentellen Faktum, daD der Wert p' fur p, der Wert p' fur p [mit der Genauigkeitsbeschrtinkung (S)] gemessen wurde. Rein mathematisch ist fur die D i r a c - J o r d a n s c h e F o r m u l i e m g der Quantenmecbanik charakteristisch, daD die Relationen zwischen p, 9, E usw. als Gleichungen zwischen sehr allgemeinen Matrizen geschrieben werden kijnnen, derart, daD irgend eine vorgegebene quantentheoretische GroDe als Diagonalmatrix erscheint. Die MItiglichkeit einer solchen Schreibweise leuchtet ein, wenn man sich die Matrizen anschaulich als Tensoren (z. B. Triigheitsmomente) in mehrdimensionalen Raumen deutet, zwischen denen mathematische Beziehungen bestehen. M m kann die Achsen des Koordinatensystems, in dem man diese mathematischen Beziehungen ausdriickt, immer in die Hauptachsen eines dieser Tensoren legen. SchlieDlich kann man die mathematische Beziehung zwischen zwei Tensoren A und B auch immer durch die T'ransformationsformeln oharakterisieren, die ein nach den Hauptachsen von A orientiertes Koordinatensystem in ein anderes uberfuhren, das nach den Hauptachsen von B orientiert ist. Die letztere Formulierung entspricht der S c h r o d i n g e r s c h e n Theorie. Als die eigentlich ,,invariante", von allen Koordinatensystemen unabhangige Formulierung der Quantenmechanik wird man dagegen die D iracsche Schreibweise der p-Zahlen ansehen. Wenn wir aus jenem mathematischen Schema physikalische Resultate ableiten wollen, so mussen wir den qnantentheoretischen GroDen, also den Matrizen (oder Tensoren im mehrdimensionalen Raum) Zahlen zuordnen. Dies ist so zu verstehen, dab in jenem mehrdimensionalen Raum eine bestimmte Richtung willkurlich vorgegeben wird (nlmlich durch die A r t des angestellten Experiments festgesetzt wird) und gefragt wird, welches der , Wert" der Matrix (z. B. in jenem Bilde der Wert des Tragheitsmoments) in dieser vorgegebenen Richtung sei. Diese Frage hat nur dann einen eindeutigen Sinn, wenn die vorgegebene Richtung mit der Richtung einer der Hauptachsen jener Matrix zusammenfut; in diesem Falle gibt es eine exakte Antwort auf die gestellte Frage. Aber auch, wenn die vorgegebene Richtung nur wenig abweicht von der einer der Bauptachsen der Matrix, so kann man noch mit einer gewissen durch die relative Neigung gegebenen Ungenauigkeit, mit einem gemissen wahrscheinlichen Fehler von dem , , W e d u der Matrix in der vorgegebenen Richtung sprechen. Man kann also sagen : Jeder quantentheoretischen GroDe oder Matrix 1aDt sich eine Zahl, die ihren ,Wert" angibt, rnit einem bestimmten wahrscheinlichen Fehler zuordnen ; der wahrscheinliche Fehler hlngt vom

182

W. Heisenberg,

Koordinatensystem a b ; fur jede quantentheoretische GroDe gibt es je ein Koordinatensystem, in dem der wahrscheinliche Fehler fur diese GrbUe verschwindet. E i n bestimmtes Experiment kann also niemals uber alle quantentheoretischen GroDen genaue Auskunft geben, vielmehr teilt es i n einer fur das Experiment charakteristischen Weise die physikalisclien Gr65en in ,bekannte" nnd ,,unbekannte" (oder: mehr und weniger genau bekannte GroDen) ein Die Resultate zweier Experimente lassen sich n u r dann exakt anseinander herleiten, wenn die beiden Experimente die phyqikalischen GroSen in gleicher Weise in , bekaniite und ,unbekannte einteilen (d. h. wenn die Tensoren in jenem mehrfdch zur Veranscliaulichung gebrauchten mehrdimensionalen Raum in beiden Experimenten von der gleichen Richtung aus ,,angesehen" werdenl. Bewirken z n ei Experimente verschiedene Einteilungen i n ,,Bekanntes" und .,Vnbekannteq , so la5t sich der Zusammenhang der Resultate jener Experimente fuglich nur statistisch angeben. 'h

Zur genaueren Diskussion dieses statistischen Zusammenhangs sei ein Gedankenexperiment vorgenommen. E i n S t e r n 4 e r l a c h s c h e r B t o m strahl werde zunachst durch ein Feld F, geschickt, das so stark inhomogen in der Strahlrichtung ist, daD es merklich viele Ubergange durch Schuttelwirkung hervorruft. Dann laufe der Atomstrahl eine Weile frei, in einem bestimmten Xbstand von Fl aber beginne ein zweiteFeld F9, ahnlich inhomogen wie F,. Zwischen Fl und 2.; und hinter F2 sei es mbglich, die Anzahl der Atome in den verschiedenen stationareii Z u s t h d e n durch ein eventuell angelegtes Magnetfeld zu messen Die Strablungskrafte der Atome seien Xu11 gebetzt. Wenn mir wissen, dai3 ein Atom im Zustand der Energie En war, bevor es F , passierte. 90 konnen wir dieses experimentelle Faktum dadurch zum husdruck bringen da13 w i r dem Atom eine Wellenfunktion - z. B. im p-Raum - mit der bestimmten Enerqie En und der u n b e s t i m m t e n Phase pli 2n,En(at,3,)

S(El2,P) = + ( & , y ) e

_~

h

zuordnen. Nach dem Durchqueren des Feldes F, n i r d sich diese Funlrtioli verwandelt haben in')

s ( E n , P>2

c m

cnm

@ (En[,P) e

-

2x 2

I., ( a f h

Bm) (7)

*) Vgl. P. D i r a c , Proc. Roy. SOC.(d) 111, 661, 1026 und If. B o r n , Zd. f. Phys. 40, 167, 1926.

Uber den anschaulichen Inhalt der quantentheoretischen Kinematik usw.

183

Hierin seien die Prn irgendwie willkiirlich festgesetzt, so daD die cnm durch F, eindeutig bestimmt sind. Die Matrix c,, transformiert die Energiewerte vor dem Durchgang durch F , auf die nach dem Durchgang durch F,. Fuhren wir hinter F, eine Beetimmung der stationaren Zustande z. B. durch ein inhomogenes Magnetfeld aus, so werden wir mit einer Wahrscheinlichkeit &,i,c, finden, daO das Atom vom Zustand n in den Zustand m ubergegangen ist. Wenn wir experimentell feststellen, dab das Atom eben i n den Zustand m wirklich ubergegangen sei, so werden wir ihm zur Berechnung alles Folgendeu nicht die Funktion

c m

C,

S,,

sondern ebeu die Funktion S, mit unbestimmter Phase zuzuordnen haben ; durch die experimentelle Feststellung: ,Zustand m u wahlen wir aus der Fulle der verschiedenen M6glichkeiten (c,,) eine bestimmte: m aus, zerstirren aber gleichzeitig, wie nachher erlautert wird, alles, was an Phasenbeziehungen noch i n den GroOen c,, enthalten war. Beim Durchgang des Stomstrahls durch F, wiederholt sich das gleiche wie bei F,. Ee seien d, rn die Koeffizienten der Transformationsmatrix, die die Energien vor F, auf die nach F, transformieren. Wenn zwischen F , und Fs keine Bestimmung des Zustandes vorgenommen wird, so verwandelt sich die Eigenfunktion nach folgendem Schema:

Es sei

m

c,,d,~

= e n l gesetzt. W i r d der stationare Zustand des

Atoms hinter F, festgestellt, so wird man mit einer Wahrscheinlichkeit enl &,l den Zustand 7 finden. Wenn dagegen zwischen F , und Fs die Feststellung: ,Zustand m y gemacht wurde, so wird die Wahrscheinlichkeit fur ,Z' hinter Fs durch d m l d,l gegeben sein. Bei mehrfacher Wiederholung des ganzen Experiments (wobei j e d e s m a 1 zwischen F , und Fs der Zustand bestimmt werde) wird man also hinter F2 den Zustand 1 mit der relativen Haufigkeit Zn = C c,, C, ,),d m l d m l beobm

achten. Dieser Ausdruck stimmt nicht uberein mit e n l Znl. J o r d a n (1. c.) h a t deswegen von einer ,Interferenz der Wahrscheinlichkeiten" gesprochen. Dem mochte ich mich aber nicht anschlieOen. Denn die beiden Experimente, die zu enlent bzw. Znlfubren, sind ia physikalisch wirklich verschieden. I n einem Falle erleidet das Atom zwischen F , und Fs keine Stbrung, im aodereu wird es durch die Apparate, die eine Feststellung des stationaren Zustandes ermbglichen, gestort. Diese Apparate haben zur Folge, dab sich die ,,PhaseYdes Atoms urn prinzipiell unkontrollier-

184

W. Heisenberg,

bare Betrage andert, ebenso, wie sich bei einer Bestimmung des Elektronenortes der Impuls andert (vgl. $ 1). Das Nagnetfeld zur Bestimmung des Zustandes zwischen F, und F, wird die Eigenwerte E verstimmen, bei der Beobachtung der Bahn des Atomstrahls werden (ich denke etwa an Wilsonaufnahmen) die Xtonie statistisch verschieden und unkontrollierbar gebremst us€. Dies hat zur Folge, daO die endgultige Transformationsmatrix e n [ (von den Energiewerten vor dem Eintreten in F , auf die nach dem Austreten aus F J nicht mehr durch c n m d n L lgegeben ist, m

sondern jedes (;lied der Summe hat noch einen unbekannten Phasenfaktor. W i r konnen also nur erwarten. daO der X i t t e l n e r t von e7,1E,l uber alle diese erentuellen Phasenanderungen gleich Z, ist. Eine einfache Rechnung ergibt, daO dies der Fall ist. - W i r konnen also nach gewissen statistischen Regeln von einem Experiment auf die moglichen Resultate eines anderen schlie0en. Das andere Experiment selbst wahlt aus der Fulle der Moglichkeiten eine ganz bestimmte aus und beschrankt dadurch fur alle spateren Experimente die Jloglichkeiten. Eine solche Deutiing der Gleichung fur die Transformationsmatrix S oder der S c h r ii d i n g e r schen Wellengleichung ist nur deshalb moglich, weil die Summe von Lijsungen wieder eine Losung darstellt. Darin erblicken wir den tiefen Sinn der Linearitat der S c h r o d i n g e r s c h e n Gleichungen; deswegen konnen sie n u r als Gleichungen fur %'ellen im Phasenraum verstanden werden und deswegen mochten wir jeden Versuch, diese Gleichungen z. B. im relativistischen Falle (bei mehreren Elektronen) durch nichtlineare zu ersetzen, fur aussichtslos halt en. $3. Der c b e r g a n g von der Mikro- zur JIakromechanik. Durch die in den vorausgehenden Ahschnitten durchgefuhrte Analyse der W o r t e ,Elektronenort", =Geschwindigkeit", ,.Energie" usw. scheinen mir die Begriffe der quantentheoretischen Kinematik und Mechanik hinreichend geklart, so daO ein anschauliches Verstandnis auch der makroskopischen Vorgange vom Standpunkt der Quantenmechanik aus miiglich sein muB. Der fibergang von der Jlikro- zur Nakromechanik iat schon von S c h r o d i n g e r ' ) behandelt worden, aber ich glaube nicht. daO die S c h r o d i n g e r s c h e i'berlegung das \Yesen des Problems trifft, und zwar aus folgenden Griinden : Kach S c h r o d i n g e r sol1 in hohen Anregungszustanden eine Summe von Eigenschwingungen ein nicht allzu groOes Wellenpaket ergeben konnen, das seinerseits unter periodischen dnderungen seiner GroOe die periodischen Bewegungen des klassischen ,, Elektroiis,* 1)

E. S c h r o d i n g e r , Saturwiss. 14, 664, 1926

o b e r den anschaulichen Inhalt der quantentheoretischen Kinematik usw.

1sj

ausfiihrt. Hiergegen ist folgendes einzuwenden: V e n n das Wellenpaket solche Eigenschaften hatte, wie sie hier beschrieben wurden, so ware die vom Atom ausgesandte Strahlung in eine Fourierreihe entwickelbar, bei der die Frequenzen der Oberschwingungen ganzzahlige Vielfache einer Grundfrequenz sind. Die Frequenzen der vom Atom ausgesandten Spektrallinien sind aber nach der Quantenmechanik nie ganzzahlige Vielfache einer Qrundfrequenz - ausgenommen den Spezialfall des harmonischen Oszillators. S c h r G d i n g e r s Uberlegung ist also nur far den von ihm behandelten harmonischen Oszillator durchfiihrbar, in a l l e n a n d e r e n Fallen breitet sich im Laufe der Zeit ein Wellenpaketiiber den ganzen Raum in der Umgebung des Atoms aus. J e hilher der Anregungszustand des Atoms ist, desto langsamer erfolgt jene Zerstreuung des Wellenpakets. Aber wenn man lange genug wartet, wird sie eintreten. Das oben angefiihrte Argument iiber die vom Atom ausgesandte Strahlung laDt sich zunachst gegen a l l e Versuche anwenden, die einen direkten Ubergang der Quantenmechanik in die klassische f u r hohe Quantenzahlen erstreben. Man h a t deshalb friiher versucht, jenem Argument durch Hinweis auf die natiirliche Strahlungsbreite der stationaren Zustande zu entgehen ; sicherlich zu Unrecht, denn erstens ist dieser Ausweg schon beim Wasserstoffatom wegen der geringen Strahlung i n hohen Zustanden versperrt, zweitens mull der n b e r g a n g der Quantenmechanik in die klassische auch ohne Anleihe bei der Elektrodynamik verstiindlich sein. Auf diese bekannten Schwierigkeiten, die einer direkten Verbindung der Quantentheorie mit der klassischen im Wege stehen, hat schon friiher B o h r ' ) mehrfach hingewiesen. W i r haben sie nur deswegen wieder so ausfuhrlich erlkutert! weil sie neuerdiugs in Vergessenheit zu geraten scheinen. Ich glaube, daO man die Entstehung der klassischen ,Bahn" p r l gnant so formulieren k a n n : D i e , B a h n Y e n t s t e h t e r s t d a d u r c h , daD w i r s i e b e o b a c h t e n : Sei z. B. ein Atom im 1000. Anregungszustand gegeben. Die Bahndimensionen sind hier schon relativ gro0, SO daD es im Sinne von $ 1 geniigt, die Bestimmung des Elektronenortes mit verh%ltnisrn&Dig langwelligem Licht vorzunehmen. Wenn die Bestimmuiig des Ortes nicht d z u ungenau sein soll, so wird der Comptonriickstob zur Folge haben, dab das -4tom sich nach dem StoD in irgend einem Zustand zwischen, sagen wir, dern 960. und 1050. befindet ; gleichzeitig kann der Impuls des Elektrons mit einer aus (1) bestimmbaren Genauigkeit aus dem Dopplereffekt geschlossen werden. Das so gegebeue ex1)

N. B o h r , Grundpostulate der Quantentheorie, I. c.

186

W. Heisenberg,

perimentelle Faktum kann man durch ein Wellenpaket - besser Wahrscheinlichkeitspaket - i m q-Raum von einer durch die Wellenlange des benutzten Lichtes gegebenen GroOe, zusammengesetzt im wesentlichen aus Eigenfunktionen zwischen der 950. und der 1050. Eigenfunktion, und durch ein entsprechendes P a k e t i m p-Raum charakterisieren. Each einiger Zeit werde eine neue Ortsbestimmung mit der gleichen Genauigkeit ausgefuhrt. Ihr Resultat laDt sich nach 5 2 nur statistisch angeben, als wahrscheinliche Orte kommen alle innerhalb des nun schon verbreiterten W-ellenpakets mit berechenbarer Wahrscheinlichkeit in Betracht. Dies ware in der klassischen Theorie keineswegs anders, denn auch in der klassischen Theorie ware das Resultat der zweiten Ortsbestimmung a e g e n der Unsicherheit der ersten Bestimmung nur statistisch angebbar : auch die Systembahnen der klassischen Theorie wiirden sich ahnlich ausbreiten wie das Wellenpaket. Allerdings sind die statistischen Gesetze selbst in der Quantenmechanik und in der klassischen Theorie verschieden. Die zweite Ortsbestimmung wahlt aus der Fulle der Moglichkeiten eine bestimmte ,,q" aus und beschrankt fur alle folgenden Bestimmungen die MGglichkeiten. Nach der zweiten Ortsbestimmung kijnnen die Resultate spaterer Messungen nur berechnet werden, indem man dem Elektron wieder ein ,kleineres" Wellenpaket der Grolle I, (Wellenlange des zur Beobachtung benutzten Lichtes) zuordnet. Jede Ortsbestimmung reduziert also das Wellenpaket wieder auf seine ursprungliche GroDe A. Die ,,Werte'* der Variablen p und q sind wahrend aller Versuche mit einer gewissen Genauigkeit bekannt. DaB die W e r t e von p und q i n n e r h a 1 b d i e s e r G e n a u i g k e i t s g r e n z e n den klassischen Bewegungsgleichungen Folge leisten, kann direlrt aus den quantenmechanischen Gesetzen

geschlossen werden. Die Bahn kann aber, wie gesagt, nur statistisch aus den Anfangsbedingungen berechnet werden, was man als Folpe der prinzipiellen Ungenauigkeit der Bnfangsbedingnngen betrachten kann. Die statistischen Gesetze sind fur die Quantenmechanik und die klassische Theorie verschieden ; dies kann unter gewissen Bedingungen zu groben makroskopischen Unterschieden zwischen klassischer und Quantenthporie fuhren. Bevor ich ein Beispiel hierfiir diskutiere, mochte ich an einem einfachen mechanischen System : der kraftefreien Bewegung elneb Massenpunktes, zeigen, wie der oben diskutierte Ubergang L u r klassischen Theorie

tfber den anschaulichen Inhalt der qnantentheoretischen Kinematik nsw.

187

mathematisch zu formulieren ist. Die Bewegungsgleichungen lauten (bei eindimensionaler Bewegung) 1

H = -pa;

1

Q = - p ; p = 0. 2m m Da die Zeit als Parameter ( a h ,c-Zahl') behandelt werden kann, wenn keine von der Zeit abhengigen auDeren Krefte vorkommen, so lautet die Ltisung dieser Gleichungen : 1 9 = ,Potfqo; P =pol (11)

wo po und qo Impuls und Ort zur Zeit t = 0 bedeuten. Zur Zeit t = 0 werlle [siehe Gleichung (3) bis ( G ) ] der Wert po = q' mit der Genauigkeit q l , p o = p' mit der Genauigkeit p 1 gemessen. Um aus den ,Weden" von po und qo auf die ,Werteu von q zur Zeit t zu schlieoen, mu0 nach D i r a c und J o r d a n diejenige Transformationsfunktion gefunden werden, die alle Matrizen, bei denen qo a l s Diagonalmatrix erscheint, in solche transformiert, bei denen q als Diagonalmatrix erscheint. pokann in dem Xatrixschema, in dem q,, als Diagonalmatix erscheint, durch h d den Operator - - ersetzt werden. Nach D i r a c [l. c. Gleichung (ll)] 2na dg, gilt dann fur die gesuchte Transformationsamplitude S (a,, q ) die Differentialgleichung :

2nimI(q--qo)d9o

h.t S (ao,q ) = const. e (1 3) S s ist also von qo unabhangig, d. h. wenn zur Zeit t = 0 po exakt bekannt ist, so sind zu irgendwelcher Zeit t 0 alle Werte von 9 gleich wahrscheinlich, d. h. die \Vahrscheinlichkeit, daD q in einem endlichen Bereich liegt, ist uberhaupt Null. Dies ist ja anschaulich auch ohne weiteres klar. Denn die exakte Bestimmung von po fuhrt zu unendlich groDem ComptonruckstoD. Das gleiche wurde natiirlich fur jedes beliebige mechanische System gelten. Wenn aber qo zur Zeit t = 0 nur mit einer Genauigkeit q1 und p o mit der Genauigkeit p1 bekannt war [vgl. Gleichung (3)]

>

S (7,qo) = const. e Zeltsohrift f i r Physik. Bd. XLLII.

( q o - q')? - 1 n i -~ h P' (rlo - 9 ' )

* 91'

1

13

188

W. Heisenberg,

so wird die Ivahrscheinlichkeitsfunktion fur y nach der Formel

J

S(7, Y) = S(7lYO) S ( Y o r Y) “ Y o zu berechnen sein.

s(7, y)

Es ergibt sich m +2

= const. J e

[ 90

t

-; P‘)

(‘I

Fiihrt man die ribkiirzung @=-

00’ I f / ’ - 90)’ -7 7 1VL2

( I 4)

dy,.

t 11

(15)

‘Lnrnyf

ein, so wird der Exponent in (14)

Das Glied mit Y ’ ~kann in den konstanten (von y unabhangigen Faktorj einbezogen werden und die Integration ergibt

S (7, q ) = const. e

2 q,Z

i l + -P

(11;

1

Das Elektron befindet sicli also zur Zeit t an der Stelle L p ’

+

)11

- y’

1 0’. Das ,,IVellenpaket.’ oder besser ,,IVahrscheinlichkeitspaket“h a t sich urn den Faktor 1 1 t p2 vergrooert. j3 ist nach (I 5 ) proportional der Zeit t , umgekehrt proportional der Masse - dies ist unmittelbar plausibel - und nmgekehrt proportional y;. Eine allzu groDe Genauigkeit in yo h a t eine groWe Ungenauigkeit in p , z u r Folge und fiihrt deshalb auch zu einer groWen Ungenauigkeit in q. Der Parameter 7, den wir oben aus formalen Grunden eingefdhrt hatten, konnte hier in allen Formeln weggelassen werden, d a er in die Rechnung nicht eingeht. Als Beispiel dafur, daW der Unterschied der klassischen statistischen Gesetze von den quantentheoretischen unter Cmstanden zu groben makroskopischen Ilnterschieden zwischen den Resultaten beider Theorien fiihrt, sei die Reflexion eines Elektronenstromes an einem Gitter kurz diskutiert. Wenn die Gitterkonstante von der GroWenordnung der mit einer Genauigkeit y1

ober den anschanlichen Inhalt der qnantentheoretischen Kinematik usw.

189

d e B r o g l i e s c h e n Wellenlange der Elektronen ist, so erfolgt die Reflexion in bestimmten diskreten Raumrichtungen, \vie die Reflexion von Licht an einem Gitter. Die klnssische Theorie gibt hier grob makroskopisch etwas anderes. Trotzdem konnen wir keineswegs an der Bahn eines einzelnen Elektrons einen Widerspruch gegen die klassische Theorie feststellen. Wir konnten es, wenn wir das Elektrnn etwa auf eine bestimmte Stelle eines Gitterstrichs lenken k o m t e n und dann feststellen, daD die Reflexion dort unklassisch erfolgt. Wenn wir den Ort des Elektrons aber SO genau bestimmen wollen, daD w i r sagen kannen, auf welche Stelle eines Gitterstrichs es trifft, so beknmmt das Elektron durch diese Ortsbestimmung eine groDe Geschwindigkeit, die d e B r o g l i e s c h e Wellenlllnge des Elektrons wird um so vie1 kleiner, daO nun die Reflexion wirklich in dieser Naherung in der klassisch vorgeschriebenen Richtung erfolgen kann und wird, ohne den quantentheoretischen Gesetzen zu widersprechen.

0 4. D i s k u s s i o n e i n i g e r b e s o n d e r e n G e d a n k e n e x p e r i m e n t e . Nach der hier versuchten anschaulichen Deutung der Quantentheorie mussen die Zeitpunkte der cbergiinge, der Quantensprunge" ebenso konkret, durch Messungen feststellbar sein, wie etwa die Energien in station#ren Z u s t h d e n . Die Genauigkeit, mit der ein solcher Zeitpunkt h ermittelbar ist, wird nach Gleichung (2) durch - - gegeben sein l ) , wenn d E

dE' die h d e r u n g der Energie beim Quantensprung bedeutet. W i r denken etwa an folgendes Experiment: Ein Atom, zur Zeit t = 0 irn Zustand 2, moge durch Strahlung in den Normalzustand 1 ubergehen. Dem Atom kann dann etwa analog zu Gleichung (7) die Eigenfunktion

zugeordnet werden, wenn wir annehmen, daO die Strahlungsdampfung eich i n einem Faktor der Form e - a t in den Eigenfunktionen auDert (die wirkliche Abhangigkeit ist vielleicht nicht so einfach). Dieses Atom werde zur Messung seiner Energie durch ein inhomogenes Magnetfeld geschickt, wie dies beim Stern-Cierlachversuch ublich ist, doch sol1 das unhomogene Feld dem Atomstrahl ein langes Stuck Weges folgen. Die jeweilige Beschleunigung wird man etwa dadurch messen, daO man die g a m e Strecke, die der Atomstrahl im Magnetfeld durchmibt, in kleine l)

Vgl. W . P a n l i , 1. c. S. 12.

13 *

190

W. Heisenberg,

Teilstrecken einteilt, a n deren Ende man jeweils die hblenkung des Strahles feststellt. J e nach der Geschwindigkeit des Atornstrahles entspricht der Einteilung in Teilstrecken am Atom eine Einteilung in kleine Zeitintervalle A t . Nach 8 1, Gleichung ( 2 ) entspricht dem Interval1 d t 1I

eine Genauigkeit in der Energie von --

At

.

Die W-ahrscheinlichkeit,

eine bestimmte Energie E zu messen, laDt sich direkt schlieoen aus S ( p , E) und wird daher im Interval1 von n d t bis (n 1) A t berechnet durch : (nf1)At -2

s ( ~ , E= )

nAt+(n+l)At

I#

+

c

. ~ ( pt ),e

dt.

At

Wenn zur Zeit (nfl) d t die Feststellung: ,Zustand 2" gemacht wird, so ist dem Atom fur alles spatere nicht mehr die Eigenfunktion (1 8) zuzuordnen, sondern eine, die aus (18) hervorgeht, wenn man t durch t - (12 1)d t ersetzt. Stellt man dagegen fest: ,,Zustand l'., so ist dem Atom von da a b die Eigenfunktion

+

niElt - P___

1L(E*,p)e

zuzuordnen. Man wird also zunachst in einer Reihe von Intervallen A t beobachten: ,,Zustand 2 " , dann dauernd ,Zustand 1". Damit eine Vnterscheidung der beiden Zustande noch moglich sei, darf At nicht h unter - herabgedruckt werden. &fit dieser Genauigkeit ist also der

AE

Zeitpunkt des Ubergangs bestimmbar., E i n Experiment von der eben geschilderten A r t meinen wir ganz i m Sinne der alten von P l a n c k , E i n s t e i n und Bo h r begrundeten Auffassung der Quantentheorie, wenn wir von der diskontinuierlichen Anderung der Energie sprechen. Da ein solches Experiment prinzipiell durchfuhrbar ist, muO eine Einigung uber seinen Ausgang moglich sein. I n B o h r s Grundpostulaten der Quantentheorie hat die Energie eines Atoms ebenso, wie die Werte der Wirkungsvariabeln J vor anderen Bestimmungsstucken (Ort des Elektrons us\\-.) den Vorzug, daW sich ihr Zahlwert stets angeben la13t. Diese Vorzugsstellung, die die Energie den anderen quantenmechanischen GroWen gegenuber einnimmt, verdankt sie indessen nur dem Umstand, daD sie bei abgeschlossenen Systemen ein Integral der Bewegungsgleichungen darstellt (Kr die Energiematrix gilt E = const) ; bei nicht abgeschlossenen Systemen wird dagegen die Energie sich vor keiner anderen quantenmechanischen

nber den anschanlichen Inhalt der qnententheoretischen Kinematik usw.

19 1

GroOe auszeichnen. Insbesondere wird man Experimente angeben konnen, bei denen die Phasen w des Atoms exakt meObar sind, bei denen dann aber die Energie prinzipiell unbestimmt bleibt, einer Relah oder J,w1 tion J w - W J = h entsprechend. Ein solches 2na Experiment stellt z. B. die Resonanzfluoreszenz dar. Bestrahlt man ein

-

Atom mit einer Eigenfrequenz, sagen wir v1a = E2

, so schwingt

das Atom in Phase mit der auDeren Strahlung, wobei es prinzipiell keinen Sinn hat, zu fragen, in welchem Zustand E, oder E2 das Atom so schwingt. Die Phasenbeziehung zwischen Atom iind BnSerer Strahlung laDt sich z. B. durch die Phasenbeziehung vieler Atome untereinander ( W o o d s Versuche) feststellen. Will man von Experimenten mit Strahlung lieber absehen, so kann man die Phasenbeziehung auch so messen, daO man genaue Ortsbestimmungen im Sinne des 9 1 des Elektrons zu verschiedenen Zeiten relativ zur Phase des eingestrahlten Lichtes (an vielen Atomen) vornimmt. Dem einzelnen Atom wird etwa die Wellen-

yon zugeordnet werden konnen; hierin hangt cg von der Starke und der Phase des eingestrahlten Lichtes ab. Die Wahrscheinlichkeit eines bestimmten Ortes y ist also

s (a, t )

rn)= 4

s2

G + (1 --

c 3 $1

lctl

Das periodische GLied in (20) ist vom unperiodischen experimentell trennbar, da die Ortsbestimmungen bei verschiedenen Phasen des eingestrahlten Lichtes ausgefiihrt werden konnen. In einem bekannten von B o h r angegebenen Gedankenexperiment werden die Atome eines S t e r n - G e r l a c h s c h e n Atomstrahls zuniichst an einer bestimmten Stelle durch eingestrahltes Licht zur Resonanzfluoreszenz erregt. Nach einem Stuck Weges durcblaufen sie ein inhomogenes Magnetfeld; die von den Atomen ausgehende Strahlung kann wahrend des ganzen Weges, vor und hinter dem Magnetfeld, beobachtet werden. Bevor die Atome in das Magnetfeld kommen, besteht gewohnliche Resonanzfluoreszenz, d. h. analog zur Dispersionstheorie muO angenommen werden, daD alle Atome in Phase mit dem einfallenden Licht Kugelwellen aussenden. Diese letzte Auffassung steht zuniichst im Gegensatz zu dem,

192

W. Heisenberg,

was eine grobe Anwendung der Lichtquantentheorie oder der quanten. theoretischen Grundregeln ergibt: denn aus ihr wiirde man schliellen, daO nur wenige Atome in den ,oberen Zustand" durch Aufnahme e k e s Lichtquants gehoben werden, die gesamte Resonanzstrahlung kiime also von wenigen intensivstrahlenden erregten Zentren. E s lag daher friiher nahe, zu sagen: die Lichtquantenauffassung darf hier nur fur die EnergieImpulsbilanz herangezogen werden, ,,in Wirklichkeit'. strahlen alle Atome im unteren Zustand schwach und kohlrent Kugelwellen aus. Nachdem die Atome das Magnetfeld passiert haben, kann aber kaum ein Zweifel sein, daO der Atomstrahl sich in zwei Strahlen geteilt hat, von denen der eine den $tomen im oberen, der andere den Atomen im unteren Zustand entspricht. Wenn nun die Atome im unteren Zustand strahlen, so lgge hier eine grobe Verletzung des Energiesatzes vor, denn die gesamte Anregungsenergie steckt in dem Atomstrahl mit den Btomen im oberen Zustand. Vielmehr kann kein Zweifel daruber sein, dall hinter dem Magnetfeld nur der eine Atomstrahl mit den oberen Zustiinden Licht und zwar unkohiirentes Licht - der wenigen intensiv strahlenden Atome im oberen Zustand aussendet. Wie B o h r gezeigt hat, macht dieses Gedankenexperiment besonders deutlich, welche Vorsicht manchmal bei der Anwendung des Begriffs: ,station&rer Zustand" notig ist. Von der hier entwickelten Auffassung der Quantentheorie aus laSt sicli eine Diskussion des Bo h r schen Experiments ohne Schwierigkeiten durchfiihren. I n dem tiulleren Strahlungsfelde sind die Phasen der h t o m e bestimmt, also hat es keinen Sinn, von der Energie des Atoms zu spreclien. Auch nachdem das Atom das Strahlungsfeld verlassen hat, kann man nicht sagen, daO es sich in einem bestimmten stationaren Zustand befande, sofern man nach den Koharenzeigenschaften der Strahlung fragt. Man kann aber Experimente anstellen, zu priifen, in welchem Zustand das Atom sei; das Resultat dieses Experiments 1aSt sich nur statistisch angeben. Ein solches Experiment wird durch das inhomogene Magnetfeld wirklich durchgefiihrt. Hinter dem Magnetfeld sind die E n e r g i e n der Atome bestimmt, also die Phasen unbestimmt. Die Strahlung erfolgt hier inkohbent und nur von den Atomen im o b e r e n Zustand. Das Magnetfeld bestimmt die Energien und zerstlirt daher die Phasenbeziehung. Das R o h r sche Gedankenexperiment ist eine sehr schone Eilauterung der Tatsache, dall auch die Energie des Atoms ,,in Wirklichkeit'. keine Zahl, sondern eine Matrix ist. Der Erhaltungssatz gilt fur die Energiematrix und deswegen auch fur den W e r t der Energie so genau, als dieser jeweils gemessen wird. Rechnerisch lallt sich die hufhebung der Phasen-

o b e r den anschaulichen Inhalt der quantentheoretischen Kinematik

USW.

193

beziehung etwa so verfolgen: Seien Q die Koordinaten des Atomschwerpunktes, so wird man dem Atom s t a t t (19) die Eigenfunktion t ) = S ( Q , ?, t ) (21) zuordnen, w o S (Q, t ) eine Funktion ist, die [wie S ( q , q ) in (Iti)] nur in einer kleinen Umgebung eines Punktes im Q-Raum von Xu11 verschieden ist und sich mit der (feschwindigkeit der Atome in der Strahlrichtung fortpflanzt. Die Wahrscheinlichkeit einer relativen Smplitude y fiir irgentlwelche Werte Q ist gegeben durch das Integral von S ( Q , q, t ) S (&, y, t ) iiber &, d. h. durch (20).

S (Ql

t) S

Die Eigenfunktion ( 2 1) wird sich aber im Jlagnetfeld berechenbar verHndern und sich wegen der verschiedenen Ablenkung der Atome im oberen und unteren Zustand hinter dem Magnetfeld verwandelt hahen in

+v1 4 -

AS,

t -2

(Vl

t ) d), (Ell n) e

'

(22)

S, ( q , t ) und S, (Q, t ) werden Funktionen des Q-Raumes sein, die nur in einer kleinen Umgebung eines Punktes von Xu11 \-erscliieden sind; aber dieser P u n k t ist fur S , ein anderer, als fur S9. SlS, ist also uberall S u l l . Die Wahrscheinlichkeit einer relativen Amplitude q und eines hestimmten Wertes Q ist dalier

s (Q, y, t ) F(Q,'1, t ) = s, s, qa + ( 1 - 1.3) s, q G.

(233) 1):is periodische Glied aus (20) ist verschwunden, und damit die lltigliclikeit, eine Phasenbeziehung zu messen. Das Resultat der statistischeii Ortshestimmung wird immer dasselbe sein, gleichgultig, hei welclier Phase des einfallenden Lichtes sie vorgenommen werde. W i r diirfeii annebmen. dafl Experimente mit Strahlung, deren Theorie ja noch nicht durchgefuhrt ist, die gleichen Resultate iiber die Phasenbeziehungen der Atome z u n ~ einfallenden I i c h t ergeben werden. Zum SchluD sei noch der Zusammenhang der Cfleirliung ( 2 ) El f , i 11 niit einem l'roblemkomplex atudiert, den E h r e n f e s t und andere Forscher ') an Hand des B o h r schen K~,rrespondenzpri,izips in zwei wichtigen Srbeiten diskutiert haben,). E h r e n f e s t und T o l m a n sprechen von schwacher Quantisierung - , weiin eine gequantelte periodisclie Bewegung durch Quantensprunge rider andere Stdrungen untert.2"

I) P. E h r e n f e s t und G. B r e i t , ZS. f . Phys. 9, 207, 192'1; und P. E h r e n f e s t nnd R. C. T o l m a n , Phys. Rev. 04, 287, 1924; siehe auch die Diskussio~~ hei N. B o h r , Grundpostulate der Quantentheorie I. c. 2 ) Auf diesen Zusammenhang hat niich Herr \V. P a u ! i hingewiesen.

194

W. Heisenberg,

brochen wird in Zeitintervallen, die nicht als sehr lange im Verhaltnis zur Periode des Systems angesehen werden konnen. Es sollen in diesem Falle nicht nur die exakten quantenmalligen Energiewerte vorkommen, sondern mit einer geringeren qualitativ angebbaren a priori-R'nhrscheinlichkeit, auch Energiewerte, die nicht allzu weit von den quantenmaoigen Werten nbweichen. I n der Quantenmechanik ist dieses Verbalten so zu deuten: D a die Energie durch die aulleren Storungen oder die Quantenspriinge wirklich verandert wird, so mull jede Energiemessung, sofern sie eindeutig sein soll, sich in einer Zeit zwischen zwei Storungen abspielen. Dadurch ist eine obere Grenze fur t , im Sinne von 5 1 gegeben. Den Energiewert E, eines gequantelten Zustandes messen wir also auch n u r h rnit einer Genauigkeit El - . Dahei h a t die Frage, ob das System

-

4

solche Energiewerte E, die von E0 abweichen, ..wirklich,b mit dem entsprechend kleineren statistischen Gewicht annehme, oder oh ihre experimentelle Feststellung n u r an der Ungenauigkeit der Messung liege, prinxipiell keinen Sinn. 1st t, kleiner als die Periode des Systems, so h a t es keinen Sinn mehr, von diskreten stationaren Zustanden oder diskreten Energiewerten zu sprechen. . E h r e n f e s t und B r e i t (1. c.) machen in ahnlichem Zusammenhang auf das folgende Paradoxon aufmerksam: E i n Rotator, den wir uns etlva als Zahnrad denken wollen, sei rnit einer Vorrichtung versehen, die nacli f Umdrehungen des Rades die Drehrichtung gerade umkehrt. Das Zahnrad greife etwa in eine Zahnstange ein, die ihrerseits zwischen zwei Kltitzen linear verschiebbar i s t ; die .Kliltze zwingen nach einer bestimmten Anzahl Drehungen die Stange und damit das R a d zur Umkehr. Die wahre Periode T des Systems ist lang im T7erhaltnis zur Umlaufszeit t des Rades; die diskreten Energiestufen liegen entsprechend dicht, und zwar um so dichter, je groller T ist. D a vom Standpunkt tier konsequenten Quantentheorie aus alle stationaren Zustande gleiclies statistisches Gewicht haben, werden fur hinreichend groDes I' praktisch alle Energiewerte rnit gleicher Haufigkeit vorkommen - im Gegensatz zu dem, was fur den Rotator zu erwarten ware. Dieses Paradoxon wird durch Betrachtung von unseren Gesichtspunkten aus zunachst noch verscharft. U m namlich festzustellen, ob 'das System die zum reinen Rot a t o r gehorigen diskreten Energiewerte allein oder besonders haufig annehmen wird, oder ob es rnit gleicher Wahrscheinlichkeit alle moglichrn Werte (d. h. Werte, die den kleinen Energiestufen

h entsprechen) an2

ober den anschaulichen Inhalt der quantentheoretischen Kinematik usw.

195

nimmt, genugt eine Zeit t,, die klein im Verhaltnis zu T (aber t) ist; d. h. obwohl die grolle Periode f u r solche Xessungen gar nicht in Wirksamkeit tritt, auflert sie sich scheinbar darin, daO alle mbglichen Energiewerte auftreten kiinnen. Wir sind der Ansicht, daO solche Experirnente zur Bestimmung der Gesamtenergie des Systems auch w i r k 1i c h a11e miiglichen Energiewerte gleichwahrscheinlich liefern wurden ; und zwar i d an diesem Ergebnis nicht die: grofle Periode T,sondern die linear verschiebbare Stange schuld. Selbst wenn sich das System einmal in einem Zustand befindet, dessen Energie der Rotatorquantelung entspricht, so kann es durch auOere Krafte, die an der Stange angreifen, leicht in solche iibergefuhrt werden, die der Rotatorquantelung nicht entsprechen *). Das gekoppelte System: Rotator und Stange, zeigt eben ganz andere Periodizitatseigenschaften als der Rotator. Die Ltlsung des Paradoxons liegt vielmehr im folgenden: Wenn wir die Energie des Rotators allein messen wollen, mussen wir erst die Kopplung zwischen Rotator und Stange losen. In der klassischen Theorie kbnnte bei hinreichend kleiner JIasse der Stange die Liisung der Kopplung ohne Energiehderung geschehen, deshalb kbnnte dort die Energie des Gesamtsystems der des Rotators (bei kleiner Masse der Stange) gleichgesetzt werden. In der Quantenmechanik ist die Wechselwirkungsenergie zwischen Stange und Rad mindestens von der gleichen GrbOenordnung, mie eine Energiestufe des Rotators (auch bei kleiner Masse der Stange bleibt fur die elastische Wechselwirkung zwischen Rad und Stange eine hohe Nullpunktsenergie !) ; bei Lbsung der Kopplung stellen sich fur Stange und Rad einzeln ihre quantenmbfligen Energiewerte her. Sofern wir also die Energiewerte des Rotators a l l e i n messen kbnnen, finden wir stets mit der durch das Experiment gegebeiien Genauigkeit die quantenmaoigen Energiewerte. Auch bei verschwindender Masse der Stange ist aber die Energie des g e k o p p e l t e n Systems von der Energie des Rotators verschieden; die Energie des gekoppelten Systems kann a l l e mbglichen (durch die Z'-Quantelung zugelassenen) Werte gleichwahrscheinlich annehmen. Die quantentheoretische Kinematik und Mechanik ist von der gewiihnlichen weitgehend verschieden. Die Anwendbarkeit der klassischen kinematischeii und mechanischen Begriffe kann aber weder aus unseren Denkgesetzen noch aus der Erfahrung gefolgert werden ; zu diesem Schlull l ) Dies kann nach E h r e n f e s t und B r e i t n i c h t otler nur sehr s e l t e n xeschehen durch Krafte, die am R a d angreifen.

196

W. Heisenberg,

gibt uns die Relation (1) p , y1 IV It das Recht. Ila Impuls, Ort, Energie usw. eines Elektrons exakt definierte Begriffe sind, braucht man sich nicht daran zu stoben, dab die fundarnentale (ileichung (1) nur eine qualitative Aussage enthalt. Da wir uns ferner die experimentellen Konsequenzen der Theorie in allen einfachen Fallen qualitativ denken konnen. wird man die Quantenmechanik nicht mehr als unanschaulich und abstrakt ') ansehen miissen. Freilich mochte man, wenn man dies zugibt, aucli die quantitativen Oesetze der Quantenmechanik direkt aus den anschaulichen Grundlagen, d. h . i m wesentlichen der Relation (1) ableiten konnen. J o r d a n h a t deswegen versuclit, die Gleichung

s (y 4")

=:

5s

((1 (1')

s ((1' y")

tl y '

als Wahrscheinlichkeitsrelation zu deuten. Dieser Deutiing kiinnen a-ir uns aber nicht anschlieoen (5 2 ) . Vielrnehr glauben wir, dall die quantitativen Gesetze aus den anschaulichen Grundlagen heraus einstweilen nur nach dem Prinzip der gri5Wtmoglichen Einfachheit verstanden n-erden k8nnen. Wenn z. B. die X-Koordinate des Elektrons keine ,,%ahl" mehr ist. wie nach Gleichung (1) experimentell geschlossen werden kann, dann i s t es die denkbar einfachste Annahme [die nicht mit (1) im If-iderspruch steht], daD diese X-Koordinate ein Diagonalglied einer Matrix sei, deren Nichtdiagonalglieder sich in einer Ungenauigkeit bzw. bei Transformationen i n anderen Weisen (vgl. z. B. 5 4) au5ern. Die Aussage, daW etwa die Geschwindigkeit in der X-Richtung ,in Wirklichkeit'. keine Zahl, sondern Diagonalglied einer N a t r i x sei, ist vielleicht nicht abstrakter und unanschaulicher, als die Feststellung, da5 die elektrische Feldstarke ,)in Wirklichkeit der Zeitanteil eines antisymmetrischen Tensors der Raumzeitwelt sei. Das Wort ..in Wirklichkeit" wird hier ebenso sehr und ebenso wenig berechtigt sein, wie bei irgend einer mathematischen Beschreibung natiirlicher Vorgange. Sobald man zugibt, daO alle quantentheoretischen GroDen ,,in M:irklichkeit'i Natrizen seien, folgen die quantitativen (;esetze ohne Schwierigkeiten. 1) S c h r o d i u g e r IieEeichnrt die Quantenmpchanik als forniale Tlieorie r o n abschreckender, ja abstoflender Cnnnschaulichkeit uud Abstraktheit. Sicher ivird man den Wert der mathematischen (und i n sof e r n anschaulichen) I)urchdriugung d e r quantenmechanischen Gesetzc, die S c h r o d i n g e r s Thcorie geleistet hat, nicht hoch genug einschatxen konnen. In den prinxipiellrn, physiknlischen Fragen hat aber meines Erachtens die populare Anschaulichkeit der Wrlleumechanik v o n ~geraden Wege abgefuhrt, der durch die Arbeiten E l n s t c i n s untl d e H r o g l i e s einerseits, durch die Arbeiteu B o h r s und die Quantenmechanik andererseits \ o r gezeichnet war.

fiber den anschaulichen Inhalt der quantentheoretischen Kinematik usw.

197

Wenn man annimmt, dall die hier versuchte Deutung der Quantenmechanik schon in wesentlichen Punkten richtig ist, so mag es erlaubt sein, in wenigen Worten auf ihre prinzipiellen Konsequenzen einzugehen. DaD die Quantentheorie im Gegensatz zur klassischen eine wesentlich statistische Theorie sei in dem Sinne, dall aus exakt gegebenen Daten nur statistische Schliisse gezogen werden konnten, haben wir nicht angenommen. Gegen solche Annahmen sprechen ja z. B. auch die bekannten Experimente von G e i g e r und B o t h e . Vielmehr gelten in allen Fallen, in denen in der klassischen Theorie Relationen bestehen zwischen GroOen, die wirklich alle exakt me5bar sind, die entsprechenden exakten Relatioiien auch in der Quantentheorie (Impuls- und Energiesatz). Aber an der scharfen Formulierung des Kausalgesetzes: .,Wenn wir die Gegeriwart genan kennen, konnen wir die Zuknnft berechnen., ist nicht der Xachsatz, sondern die Voraussetzung falsch. Wir k o n n e n die (:egenwart in alleii Bestimmungsstiicken prinzipiell n i c h t kennenlernen. Deshalb ist alles Wahrnehmen eine Auswahl aus einer E'iille von Mbglichlteiten und eine Ueschrankung des zukunftig IIIijglichen. I>a nun der statistische Charakter der Quantentheorie so eng an die Ungenauigkeit aller Wnhrnehmung gekniipft ist, konnte man zu der Vermutung verleitet Iverden, daD sich hinter der wahrgenommeneii statistischen Welt noch eine ,,wirkliche" Welt verberge, in der das Kausalgesetz gilt. Aber solche dpekulationen scheinen uns, das betonen wir ausdriicklicll, unfruchtbar und sinnlos. Die Physilc sol1 nur den Zusanlmenhang der Wahrnehmungen formal beschreiben. Vielniehr kann man den nnliren Sachverhalt vie1 besser so charakterisieren : Weil alle Esperimente den Gesetzen der (tuantenmechanik und damit tler Gleichung (1) unterworfeii s i n 4 SO wird durch die Quantenniechanik die Ungiiltiglteit des K a u d gesetzes definitiv festgestellt. Y a c h t r a g b e i d e r K o r r e k t u r . Sach AbschluW der vorliegenden -4rheit haben neuere Untersuchungen yon I3oli r zu (~esiclitspunktetig+ fiilirt, die eine wesentliclie Vertiefung und Verfeincrung cler in diesrr Arbeit versuchten *%nalyse der quantenineclianischen Zusaitiiiienliiinge zdilssen. In dieseiii Zusammenliang hat tnicli Bo 11 r darauf aufmerksam ,genincht, dab ich in einigen Diskussionen dieser Arheit wesentliche I'unktc iiberselien hatte. Vor allem beruht die Cnsicherlieit i n der 13eol)irchtung iiicht ausschlielllicli auf den1 Vorkommeii von Diskontiiiuitiiten, sondcrn hiingt direkt zusammen mit der Forderung, den wrschiedenen Erfahrungeti Kleichzeitig gerecht zu werden, die ill der hJrpuskulartheorie cinerseits,

198

W. Heisenberg, fiber den anschaulichen Inhalt usw.

der Wellentheorie andererseits zum Ausdruck kommen. Z. B. ist bei Strahlmikroskops die notwendige Divergenz Benutzung eines gedachten des Strahlenbundels in Betracht zu ziehen; diese erst h a t zur Folge, daD bei der Beobachtung des Elektronenortes die Richtung des ComptonriickstoDes nur mit einer Ungenauigkeit bekannt ist, die dann zur Relation ( I ) fuhrt. Ferner ist nicht geniigend betont, daD die einfache Theorie des Comptoneffekts i n Strenge nur auf freie Elektronen anwendbar ist. Die daraus folgende Vorsicht bei Anwendung der Unsicherheitsrelation ist. wie Prof. B o h r klargestellt hat, unter anderem wesentlich fur eine allseitige Diskussion des Ubergangs von Mikro- zu Makromechanik. SchlieDlich sind die Betrachtungen uber die Resonanzfluoreszenz nicht ganz korrekt, weil der Zusammenhang zwischen der Phase des Lichtes nnd der der Elektronenbewegung nicht so einfach ist, wie angenommen. Dafiir, d d ich die genannten neueren Untersucliungen B o h r s , die in einer Arbeit uber den begrifflichen Aufbau der Quantentheorie bald erscheinen werden, im Entstehen kennenlernen und diskutieren durfte, bin ich Herrn Prof. B o h r zu herzlichem Danke verpflichtet.

r-

K o p e n h a g e n , Institut fur tlieoret. Physik der Universitat.

INTRODUCTION bY

JQRGEN KALCKAR

1.

CONTRIBUTION TO THE PLANCK JUBILEE ISSUE

To celebrate Planck’s 50th doctoral anniversary, a special jubilee issue of Naturwissenschaften appeared on June 28, 1929. Bohr contributed a paper entitled “Wirkungsquantum und Naturbeschreibung”’, which represents his first attempt at tracing the bearing of what he used to call “the general lesson of quantum theory” within a wider epistemological context. The main theme is the complementary aspects arising in a general philosophical or psychological analysis of ourselves as conscious beings. In his later writings Bohr was often to return to the analogy between the movable partition between the perceiving subject and the content of our consciousness, on the one hand, and the corresponding movable partition between the measuring agency and the atomic object under investigation, on the other. In a letter from November 1928 to his friend, the Swedish physicist Carl W. Oseen, many of the general ideas of this paper are already foreshadowed: [Copenhagen,] November 5 , [ 19128 Dear friend, Believe me, I was happy for your kind letter and to learn that you are interested in the same questions that at present occupy us down here. I hope that you have received the reprints, which I sent a couple of days ago. I am very

’

Naturwiss. 17 (1929) 483-486. Reprinted in Atomtheorie und Naturbeschreibung, Julius Springer Verlag, Berlin 1931. English translation (not by Bohr) in Atomic Theory and the Description of Nature, Cambridge University Press, 1934 (reprinted 1961). For the Danish version, vide infra, p. [197]. The German version is reproduced on p. [203]; the English version, The Quantum of Action and the Description of Nature, is reproduced on p. [208].

Bohr to Oseen, 5 UOL28 .~ Danish t e x t

on p. [430]

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much ashamed for not having written before to thank you for your letter, but just in these days we have been very much occupied with discussions of the paradoxes that have come to light through the endeavours to unify the theory of relativity and the quantum theory, and I should like very much to tell you a little about this. However, this has still to wait a little while, since we have not yet been able to clarify our thoughts. My article in Naturwissenschaften is of course concerned with a general attitude, which I have had at heart during all the years that I have been occupied with the quantum theory, but which we have only got the means to express through the great development of recent years, which has made possible a consistent representation of the experimental evidence. As we already discussed years ago, the difficulty in all philosophy is the circumstance that the functioning of our consciousness presupposes a requirement as regards the objectivity of the content, while on the other hand the idea of the subject, of our own ego, forms a part of the content of our consciousness. This is exactly the kind of difficulties of which we have got such a clear example in the character of the description of nature required by the essence of the quantum postulate. Far from bemoaning the fact that in atomic physics our usual wishes with respect to the description of nature cannot be fulfilled, I believe that we ought to rejoice at the new lesson concerning the limitation in the human forms of visualization that is implied by the discovery of the quantum of action. The very simplest manner of comprehending the incompatibility of space-time pictures with the requirement of momentum and energy conservation is perhaps to bear in mind that the basis of space-time description is the use of rigid bodies and imperturbable clocks, which because of their very nature imply a renunciation as regards momentum and energy exchange. In order to be able to say that a particle at a given time has been at a given position, we must know that some more closely specified diaphragm has been opened and closed at specified times. It is now easy to demonstrate that the uncertainty in the description, arising from the diffraction of the matter or light waves at the diaphragm and the disturbance of the spectral composition of the wave system by the opening and closing, corresponds exactly to our ignorance regarding respectively the momentum exchange between the individual and the rigid bodies, to which the diaphragm is fixed, and the energy exchange with the machinery or the clock responsible for the opening and the closing. For the space-time determination we therefore evidently pay by the rupture in the momentum-energy description. It is of course not my intention here to go through all this old stuff. I hope that you will excuse my enthusiasm that is kept alive not least through the dissatisfaction still being voiced in various quarters about the state of affairs

P A R T 11: F U R T H E R E L U C I D A T I O N O F T H E C O M P L E M E N T A R I T Y A R G U M E N T

in atomic physics. Even Einstein has hitherto taken a somewhat cool attitude towards the latest development of quantum theory. In connection with a discussion of the admittedly exaggerated form of statistics proposed here some years ago, Einstein uttered to Pauli words to the effect that he was prepared for many surprises, but he did not like particularly the idea ‘dass der liebe Gott wiirfelt’. Naturally, I was keenly struck by this. But then when I met Einstein in Brussels last winter I attempted to say as a rejoinder that the old Jewish prophets already knew very well how difficult it was to describe the nature of God on the basis of our human concepts. Only in so far as we insist on referring everything to space-time pictures can there be any question of comparing the laws of nature with a game of dice. It is just the recognition of the limitation in our forms of visualization, implied by these very laws, that permits us to apply the concept of causality to its ultimate limit. Indeed, one might perhaps summarize the wonderful progress of recent years in the form and content of the quantum theory by stating that it has turned out to be possible to apply all classical concepts in a consistent manner. As already mentioned at the beginning of the letter, the satisfaction with the formulation of the quantum theory applies only in so far as we disregard the requirement of relativity. In spite of the wonderful progress, which we owe to Dirac, with respect to the endeavours to pay due attention to this requirement, new difficulties are revealed every day. As I guess Waller has told you, a new particularly instructive example has recently been brought to light by Klein, who is extremely familiar with the symbolic methods that at the present form the basis of work in these areas. It appears as if the difficulties, with which we are confronted here, are very deep-rooted and demand an even greater renunciation as regards visualizability than the one to which we have already had to accustom ourselves. Still, as already mentioned, I do not yet have anything decisive to relate. I long very much to see you and to really talk with you, and I have been tentatively thinking of coming for a short visit to Sweden if I can find the time, and if I have something new to tell. With many kind regards to your wife and yourself and to all mutual friends in Uppsala from my wife and Your devoted friend, [Niels Bohr] Two papers from the beginning of 1929 were to influence the elaboration of Bohr’s contribution to the Planck issue. The first was the publication of Schrodinger’s inauguration lecture in Zurich, dating back to December 9, 1922, “Was ist ein Naturgesetz?’’2. In this lecture Schrodinger emphasizes the rale of 2Naturwiss. 17 (1929) 9-11.

P A R T 11: F U R T H E R E L U C I D A T I O N O F T H E C O M P L E M E N T A R I T Y A R G U M E N T

Planck to Bohr, 14 Jul) 29 German text on p. [4561

statistics in quantum theory, and, in accordance with the views of his teacher Franz Exner, he expresses doubt as regards the applicability of the notion of causality within the domain of atomic physics. In contrast to Schrodinger, Joseph Petzoldt advocated in a note3 that the requirement of causality be maintained and criticized the view that atomic processes were in principle governed by chance. As had happened before and as was to happen often later when Bohr came across points of view that he considered untenable, he began to draft a reply. In the present case Bohr planned an article entitled “Kausalitat und Objekt i ~ i t a t ” ~but , it was never completed. Instead, some of the arguments were elaborated and included in “Wirkungsquantum und Naturbeschreibung” . In particular the manuscript alluded to in reference 4 contains a paragraph on the subject-object partition covering essentially the last half of page 96 (p. [212]) and the beginning of page 97 (p. [213]) of the English version of the Planck article. This section is an elaboration of the remarks at the beginning of the letter to Oseen. A curious feature of the Planck jubilee paper is that Bohr has here abandoned the word “complementarity” in favour of “ r e c i p r ~ c i t y ~As ’ ~ .borne out by the exchange of letters with Pauli quoted below, Bohr felt some misgivings as regards the wisdom of this step, and in all his later works he returned to the phrase “complementarity”. However, I remember some conversations, during which Bohr said that perhaps he should have stuck to the word “reciprocity” after all, since it had fewer literary and philosophical overtones than “complementarity”. I could understand his point. In fact, “reciprocity” points more directly to the purely logical relationship that Bohr had in mind, also when he was extending the notion to areas outside the domain of physics. As already mentioned, the jubilee issue appeared on June 2 8 , 1929. A couple of weeks later Planck wrote Bohr a letter of thanks for his contribution. However, as one could anticipate, Planck was much too cautious to commit himself as regards Bohr’s far-reaching conclusions. He writes: “The content of your article - like everything that you write - is so deeply thought out that I for my part shall not now attempt to comment on details. That could not be done without a longer discussion. Perhaps an occasion for this will present itself later. There still remains a rich field for reasoning here. ” J . Petzoldt, Kausalitat und Wahrscheintichkeit, Naturwiss. 17 (1929) 51-52. Various versions of a typed draft with corrections in Bohr’s hand are preserved in the Niels Bohr Archive. Bohr MSS, microfilm no. 12. The reprinted Danish edition of the paper contains a footnote by Bohr explaining this point.

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Niels Bohr and Max Planck in the old lecture room of the Copenhagen Institute.

“Wirkungsquantum und Naturbeschreibung” is Bohr’s first purely “philosophical” paper, and he felt certain qualms in this connection, as we learn from the following letter to Pauli: “However, the fact is that during recent months I have worked rather diligently on an investigation of the statistical problems in quantum theory. I really feel that I can now present the question of observation substantially more clearly. Thus one can pursue the reciprocity of the concept of individuality and the superposition principle to far-reaching consequences. One can show in general that any use of the former concept limits the application of the latter principle as an immediate consequence of the loss of phase resulting from every observation. Conversely, any consistent application of the superposition principle limits the possibilities of a visualizable interpretation based on the principle of individuality, as it finds expression above all in the quantum theoretical treatment of systems with several identical particles. All this contains of course nothing really new. Yet, as already said, I feel that I can present it in a manner from which others also - perhaps even you? - may learn a little. I had intended to write about it in the Planck issue, but it got much too long and took so much time that at the last moment I had to leave

BohrtoPauli, I J u l y 29 Dan,rh on

[4411

Translat’on On

[3J31

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all physics out of my article and stick to pure philosophy, and even that only by way of allusions. I hope however that the physicists will look upon it with indulgence and that Planck himself at least will appreciate how sincerely my little article was meant.” At the end of the letter Bohr adds in parenthesis:

“I do hope that you are not too angry with the new change of name6. As far as I can see, however, it is well justified, factually as well as pedagogically.” In his reply Pauli comforts Bohr in his characteristic manner. He compliments Bohr on the exciting originality of his general philosophical conclusions, while in the same breath delicately indicating that they may be quite wrong: Pauli to Bohr, 17 July 29 German text o n p. [4441 Translation on p , [4461

“I was so very satisfied with your article in the Planck issue, precisely because all physics was omitted. For once this was something new, original and exciting! (Whether it is correct, I also must leave to the judgement of the psychologists and the professional philosophers. I feel myself here just as much a layman as you do - or perhaps even more.) I have no objections at all as regards the change of name to reciprocity instead of complementarity, only I should have liked a more detailed explanation in the article of the reasons that have led you to this change. At the beginning of August I shall probably go with Hecke to southern Sweden. The temptation is great to visit you then around the middle of August in your house without water pipes (I would bring plenty of candles along). However I will not cause you any trouble; I could easily stay a couple of days in a hotel in Tisvilde! By the wayTI assume that Klein will be staying nearby in another house without water pipes and without light (except for the case that the house belongs to Mrs Maar), and it would also be a great pleasure for me to see him.” Notwithstanding Pauli’s conciliatory reception of the introduction of the new term “reciprocity”, Bohr now regarded it as a mistake. However, as always he was thrilled at the prospect of being together with his friend: Lynghuset, Tibirkelunde’ pr. Tisvildeleje, July 31, 1929

Draft. Bohr to Pauli, 31 July 29 Danish text on p. 14481

Dear Pauli, Thanks for your nice long letter which I found here in Tisvilde on my return I.e., “reciprocity” in place of “complementarity”.

’Since the letter was written in Bohr’s summer cottage in Tisvilde there is no typed copy. What is preserved is a draft in Bohr’s handwriting.

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from a sailing and walking tour in Jutland. How nice it would be, if you could visit us here for some days in connection with your holiday trip to Sweden. I am longing to discuss many questions with you. As regards the little article for the Planck issue, I am afraid that the change of name was a blunder. The problem has so many aspects, from the purely pedagogical point of view, and I contemplated for a little while whether to send a letter to Naturwissenschaften to explain that my preference for artificial words is due, not so much to an urge for mysticism, as to the endeavour to avoid this by the help of language itself. I dare say that you will not find this very clear, but we may talk in more detail about it when you come. It would suit us fine if you are going to work on a paper here. You need not bring along “candles”; we have now got electric light out here. I hope you are going to stay for a long time. You know also that you can always live at the Institute. Many kind regards from Margrethe and the whole family, Yours, N. Bohr

2.

THE MEETING OF SCANDINAVIAN SCIENTISTS AND THE UNIVERSITY YEAR BOOK

From August 26 to 3 1, 1929, a gathering of Scandinavian scientists took place in Copenhagen (18. skandinaviske Naturforskerm~de).On the first day of the meeting Bohr delivered a lecture with the title “The Atomic Theory and the Fundamental Principles Underlying the Description of Nature”. The Danish version was published in the transactions of the conferences as well as in Fysisk Tidsskriftg. Christian M ~ l l e rundertook a German translationlo, whereas the English translation” was prepared by Rud Nielsen and Urquhart. It must be conceded that this version is not entirely successful. This circumstance has been noticed, e.g., by John Honner12 who writes: “ ... the collection of essays that

N. Bohr, Atomteorien og Grundprincipperne f o r Naturbeskrivelsen, in: Beretning om det 18. skandinaviske Naturforskermnde i Knbenhavn 26.-31. August 1929, Frederiksberg Bogtrykkeri, Copenhagen 1929, pp. 71-83. Reproduced on p. [219]. Fys. Tidsskr. 27 (1929) 103-114. l o N. Bohr, Die Atomtheorie und die Prinzipien der Naturbeschreibung, Naturwiss. 18 (1930) 73-78. I ‘ Included in Atomic Theory and the Description of Nature, Cambridge University Press, 1934. Reprinted 1961. This version is reproduced on p. [236]. J. H o m e r , The Transcendental Philosophy of Niels Bohr, Stud. Hist. Phil. Sci. 13 (1982) 1-29.

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appeared in 1934 was not in Bohr’s own inventive English, and the force and subtlety of his point is obscured by translation”. For this reason we decided to include the original Danish version of the lecture in this volume. As in the Planck article, Bohr stresses the very general character of the lesson received from the development of quantum theory. This lesson throws new light on the old philosophical question of the objective existence of phenomena independent of observations. Besides the general philosophical implications, in this lecture Bohr points for the first time to a possible analogy with the situation in biology. In particular he suggests that “the distinction between the living and the dead escapes comprehension in the ordinary sense of the word” - a theme that was later taken up in “Light and Life” (cf. Vol. 10). It may be mentioned in passing that in the closing sentence of the lecture we encounter for the first time, I believe, what was to become one of Bohr’s favourite similes: the admonition that we are both spectators and actors in the great drama of existence. On December 7 , Bohr announced in a letter to Kramers that a copy of the proof would follow “in a few days”: Bohr to Kramers, 7 Dec 29 Danish text on p 14281 Translation on p 14291

“Since you were last in Copenhagen, I have pondered the general questions that you touch upon in your letter, and in a few days I am going to send you the proofs of the lecture that I gave at the gathering of scientists this summer, where I discuss the causality problem a little more closely. By the way, I gave a lecture about this to a Copenhagen association, which calls itself the Society for Philosophy and Psychology, and I learned a great deal from the ensuing discussion. In particular, I now know better which points non-physicists resent, and I also believe that for this very reason I found on this occasion better words than previously to answer the objections.” The lecture alluded to took place on November 11, and had the title: “Remarks on the Position of Modern Physics with Regard to the Law of Causality”13. The manuscript is lost, but an announcement of the lecture is preserved in the Niels Bohr Archive.

*** Every year in November, on the occasion of its anniversary, the Copenhagen University issues a “Festskrift” or Year Book, in which one of the staff members surveys recent scientific developments in his field. In 1929 the university celebrated its 450th anniversary, and Bohr had agreed to contribute to the Year Book. Owing to the heavy pressure of work, he had to give up writing a new arti” Bemzrkninger

om den nyere Fysiks Stilling ti1 Aarsagssztningen.

P A R T 11: F U R T H E R E L U C I D A T I O N O F T H E C O M P L E M E N T A R I T Y A R G U M E N T

Hendrik A . Kramers.

cle as he had intended. Instead he collected Danish translations of the three earlier papers: “Atomic Theory and Mechanics”, the Como Lecture and the Planck article and furnished an “Introductory Survey”. In this form the collection appeared in the Year Book under the title “Atomic Theory and the Description of Nature”14. Bohr sent a copy of the Year Book to Kramers and wrote: “Long ago I should have answered your nice letter that gave me great pleasure. However, I have been so busy with my article for the University Year Book and other obligations that I never got around to it. As you will see from the copy of the Year Book, which I am forwarding simultaneously, I have not been able to find time to work out a new, coherent exposition of the problems of atomic theory, as I had actually intended, but I have used a Danish translation of my most recent general articles. During this work I have often recalled with gratitude all your help through the many years and not 14 N. Bohr, Atomteori og Nuturbeskrivelse. Festskrift udgivet af K~benhuvnsUniversitet i Anledning uf Universitetets Aursfest November 1929, Copenhagen 1929. The Introductory Survey is reproduced on p. [277] from the translation given in ref. 11. For reasons already explained above, the original Danish version is also included in this volume (p. [ 2 5 5 ] ) .

BohrroKramers, 30 No) 29 D a n l s h t e x t o n p [4251 On

P [42il

P A R T 11: F U R T H E R E L U C I D A T I O N O F T H E C O M P L E M E X T A R I T Y A R G U M E N T

least our toil together with the first of the articles, which nearly came to mark a sort of termination of our direct collaboration. In the introduction I have tried to give expression to the mood that presently holds me under its sway, and which, notwithstanding all difference in time and conditions, is not so very different from the mood that stirred us both during the very first years of our collaboration.’’ In the later German15,English” and Danish16 editions Bohr included the lecture read before the Scandinavian scientists, and furnished the Introductory Survey with an addendum in which he commented on this lecture. With the publication of this collection, the history of the genesis of the complementarity argument is brought to an end.

I5Atomtheorie und Naturbeschreibung, Julius Springer Verlag, Berlin 1931. l6

Atomteori og naturbeskrivelse, Schultz, Copenhagen 1958.

I. QUANTUM THEORY AND RELATIVITY KVANTETEORI OG RELATIVITET Overs. Dan. Vidensk. Selsk. Forh. Juni 1928 - Maj 1929, p. 24 QUANTUM THEORY AND RELATIVITY Nature 123 (1929) 434 Communication to the Royal Danish Academy on 19 October 1928 ABSTRACT

X I E L ? B O H Rgay e n Rleddelelse: Kvcrriteteori 09 Relafiuitc~t. En Undersugelse af d e Yanskeligheder, son1 1:orsugene paa a t hringe KLantepostulatet i H a r m o n i meti det nlmindelige Relnti\ itetskrav liar aabenharet, sync5 a t forlange en 3 derligere Re\ ision af yore fysiske Grundbegreber, h\ ad deres hnvenclellghed paa d e atoniistiske Fxmoniener angaar

Royal Danish A c a d e m y of Science a n d Letters, Oct. 10.-Niels Bohr : Quantum theory and relativity. An examination of the difficulties brought t o light by the attempts a t reconciliation of the quantum postulate with the idea of relativity seems t o require a further revision of our fundamental physical concepts as regards their application to atomic phenomena.

'[The correct date is October 19.1

11. THE QUANTUM OF ACTION AND THE DESCRIPTION OF NATURE WIRKUNGSQUANTUM UND NATURBESCHREIBUNG Naturwiss. 17 (1929) 483-486 THE QUANTUM OF ACTION AND THE DESCRIPTION OF NATURE “Atomic Theory and the Description of Nature”, Camb. Univ. Press, 1934 (reprinted 1961), pp. 92-101

See Introduction

to Part 11,

sect. 1.

P A R T 11: F U R T H E R E L U C I D A T I O N O F T H E C O M P L E M E N T A R I T Y A R G U M E N T

THE QUANTUM O F ACTION AND THE DESCRIPTION OF NATURE (1929) Versions published in German, English and Danish

German: Wirkungsquantum und Naturbeschreibung A Naturwiss. 17 (28 June 1929) 483-486 B “Atomtheorie und Naturbeschreibung” , Julius Springer Verlag, Berlin 1931, pp. 60-66 English: The Quantum of Action and the Description of Nature C “Atomic Theory and the Description of Nature”, Camb. Univ. Press, 1934 (reprinted 1961), pp. 92-101 Danish: Virkningskvantet og Naturbeskrivelsen D “Atomteori og Naturbeskrivelse”, Festskrift udgivet af K~benhavns Universitet i Anledning af Universitetets Aarsfest 1929, Bianco Lunos Bogtrykkeri, Copenhagen 1929, pp. 69-76 E “Atomteori og naturbeskrivelse”, J.H. Schultz Forlag, Copenhagen 1958, pp. 77-84 A and B agree, apart from a few trivial changes of spelling and a reference ( A , p. 484) which is left out in B . C and D correspond to A and B . E has a footnote (p. 79) concerning the designations “complementary” and “reciprocal”, with a reference to later papers published in “Atomic Physics and Human Knowledge” (see Vol. 7 ) .

Sonderdruck aus Die Naturwissenschaften. 17.Jahrg., Heft 26 (Verlag von Julius Springer, Berlin W 9 )

Wirkungsquantum und Naturbeschreibung. Von N. BOHR,Kopenhagen. In der Geschichte der Wissenschaft gibt es sich in bezug auf Schonheit und iiineren Zusammenwohl wenige Ereignisse, die in der kurzen Zeit- hang wohl vergleichen laBt. spanne eines Menschenalters so aui3erordentliche Zwar ist dieses Ziel nicht ohne Verzicht erEntdeckung reicht worden, was die kausale raumzeitliche BeFolgen gehabt haben wie PLANCKS des elementaren Wirkungsquantums. Nicht nur schreibungsweise betrifft, die das Merkmal der bildet diese Entdeckung in immer hoherem Grade klassischen physikalischen Theorien bildet, welche die Grundlage fur die Einordnung der Erfahrungen eine so tiefgehende Klarung durch die Relativitatsiiber die atomaren Erscheinungen, die eben in den theorie erfahren haben. In dieser Hinsicht bedeutete die Quantentheorie insofern eine Entletzten dreil3ig Jahren sich so ungeheuer vermehrt haben, sondern sie hat gleichzeitig eine vollige tauschung, als die Atomtheorie gerade aus der Umformung der Grundlage der Beschreibung der Bestrebung entstanden war, eine solche BeschreiNaturphanomene hervorgebracht. Wir stehen hier bungsweise auch bei Erscheinungen durchzufuhren, vor einer ununterbrochenen Entwicklung von Ge- die den unmittelbaren Sinneseindriicken gegeniiber sichtspunkten und begrifflichen Hilfsmitteln, die nicht als Bewegungen materieller Korper erscheimit den grundlegenden Arbeiten von PLANCKnen. Von jeher war man aber darauf gefaBt, eben iiber die Hohlraumstrahlung anfangend in den hier auf ein Versagen unserer den Sinneswahrletzten Jahren in der Formulierung einer sym- nehmungen angepaBten Anschauungsformen zu bolischen Quantenmechanik gegipfelt hat, die stoBen. Wir wissen jetzt, daB die oft geauBerte als eine ungezwungene Verallgemeinerung der Skepsis hinsichtlich der Realitat der Atome iiberklassischen Mechanik aufzufassen ist, mit der sie trieben war, d a ja die wunderbare Entwicklung

4 84

BOHR:Wirkungsquantum und Naturbeschreibung.

der Experimentierkunst uns erlaubt, die Wirkungen einzelner Atome zu konstatieren. Nichtsdestoweniger hat eben die Erkenntnis der durch das Wirkungsquantum symbolisierten begrenzten Teilbarkeit der physikalischen Vorgange den alten Zweifel an die Tragweite unserer gewohnlichen Anschauungsformen den atomaren Erscheinungen gegeniiber zu ihrem Recht gebracht. Indem jede Wahrnehmung dieser Erscheinungen rnit einer nicht zu vernachlassigenden Wechselwirkung zwischen Gegenstand und Beobachtungsmittel verbunden ist, riickt dieFrage nach den Beobachtungsmoglichkeiten wieder in den Vordergrund. In neuer Beleuchtung begegnen wir hier dem Problem der Objektivitat der Erscheinungen, das in der philosophischen Diskussion stets so viel Aufmerksamkeit beansprucht hat. Bei dieser Sachlage kann es nicht wunder nehmen, daB es sich bei allen sinngemaBen Anwendungen der Quantentheorie stets um wesentlich statistische Probleme gehandelt hat. In den urspriinglichen Arbeiten von P L A S C K war es ja zunachst die Notwendigkeit der Modifikation der klassischen statistischen Mechanik, welche die Einfiihrung des Wirkungsquantums veranlaBte. Dieser fur dieQuantentheorie eigentiimliche Charakter kommt in eindrucksvoller Weise zum Ausdruck in der erneuten Diskussion iiber das Wesen des Lichtes und der Bausteine der hlaterie. Wahrend diese Fragen im Rahmen der klassischen Theorien eine scheinbar endgiiltigeLosung bekommen hatten, so wissen wir jetzt, daB sowohl fur das Licht wie fur die materiellen Teilchen verschiedenartige Bilder notwendig sind, um die Erscheinungen allseitig zum Ausdruck zu bringen und eine eindeutige Formulierung der statistischen Gesetze, welche die Beobachtungsergebnisse regeln, zu gewahren. Je klarer die Unmoglichkeit einer einheitlichen Formulierung des Inhalts der Quantentheorie mit Hilfe von klassischen Vorstellungen hervortritt, gliickliche um so mehr bewundern wir PLANCKS Intuition bei der Wahl der Bezeichnung Wirkungsquantum, die direkt auf ein Versagen des Wirkungsprinzipes hinweist, dessen zentrale Stellung in der klassischen Naturbeschreibung er selber bei mehreren Gelegenheiten betont hat. Dieses Prinzip symbolisiert sozusagen die eigentiimlich reziproke symmetrische Beziehung zwischen der RaumZeitbeschreibung und den Gesetzen der E r h a h n g von Energie und Impuls, deren groBe Fruchtbarkeit schon in der klassischen Physik damit zusammenhangt, daB diese Gesetze weitgehend unabhangig von der raum-zeitlichen Verfolgung der Erscheinungen angewandt werden konnen. E s ist eben diese Reziprozitat, die auf gliicklichste Weise in dem Formalismus der Quantenmechanik verwertet worden ist. I n der T a t tritt hier das Wirkungsquantum nur in Beziehungen auf, in denen die im Sinne von HAMILTONkanonisch konjugierten Raum-ZeitgroBen und Impuls-Energiegroaen in symmetrischer und reziproker Weise eingehen. Auch die Analogie zwischen Optik und Mechanik, die fur die neueste Entwicklung der Quantentheorie sich so iruchtbar erwiesen hat,

hangt rnit diesen Verhaltnissen in engster Weise zusammen. E s liegt im Wesen einer physikalischen Beobachtung, daB alle Erfahrungen schlieBlich mit Hilfe der klassischen Begriffe unter Vernachlassigung des Wirkungsquantums ausgedriickt werden mussen. Es ist deshalb eine unvermeidbare Folge der begrenzten. Anwendbarkeit klassischer Vorstellungen, daB die durch jede RIessung atomarer GroBen erreichbaren Ergebnisse einer ihnen innewohnenden Begrenzung unterliegen. Eine weitgehende Klarung dieser Frage wurde neulich durch das von HEISENBERG formulierte allgemeine quantenmechanische Gesetz gebracht, wonach das Produkt der mittleren Fehler, rnit denen zwei kanonisch konjugierte mechanische GroBen gleichzeitig gemessen werden konnen, nie kleiner als das Wirkungsquantum sein kann. Mit Recht hat HEISENBERG die Bedeutung dieses reziproken Unsicherheitsgesetzes fur die Beurteilung der Widerspruchsfreiheit der Quantenmechanik rnit der Bedeutung der Unmoglichkeit einer Oberlichtgeschwindigkeit von Signalen fur die Widerspruchsfreiheit der Relativitatstheorie verglichen. Zur Beurteilung der bekannten Paradoxien, denen wir in der Quantentheorie des Atombaus begegnen, ist es in dieser Verbindung wesentlich, daran zu erinnern, daB die Eigenschaften der Atome imnier durch ihre Reaktionen gegeniiber StoBen und Strahlung zur Beobachtung gelangen, und daB die in Frage stehende Begrenzung der Messungsmoglichkeiten direkt mit den scheinbaren Gegensatzen zusammenhangt, welche die Diskussion iiber das Wesen des Tichtes und der materiellen Teilchen entschleiert hat. Um zu betonen, daB es sich hier nicht um eigentliche Gegensatze handelt, wurde in einem friiheren Artikel des Verfassers (Naturwiss. 16, 245, [1928]) die Bezeichnung Komplementaritat vorgeschlagen. I n Anbetracht der oben beriihrten schon in der klassischen 3lechanik vorkommenden reziproken Symmetrie diirfte die Bezeichnung Reziprozitat jedoch besser geeignet sein, um den Sinn des in Frage stehenden Sachverhaltes auszudriicken. I n dem genannten Artikel wurde am SchluB hingewiesen auf die nahe Beziehung des Versagens unserer Anschauungsformen, die in der Unmoglichkeit einer strengen Trennung von Phanomen und Beobachtungsmittel wurzelt, zu den mit der Unterscheidung zwischen Subjekt und Objekt zusammenhangenden allgemeinen Grenzen der menschlichen Begriffsbildung. Zwar fallen die hier in Betracht kommenden erkenntnistheoretischen und psychologischen Fragen vielleicht auBerhalb des Rahmens der eigentlichen Physik. Doch mochte ich mir gern bei dieser besonderen Gelegenheit erlauben, etwas naher auf diese Gedanken einzugehen. Das in Frage stehende Erkenntnisproblem 1aBt sich wohl kurz dahin kennzeichnen, daB einerseits die Beschreibung unserer Gedankentatigkeit die Gegenuberstellung eines objektiv gegebenen Inhalts und eines betrachtenden Subjekts verlangt, wahrend andererseits - wie schon aus einer solchen

BOHR:Wirkungsquantuni und Naturbeschreibung. Aussage einleuchtet - keine strenge Trennung zwischen Objekt und Subjekt aufrecht zu erhalten ist, da ja auch der letztere Begriff dem Gedankeninhalt angehort. Aus dieser Sachlage folgt nicht nur die relative von der Willkur in der Wahl des Gesichtspunktes abhangige Bedeutung eines jeden Begriffes, oder besser jeden Wortes, sondern wir mussen im allgemeinen darauf gefaBt sein, daD eine allseitige Beleuchtung eines und desselben Gegenstandes verschiedene Gesichtspunkte verlangen kann, die eine eindeutige Beschreibung verhindern. Streng genommen steht ja die bewuBte Analyse eines jeden Begriffes in einem ausschlieBenden Verhaltnis zu seiner unmittelbaren Anwendung. Nit der Notwendigkeit, zu einer in diesem Sinn komplementaren oder besser reziproken Beschreibungsweise Zuflucht zu nehmen, sind wir wohl besonders durch psychologische Probleme vertraut. Demgegeniiber durfte gewohnlich das Merkmal der sog. exakten Wissenschaften in dem Bestreben gesehen werden, Eindeutigkeit durch Verineiden jeden Hinweises auf das betrachtende Subjekt zu erreichen. Diesem Bestreben begegnen wir vielleicht am bewuotesten in der mathematischen Symbolik, die uns ein Ideal von Objektivitat vor Augen halt, dessen Erreichung, bei j edem in sich geschlossenenAnwendungsgebiet der Logik, kaum Grenzen gesetzt sind. In den eigentlichen Naturwissenschaften aber kann jedoch von keinen streng abgeschlossenen Anwendungsgebieten der logischen Prinzipien die Rede sein, da wir immer mit neu liinzukommenden Tatsachen rechnen miissen, deren Einordnung in den Kahmen der fruheren Erfahrungen eine Revision unserer begrifflichen Hilfsmittel verlangen kann. Eine derartige Revision haben wir kiirzlich mit der Entstehung der Relativitatstheorie erlebt, die eben durch eine weitgehende Vertiefung des Beobachtungsproblems den subjektiven Charakter aller Begriffe der klassischen Physik offenbaren sollte. Ungeachtet der hohen Anforderungen, die sie an unser Abstraktionsvermogen stellt, kommt jedoch die Relativitatstheorie dem klassischen Ideal von Einheitlichkeit und Ursachenzusammenhang in der Naturbeschreibung in besonders hohem MaBe entgegen. Vor allem wird dabei die Vorstellung der objektiven Realitat der zur Beobachtung gelangenden Phanomene noch in Strenge aufrechtbetont, ist es ja eine erhalten. \Vie von EIXSTEIS fur die ganze Relativitatstheorie grundlegende Annahme, daB jede Beobachtung schlieBlich auf ein Zusammentreffen von Gegenstand und Me& korper in demselben Raum-Zeitpunkt beruht und insofern von dem Bezugssystem des Beobachters unabhangig definierbar ist. Kach der Entdeckung des Wirkungsquantums wissen wir aber, daB das klassische Ideal bei der Beschreibung atomarer Vorgange nicht erreicht werden kann. Insbesondere fiihrt jeder Versuch einer raum-zeitlichen Einordnung der Individuen einen Bruch der Ursachenkette rnit sich, indem er mit einem nicht zu vernachlassigenden Austausch von Impuls und Energie mit den zum Vergleich benutzten MaDstaben und Uhren verbunden ist, dem keine Rechnung getragen

485

werden kann, wenn diese MeDmittel ihren Zweck erfullen sollen. Umgekehrt verlangt jeder eindeutige auf die strenge Erhaltung von Energie und Impuls begriindete SchluD iiber das dynamische Verhalten der Individuen offenbar einen volligen Verzicht auf deren TTerfolgungin Raum und Zeit. Uberhaupt konnen wir sagen, daB die ZweckmaDigkeit der kausalen Raum-Zeitbeschreibung bei der Einordnung der iiblichen Erfahrungen nur in der Kleinheit des Wirkungsquantums im Vergleich mit den fur die gewohnlichen Wahrnehmungen in Betracht kommenden Wirkungen begriindet Entdeckung hat uns hier vor eine ist. PLAXCKS ahnliche Situation gestellt wie die, welche die Entdeckung der Endlichkeit der Lichtgeschwindigkeit gebracht hatte; beruht ja die ZweckmaUigkeit der scharfen von unseren Sinnen verlangte Trennung zwischen Raum und Zeit lediglich auf der Kleinheit der Geschwindigkeiten, mit denen wir im taglichen Leben zu tun haben, verglichen mit der Lichtgeschwindigkeit. In der T a t darf bei der Frage der Kausalitat der atomaren Erscheinungen die Reziprozitat der Nessungsergebnisse ebensowenig vergessen werden wie bei der Frage der Gleichzeitigkeit die Relativitat der Beobachtungen. Bei der Resignation hinsichtlich der \Yiinsche nach Anschaulichkeit, die unserer ganzen Sprache ihr Geprage gibt, z u der uns die besprochene Situation zwingt, ist es besonders lehrreich, daU Grundzuge nicht nur der relativistischen, soiiderii auch der reziproken Betrachtungsweise uns schon bei einfachen psychologischen Erfahrungen begegnen. Der Relativitat unserer \Vahrnehmungen voii Bewegung, die jedem schon aus der Kindlieit durch Schiff- oder Wagenfahrten vertraut ist, entsprechen alltagliche Erfahrungen iiber die Reziprozitat der Beruhrungswahrnehmungen. Hier sei an die von Psychologen oft herangezogene Enip. findung erinnert, die jeder erlebt hat bei dem Versuch in einem dunklen Zimmer sich durch Tasten mittels eines Stockes zu orientieren. Wahrend der Stock bei losem Xnfassen dem Beruhrungssinn als Objekt erscheint, verlieren wir bei festem -4nfassen die Vorstellung eines Fremdkorpers und die LVahrnehmung der Beruhrung wird unmittelbar in dem Punkt lokalisiert, wo der Stock an den zu untersuchenden Korper stol3t. Es ist kaum eine fibertreibung, wenn man schon aus psychologischen Erfahrungen behaupten wollte, dal3 die Begriffe Raum und Zeit ihrem Wesen nach erst durch die Moglichkeit der Vernachlassigung der Wecliselwirkung mit den MeDmitteln einen Sinn bekommen. Allgemein zeigt uns die Analyse der Sinnesempfindungen eine bemeikenswerte Unabhangigkeit bezuglich der psychologischen Grundlage der Wahrnehmungen von Raum und Zeit einerseits und der auf Kraftwirkungen zuriickgehenden \Vahrnehmungen von Energie und Impuls andererseits. Vor allem wird aber dieses Gebiet, \vie schon beruhrt, durch Reziprozitatsverhaltnisse gekennzeichnet, die mit dem einheitlichen Charakter des BewuBtseins zusammenhangen und eine auffallende khnlichkeit zeigen mit den physikalischen Konsequenzen des Wirkungsquantums. Es handelt sich

486

BOHR : Wirkungsquantum und Naturbesclircibung.

liier u n i allbekannte Eigentumlichkeiten des Gefiihls- und Willenlebens, die sich ganzlich der Darstellung durch anschauliche Bilder entziehen. Insbesondere findet der scheinbare Gegensatz zwischen dem kontinuierlichen Fortschreiten des assoziativen Denkens und der Bewahrung der Einheit der Personlichkeit eine eindrucksvolle Analogie in dem Verhaltnis der von dem Superpositionsprinzip beherrschten Wellenbeschreibung des Verhaltens materieller Teilchen zu deren unzerstorbarer Individualitat. Die unvermeidbare Beeinflussung der atomaren Erscheinungen durch deren Beobachtung entspricht hier der wohlbekannten Knderung der Farbung des psychischen Geschehens, welche jede Lenkung der Aufmerksamkeit auf ihre verschiedenen Elemente begleitet. Es sei hier noch erlaubt kurz auf die Beziehung Iiinzuweisen, die zwischen den GesetzmaBigkeiten auf psychischem Gebiet und dem Problem der Kausalitat der physikalischen Erscheinungen besteht. I n Betracht des Kontrastes zwischen dem Gefuhl des freien Willens, das das Geistesleben beherrscht, und des scheinbar ununterbrochenen Ursachszusammenhanges der begleitenden physiologischen Prozesse ist es ja den Denkern nicht entgangen, daB es sich hier um ein unanschauliches Komplementaritatsverhaltnis handeln kann. So ist ofters die Ansicht vertreten worden, daD eine wohl nicht ausfuhrbare, aber doch denkbare, ins einzelne gehende Verfolgung der Gehirnprozesse eine Ursachskette entschleiern wurde, die eine eindeutige Abbildung des gefuhlsbetonten psychischen Geschehens darbieten wiirde. Ein solches Gedankenexperiment kommt aber jetzt in ein neues Licht, indem wir nach der Entdeckung des Wirkungsquantums gelernt haben, daB eine ins einzelne gehende kausale Verfolgung atomarer Prozesse nicht moglich ist, und daB jeder Versuch, eine Kenntnis solcher Prozesse zu erwerben, mit einem prinzipiell unkontrollierbaren Eingreifen in deren

Die Natur-

wissenschafteii

Verlauf begleitet sein wird. Nach der erwahnten Ansicht uber das Verhaltnis der Gehirnvorgange und des psychischen Geschehens miissen wir also darauf gefaBt sein, daB ein Versuch erstere z u beobachten eine wesentliche Anderung des begleitenden Willengefuhls rnit sich bringen wurde. Obwohl es sich hier zunachst nur um mehr oder weniger zutreffende Analogien handeln kann, so wird man sich schwerlich von der uberzeugung freimachen konnen, daB wir in den1 von der Quantentheorie entschleierten, unserer gewohnlichen Anscliauung unzuganglichen Tatbestand ein Mittel in die Hande bekommen haben zur Beleuchtung allgemeiner Fragestellungen menschlichen Denkens. Die besondere Gelegenheit moge entschuldigen, daB ein Physiker sich auf fremde Gebiete wagt. Meine Absicht:war ja vor allem der Begeisterung Ausdruck zu geben fur die Aussichten, die sich unserer gesamten Wissenschaft durch die P ~ a ~ c ~ s c h e Entdeckung geoffnet haben. Auch lag es mir am Herzen nach bestem Verrnogen A-achdruck zu legen auf die mit der neuen Erkenntnis folgende Erschutterung der Grundlagen der Begriffsbildung auf der nicht nur die klassische Darstellung der Physik, sondern auch unsere gewohnlichc Denkweise beruht. Eben der hierdurch gewonnenen Befreiung verdanken wir den wunderbaren Fortschritt unserer Einsicht in die Naturerscheinungen, die wir wahrend des letzten Menschenalters errungen haben ; ein Fortschritt, der alle Hoffnungen iibertrifft, die man bis vor wenigen Jahren zu hegen wagte. Die jetzige Lage der Physik ist vielleicht am besten dadurch gekennzeichnet, da13 fast alle Gedanken, die sich je in der Xaturforschung als erfolgreich erwiesen hatten, in einer gemeinsamen Harmonie zu ihrem Recht gekommen sind, ohne dabei an Fruchtbarkeit verloren zu haben. In Dankbarkeit fur die Arbeitsmoglichkeiten die er uns geschenkt hat, feiern seine Fachgenossen heute den Schopfer der Quantentheorie.

Niels Bohr in his study in the villa on Blegdamsvej.

I11 The Quantum of Action and the Description of Nature (1 929)

In the history of science there are few events which, in the brief span of a generation, have had such extraordinary consequences as Planck’s discovery of the elementaryquantum of action. Not only does this discovery, to an ever increasing degree, form the background for the ordering of our experience concerning atomic phenomena, the knowledge of which has been so amazingly extended in the last thirty years, but, at the same time, it has brought about a complete revision of the foundations underlying our description of natural phenomena. We are dealing here with an unbroken development of points of view and conceptual aids which, beginning with the fundamental works of Planck on black body radiation, has reached a temporary climax, in recent years, in the formulation of a symbolic quantum mechanics. This theory may be regarded as a natural generalization of the classical mechanics with which in beauty and self-consistency it may well be compared. This goal has not been attained, still, without a renunciation of the causal space-time mode of description that characterizes the classical physical theories which have experienced such a profound clarification through the theory of relativity. I n this respect, the quantum theory may be said to be a disappointment, for the atomic theory arose just from the attempt to ac-

93 complish such a description also in the case of phenomena which, in our immediate sense impressions, do not appear as motions of material bodies. From the very beginning, however, one was not unprepared in this domain to come upon a failure of the forms of perception adapted to our ordinary sense impressions. We know now, it is true, that the often expressed scepticism with regard to the reality of atoms was exaggerated ; for, indeed, the wonderful development of the art of experimentation has enabled us to study the effects of individual atoms. Nevertheless, the very recognition of the limited divisibility of physical processes, symbolized by the quantum of action, has justified the old doubt as to the range of our ordinary forms of perception when applied to atomic phenomena. Since, in the observation of these phenomena, we cannot neglect the interaction between the object and the instrument of observation, the question of the possibilities of observation again comes to the foreground. Thus, we meet here, in a new light, the problemof the objectivity of phenomena which has always attracted so much attention in philosophical discussion. This being the state of affairs, it is not surprising that, in all rational applications of the quantum theory, we have been concerned with essentially statistical problems. Indeed, in the original researches of Planck, it was, above all, the necessity for modifying the classical statistical mechanics which gave rise to the introduction of the quantum of action. This feature, which is characteristic of the quantum theory, is strikingly expressed in the recently renewed discussion on the nature of light and of the elementary particles of matter. Although these questions had apparently found their final solution THE QUANTTJM O F A C T I O N

94

THE ATOMIC THEORY

within the compass of the classical theories, we know now that for material particles as well as for light different conceptual pictures are necessary to account completely for the phenomena and to furnish a unique formulation of the statistical laws which govern the data of observation. The more clearly it appears that a uniform formulation of the quantum theory in classical terms is impossible, the more we admire Planck’s happy intuition in coining the term “ quantum of action ” which directly indicates a renunciation of the action principle, the central position of which in the classical description of nature he himself has emphasized on more than one occasion. This principle symbolizes, as it were, the peculiar reciprocal symmetry relation between the spacetime description and the laws of the conservation of energy and momentum, the great fruitfulness of which, already in classical physics, depends upon the fact that one may extensively apply them without following the course of the phenomena in space and time. It is this very reciprocity which has been made use of in a most pregnant way in the quantum-mechanical formalism. As a matter of fact, the quantum of action appears here only in relations in which space-time co-ordinates and momentum-energy components, which are canonically conjugate quantities in the Hamiltonian sense, enter in a symmetrical and reciprocal manner. I n addition, the analogy between optics and mechanics, which has proved to be so fruitful for the recent development of the quantum theory, depends intimately upon this reciprocity. It lies in the nature of physical observation, nevertheless, that all experience must ultimately be expressed in terms of classical concepts, neglecting the quantum of

THE QUANTUM OF ACTION

95

action. It is, therefore, an inevitable consequence of the limited applicability of the classical concepts that the results attainable by any measurement of atomic quantities are subject to an inherent limitation. A profound clarification of this question was recently accomplished with the help of the general quantum-mechanical law, formulated by Heisenberg, according to which the product of the mean errors with which two canonically conjugate mechanical quantities may be simultaneously measured can never be smaller than the quantum of action. Heisenberg has rightly compared the significance of this law of reciprocal uncertainty for estimating the self-consistency of quantum mechanics with the significance of the impossibility of transmitting signals with a velocity greater than that of light for testing the selfconsistency of the theory of relativity. In considering the well-known paradoxes which are encountered in the application of the quantum theory to atomic structure, it is essential to remember, in this connection, that the properties of atoms are always obtained by observing their reactions under collisions or under the influence of radiation, and that the above-mentioned limitation on the possibilities of measurement is directly related to the apparent contradictions which have been revealed in the discussion of the nature of light and of material particles. In order to emphasize that we are not concerned here with real contradictions, the author suggested in an earlier article the term " complementarity ". I n consideration of the above-mentioned reciprocal symmetry which occurs already in classical mechanics, perhaps the term " reciprocity " is more suitable for expressing the state of affairs with which we are dealing. At the con-

96 T H E ATOMIC THEORY clusion of the paper referred to, it was pointed out that a close connection exists between the failure of our forms of perception, which is founded on the impossibility of a strict separation of phenomena and means of observation, and the general limits of man’s capacity to create concepts, which have their roots in our differentiation between subject and object. Indeed, the epistemological and psychological questions which arise here lie perhaps outside the range of physics proper. Yet, on this special occasion, I should like to be permitted to go somewhat more deeply into these ideas. The epistemological problem under discussion may be characterized briefly as follows: For describing our mental activitv, we require, on one hand, an objectively given content to be placed in opposition to a perceiving subject, while, on the other hand, as is already implied in such an assertion, no sharp separation between object and subject can be maintained, since the perceiving subject also belongs to our mental content. From these circumstances follows not only the relative meaning of every concept, or rather of every word, the meaning depending upon our arbitrary choice of view point, but also that we must, in general, be prepared to accept the fact that a complete elucidation of one and the same object may require diverse points of view which defy a unique description. Indeed, strictly speaking, the conscious analysis of any concept stands in a relation of exclusion to its immediate application. The necessity of taking recourse to a complementary, or reciprocal, mode of description is perhaps most familiar to us from psychological problems. I n opposition to this, the feature which characterizes the so-called exact sciences is, in general,

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the attempt to attain to uniqueness by avoiding all reference to the perceiving subject. This endeavour is found most consciously, perhaps, in the mathematical symbolism which sets up for our contemplation an ideal of objectivity to the attainment of which scarcely any limits are set, so long as we remain within a self-contained field of applied logic. I n the natural sciences proper, however, there can be no question of a strictly self-contained field of application of the logical principles, since we must continually count on the appearance of new facts, the inclusion of which within the compass of our earlier experience may require a revision of our fundamental concepts. We have recently experienced such a revision in the rise of the theory of relativity which, by a profound analysis of the problem of observation, was destined to reveal the subjective character of all the concepts of classical physics. I n spite of the great demands that it makes upon our power of abstraction, the theory of relativity approaches, in a particularly high degree, the classical ideal of unity and causality in the description of nature. Above all, the conception of the objective reality of the phenomena open to observation is still rigidly maintained. As Einstein has emphasized, the assumption that any observation ultimately depends upon the coincidence in space and time of the object and the means of observation and that, therefore, anyobservation is definable independently of the reference system of the observer, is, indeed, fundamental for the whole theory of relativity. However, since the discovery of the quantum of action, we know that the classical ideal cannot be attained in the description of atomic phenomena. B

7

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I n particular, any attempt at an ordering in space-time leads to a break in the causal chain, since such an attempt is boundup with anessential exchangeof momentum and energy between the individuals and the measuring rods and clocks used for observation; and just this exchange cannot be taken into account if the measuring instruments are to fulfil their purpose. Conversely, any conclusion, based in an unambiguous manner upon the strict conservation of energy and momentum, with regard to the dynamical behaviour of the individual units obviously necessitates a complete renunciation of following their course in space and time. I n general, we may say that the suitableness of the causal space-time mode of description for the ordering of our usual experiences depends only upon the smallness of the quantum of action relative to the actions with which we are concerned in ordinary phenomena. Planck’s discovery has brought before us a situation similar to that brought about by the discovery of the finite velocity of light. Indeed, the suitability of the sharp distinction between space and time, demanded by our senses, depends entirely upon the smallness of the velocities with which we have to do in daily life compared with the velocity of light. I n fact, in the question of the causality of atomic phenomena, the reciprocal character of the results of measurements may no more be neglected than can their relativity in the question of simultaneity. In considering the resignation with regard to the desires for visualization which give our whole language its character, to which we are compelled by the situation discussed above, it is very instructive that already in simple psychological experiences we come upon

THE QUANTUM OF ACTION 99 fundamental features not only of the relativistic but also of the reciprocal view. The relativity of our perception of motion, with which we become conversant as children when travelling by ship or by train, corresponds to common-place experiences on the reciprocal character of the perception of touch. One need only remember here the sensation, often cited by psychologists, which every one has experienced when attempting to orient himself in a dark room by feeling with a stick. When the stick is held loosely, it appears to the sense of touch to be an object. When, however, it is held firmly, we lose the sensation that it is a foreign body, and the impression of touch becomes immediately localized at the point where the stick is touching the body under investigation. It would scarcely be an exaggeration to maintain, purely from psychological experiences, that the concepts of space and time by their very nature acquire a meaning only because of the possibility of neglecting the interaction with the means of measurement. On the whole, the analysis of our sense impressions discloses a remarkable independence of the psychological foundations of the concepts of space and time, on the one hand, and the conceptions of energy and momentum, based upon actions of force, on the other hand. Above all, however, this domain, as already mentioned, is distinguished by reciprocal relationships which depend upon the unity of our consciousness and which exhibit a striking similarity with the physical consequences of the quantum of action. We are thinking here of well-known characteristics of emotion and volition which are quite incapable of being represented by visualizable pictures. In particular, the apparent contrast between the continuous onward flow 7-2

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of associative thinking and the preservation of the unity of the personality exhibits a suggestive analogy with the relation between the wave description of the motions of material particles, governed by the superposition principle, and their indestructible individuality. The unavoidable influence on atomic phenomena caused by observing them here corresponds to the well-known change of the tinge of the psychological experiences which accompanies any direction of the attention to one of their various elements. It might still be permitted here briefly to refer to the relation which exists between the regularities in the domain of psychology and the problem of the causality of physical phenomena. When considering the contrast between the feeling of free will, which governs the psychic life, and the apparently uninterrupted causal chain of the accompanying physiological processes, the thought has, indeed, not eluded philosophers that we may be concerned here with an unvisualizable relation of complementarity. Thus, the opinion has often been expressed that a detailed investigation of the processes of the brain, which, although not practicable, is, nevertheless, thinkable, would reveal a causal chain that formed a unique representation of the emotional mental experience. However, such an idealized experiment now appears in a new light, since we have learned, by the discovery of the quantum of action, that a detailed causal tracing of atomic processes is impossible and that any attempt to acquire a knowledge of such processes involves a fundamentally uncontrollable interference with their course. According to the above-m.entioned view on the relation between the processes in the brain and the psychical

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experiences, we must, therefore, be prepared to accept the fact that an attempt to observe the former will bring about an essential alteration in the awareness of volition. Although, in the present case, we can be concerned only with more or less fitting analogies, yet we can hardly escape the conviction that in the facts which are revealed to us by the quantum theory and lie outside the domain of our ordinary forms of perception we have acquired a means of elucidating general philosophical problems. I hope that the special occasion will excuse a physicist for venturing into a foreign field. Above all, my purpose has been to give expression to our enthusiasm for the prospects which have been opened up for the whole of science. I n addition, it has been my desire to emphasize as strongly as possible how profoundly the new knowledge has shaken the foundations underlying the building up of concepts, on which not only the classical description of physics rests but also all our ordinary mode of thinking. It is above all to this emancipation that we owe the wonderful progress in our insight into the phenomena of nature which has been made during the last generation, a progress far exceeding all the hopes which one ventured to cherish just a few years ago. Perhaps the most distinguishing characteristic of the present position of physics is that almost all the ideas which have ever proved to be fruitful in the investigation of nature have found their right place in a common harmony without thereby having diminished their fruitfulness. I n gratitude for the possibilities of research which he has opened up before us, his colleagues celebrate to-day the creator of the quantum theory.

111. THE ATOMIC THEORY AND THE FUNDAMENTAL PRINCIPLES UNDERLYING THE DESCRIPTION OF NATURE ATOMTEORIEN OG GRUNDPRINCIPPERNE FOR NATURBESKRIVELSEN Beretning om det 18. skandinaviske Naturforskermerde i Kerbenhavn 26.-3 1. August 1929, Frederiksberg Bogtrykkeri, Copenhagen 1929, pp. 71-83 THE ATOMIC THEORY AND THE FUNDAMENTAL PRINCIPLES UNDERLYING THE DESCRIPTION OF NATURE “Atomic Theory and the Description of Nature”, Camb. Univ. Press, 1934 (reprinted 1961), pp. 102-119 Address to the 18th Meeting of Scandinavian Scientists given on 26 August 1929

See Introduction to Part 11, sect. 2 .

P A R T 11: F U R T H E R E L U C I D A T I O N O F T H E C O M P L E M E K T A R I T Y A R G U M E N T

THE ATOMIC THEORY AND THE FUNDAMENTAL PRINCIPLES UNDERLYING THE DESCRIPTION OF NATURE (1929)

Versions published in Danish, German and English

Danish: Atomteorien og Grundprincipperne f o r Naturbeskrivelsen A Beretning om det 18. skandinaviske Naturforskermode i Krzlbenhavn 26.-31. August 1929, Frederiksberg Bogtrykkeri, Copenhagen 1929, pp. 71-83 B Fys. Tidsskr. 27 (1929) 103-114 C “Atomteori og naturbeskrivelse”, J.H. Schultz Forlag, Copenhagen 1958, pp. 85-96 German: Die Atomtheorie und die Prinzipien der Naturbeschreibung D Naturwiss. 18 (24 January 1930) 73-78 E “Atomtheorie und Naturbeschreibung”, Julius Springer Verlag, Berlin 1931, pp. 67-77 English: The A tomic Theory and the Fundamental Principles underlying the Description of Nature F “Atomic Theory and the Description of Nature”, Carnb. Univ. Press, 1934 (reprinted 1961), pp. 102-119 All of these versions agree with each other - with one important exception to be mentioned below - apart from the introductory footnote which naturally has a special formulation in the Congress Report A (and is left out in the book versions). On Bohr’s reprints the misleading word “Referatet” in this footnote has been corrected (in Mrs Schultz’ handwriting) to “Foredraget”. This correction has been introduced in version B . One sentence has been completely changed from one German version to the other: D , p. 77 “Wahrend das Gefuhl der Willensfreiheit das Geistesleben beherrscht, liegt die Forderung der Kausalitat der Einordnung der Sinnesbeobachtungen zugrunde.” E , p. 76 “Wahrend die Willensfreiheit die Erlebnisforrn der Subjektivitat darstellt, ist die Kausalitat die Anschauungsforrn f u r die Einordnung der Sinneswahrnehmungen.”

P A R T 11: F U R T H E R E L U C I D A T I O N O F T H E C O M P L E M E K T A R I T Y A R G U M E U T

The Danish version has: A , p. 82 “Ligesom Viljesfriheden er Oplevelsesform for Sjzlelivet, t O r B , p. 113 Aarsagssammenhzngen betragtes som Anskuelsesform for c, p. 95 Indordningen af Sanseiagttagelserne.” The English version has: F , pp. 116-1 17 “Just as the freedom of the will is an experiential category of our psychic life, causality may be considered as a mode of perception by which we reduce our sense impressions to order.”

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Professor, Dr. Niels Bohr holdt derefter folgende Foredrag*) :

Atomteorien og Grundprincipperne for Naturbeskrivelsen. De Xaturfanomener, der fremstiller sig for vore Sanser, udviser ofte en stor Foranderlighed og Ubestandighed. For at forklare dette Forhold har man fra gammel Tid antaget, at FEnomenerne fremkommer som Fslge af en Samvirken mellem et stort Antal Smaadele, Atomerne, der i sig selv er uforanderlige og bestandige, men paa Grund af deres Lidenhed unddrager sig den umiddelbare Sanseiagttagelse. Rent bortset *) Nogle af de nedenfor medtagne Enkeltheder maatte p a a Grund af den begmnsede Tid ved Aabningsmodet udelades af Referatet.

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fra det principielle S p ~ r g s m a a l om , vi er berettiget ti1 at kreve anskuelige Billeder paa Omraader, der ligger udenfor vore Sansers Rekkevidde, maatte Atomteorien oprindelig have og antages altid at ville bevare en hypotetisk Karakter, idet man mente, at det efter Sagens Natur aldrig vilde v e r e muligt at opnaa et direkte Indblik i Atomernes Verden. Det er imidlertid gaaet her som paa saa mange Omraader, at G r m x e n for Iagttagelsesmulighederne stadig h a r forskudt sig som Falge af H j d pemidlernes Udvikling. Vi behover blot at tEnke paa det Indblik i Verdensbygningen, som vi h a r vundet ved Kikkertens og Spektroskopets H j d p , eller det Indblik i Organismernes finere Strukturer, som Mikroskopet h a r skenket 0 s . Den fysiske Eksperimenterkunsts overordentlige Udvikling h a r d a ogs a a gjort 0s bekendt med et stort Antal Faenomener, der paa direkte Maade belerer 0s om Atomernes Bevegelser og deres Antal. Vi kender endog Fenomener, der med Sikkerhed tOr antages at hidrare f r a Virkningen af et enkelt Atom, ja af en Del af et saadant. Samtidig med at enhver Tvivl om Atomernes Realitet e r bortjaget, og vi tilmed h a r vundet et n0je Kendskab ti1 Atomernes indre Bygning, e r vi imidlertid paa lererig Maade blevet paamindet om vore Anskuelsesformers naturlige Begrensning. Det er denne ejendommelige Situation, som jeg her skal fors0ge at skildre. Tiden tillader mig ikke i Enkeltheder at beskrive den overordentlige Udridelse af vort Erfaringsomraade, som det her drejer sig om, og som kendetegnes ved Opdagelserne af Katodestraaler, Rantgenstraaler og radioaktive Stoffer. Jeg skal blot minde om Grundtrekkene i det Billede af Atomet. vi derigennem h a r erhvervet. Som en felles Byggesten i alle Stoffers Atomer indgaar de saakaldte Elektroner, negativt elektriske, lette Smaadele, der fastholdes i Atomet ved Tiltrekningen fra den langt tungere, positivt elektriske Atomkerne. Kernens Masse er bestemmende for Stoffets Atomvegt, men har iovrigt kun ringe Indflydelse paa Stoffernes Egenskaber, der i forste Linie bestemmes af Kernens elektriske Ladning, som paa Fortegnet naer altid er et helt Antal Gange Elektronens Ladning. Dctte hele Tal, der altsaa bestemmer Antallet af Elektroner i det neutrale Atom, har nu vist sig netop at vaxe lig Atomnummeret, der angiver vedkommende Grundstofs Plads i det saakaldte naturlige System, hvori Stoffernes ejen-

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dommelige Slaegtskabsforhold, hvad deres fysiske og kemiske Egenskaber angaar, h a r fundet saa traeffende Udtryk. Denne Tydning af Atomnummeret t O r siges at betegne et vigtigt Skridt ti1 Lmsningen af en Opgave, der lenge h ar h0rt ti1 Naturvidenskabens dristigste Dromme, at opbygge en Forstaaelse af Naturens Lovmmsigheder paa Betragtninger af rene Tal. Ved den omhandlede Udvikling er Atomteoriens Grundforestillinger jo unegtelig undergaaet en vis Forandring. I Stedet f o r Antagelsen om Atomernes Uforanderlighed treder nu Antagelsen om Atomdelenes Bestandighed. I Saxdeleshed beror Grundstoffernes store Bestandighed paa den Omstaendighed, at Atomkernen ikke bermes ved saedvanlige fysiske og kemiske Indgreb, der kun medfmer ,!Endringer i den Maade, hvorpaa Elektronerne er bundne i Atomet. Medens alle Erfaringer bestyrker Antagelsen om Elektronernes Uforanderlighed, ved vi imidlertid, at Atomkernernes Bestandighed h a r en mere begmenset Karakter. I de radioaktive Stoffers ejendommelige Straaling har vi jo netop Vidnesbyrd om en Ssnderdeling af Atomkerner, hvorved Elektroner eller positivt ladede Kernedele udslynges med stor Energi. Efter alt at dnmme finder disse SDnderdelinger Sted uden ydre Foranledning. Har vi et vistAntal Radiumatomer, kan vi kun sige, at der vil v ere l disse sonderen bestemt Sandsynlighed for, at en vis B r ~ k d e af deles i det naeste Sekund. Ti1 den ejendommelige Svigten af Aarsagsbeskrivelsen, som vi her mader, og som h a r n0je Forbindelse med Grundtraek i vor Beskrivelse af Atomfenomenerne. skal vi i det folgende komnie tilbage. Her skal jeg blot endnu minde om Rutherfords vigtige Opdagelse, at en Sonderdeling af Atomkernen under saerlige OmstEndigheder kan frembringes ved ydre Paavirkning. Som alle ved, lykkedes det ham at vise, at Atomkernerne af visse ellers bestandige Grundstoffer kan smderdeles, naar de rammes af de Smaadele, der udslynges fra de radioaktive Atomkerner. Dette f ~ r s t e Tilfelde af en af Mennesker reguleret Grundstofforvandling tsr siges at danne Skel i Naturvidenskabens Historie og aabne et helt nyt Felt for Fysikken, nemlig Udforskningen af Atomkernernes Indre. Jeg skal dog ikke udmale disse Perspektiver naermere, men nsjes med at omtale den almindelige Belming, som Bestraebelserne paa at g0re Rede for Grundstoffernes saedvanlige fysiske og kemiske Egenskaber ud fra de nevnte Forestillinger om Atombygningen har bragt.

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I forste 0jeblik kunde det se ud, som om den omhandlede Opgaves L ~ s n i n gvilde vaere ganske ligetil. Det Billede af Atomet, som det drejer sig om. viser 0 s et lille mekanisk System, der endda i visse Hovedtraek minder om vort Solsystem, ved hvis Beskrivelse Mekanikken h a r fejret saa store Triumfer og givet 0s et Hovedeksempel paa Opfyldelsen af Aarsagskravet i den saedvanlige Fysik. F r a Kendskabet ti1 Planeternes ojeblikkelige Steder og Bevegelser kan vi jo med tilsyneladende ubegraenset No jagtighed beregne deres Steder og Bevaegelser ti1 senere Tider. Muligheden for ved en saadan mekanisk Beskrivelse at vaelge en vilkaarlig Begyndelsestilstand bereder imidlertid Spmgsmaalet om Atombygningen store Vanskeligheder. Hvis vi skal regne rned en uendelig Mangfoldighed af kontinuerligt varierede Bevaegelsestilstande af Atomerne, kommer vi nemlig i aabenbar Modstrid med Erfaringerne om Stoffernes bestemte Egenskaber. Man kunde maaske tro, at Stoffernes Egenskaber ikke bringer 0s umiddelbart Bud om de enkelte Atomers Opforsel, men at vi altid kun havde at g m e med statistiske Lovmaessigheder gaeldende for mange Atomers Gennemsnitsforhold. I den mekaniske Varmeteori, der ikke alene tillader 0 s at g me Rede for Varmelaerens Hovedsaetninger, men ogsaa at forstaa mange af Stoffernes almindelige Egenskaber, h ar vi netop et velkendt Eksempel paa statistiske mekaniske Betragtningers Frugtbarhed indenfor Atomteorien. Grundstofferne besidder imidlertid andre Egenskaber, der tillader mere direkte Siutninger om Atomdelenes Bevsegelsestilstande. Fremfor alt maa Beskaffenheden af det Lys, som Stofferne under Omstrendigheder udsender, og som er saeregent for hvert Grundstof, antages at vaere vaesentlig betinget af Forholdene i det enkelte Atom. Ligesom Radiobolgerne fortaeller 0s om Arten af de elektriske Svingninger i Afsenderstationens Xpparater, maatte efter den elektromagnetiske Lysteori Svingningstallene for de enkelte Linier i Grundstoffernes karakteristiske Spektre forventes at give os Oplysning om Elektronbevaegelserne i htomet. Fo r Tydningen af disse Oplysninger giver Mekanikken 0s imidlertid intet tilstrekkeligt Grundlag; ja, vi kan paa Grund af den naevnte Variationsmulighed af de mekaniske Bevsegelsestilstande ikke engang forstaa Fremkomsten af skarpe Spektrallinier. Det manglende Trek i Naturbeskrivelsen, som Redegorelsen

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for Atomernes Forhold aabenbart krever, h a r v i imidlertid faaet gennem Plancks Opdagelse af det saakaldte Virkningskvantum. Udgangspunktet f o r denne Opdagelse var Varmestraalingsfaenomcnerne, hvis almene, af Stoffernes specielle Egenskaber uafhaengige Karakter tilbad en afgsrende Prsve paa Rzkkevidden af den mekaniske Varmeteori og den elektromagnetiske Straalingsteori. Det var netop disse Teoriers Svigten ved Redegsrelsen for Varmestraalingsloven, der ledte Planck ti1 Erkendelsen af et hidtil upaaagtet, almindeligt Traek hos Naturlovene, der vel ikke gar sig umiddelbart g d d en d e ved Beskrivelsen af de saedvanlige fysiske Fenomener, men har medfsrt en he1 Omveltning i vor Redegsrelse for saadanne Forhold, der afhwnger af enkelte Atomer. I Modsaetning ti1 det Krav om Kontinuitet, der er den saedvanlige Naturbeskrivelses Kendemaerke, krever saaledes Virkningskvantets Udelelighed Indfsrelsen af et vaxentligt Element af Diskontinuitet i Beskrivelsen af Atomfenomenerne. Vanskeligheden ved at forbinde den nye Erkendelse med vor tilvante fysiske Forestillingskreds fremgik isaer gennem den af Einstein i Forbindelse med Forklaringen af den photoelektriske Effekt rejste, fornyede Diskussion af Spargsmaalet om Lysets Xatur, der efter alle tidligere Erfaringer at damme havde fundet en fuldt tilfredsstillende Besvarelse indenfor den elektromagnetiske Teoris Rammer. Den Situation, vi her mader, kendetegnes ved, at vi tilsyneladende tvinges ti1 at skulle velge mellem to modstridende Billeder af Lysforplantningen, paa den ene Side Forestillingen om Lysbalger, paa den anden Side Lyskvanteteoriens korpuskulare Opfattelse. der hver for sig giver Udtryk for vmentlige Sider af Erfaringerne. Som vi i det falgende skal se, er dette tilsyneladende Dilemma Udtryk for en ejendommelig, med Virkningskvantet sammenhzngende Begraensning af vore Anskuelsesformer, som den naermere Analyse af de fysiske Grundbegrebers Anvendelighed ved Beskrivelsen af Atomfwnomenerne bringer for Dagen. Det var ogsaa kun ved bevidst at resignere paa de saedvanlige Fordringer ti1 Bnskuelighed og Aarsagssammenhaeng, at det lykkedes at frugtbargnre Plancks Opdagelse ved Redegarelsen for Stoffernes Egenskaber paa Grundlag af Kendskabet ti1 Atomernes Byggestene. Med Antagelsen om Virkningskvantets Udelelighed som Udgangspunkt foreslog saa-

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ledes Foredragsholderen, at enhver a n d r i n g i et Atoms Tilstand betragtes som en individuel, ikke nErmere beskrivelig Proces, hvorved Atomet g a a r over fra en saakaldt s t a t i o n m Tilstand ti1 en anden. Efter denne Opfattelse b e l e r e r Stoffernes Spektre os ikke umiddelbart om Atomdelenes Bcvegelser. men hver enkelt Spektrallinie knyttes ti1 en Overgangsproces mellem to stationere Tilstande, saaledes at Produktet af Svingningstal og Virkningskvantum angiver Atomets Energiamdring ved Processen. Det viste sig p a a denne Maade muligt at opnaa e n simpel Tydning af de almindelige spektroskopiske L o v m ~ s sigheder, som det v a r lykkedes Balmer, Rydberg og Ritz at udlede fra det eksperimentelle Materiale. Den paageldende Opfattelse af Spektrenes Oprindelse fandt ogsaa en direkte S t d t e i de bekendte Fors0g af Franck og Hertz over Sammenst0d mellem Atomer og frie Elektroner. D e Energimxngder. der kan omsettes ved saadanne S a m m e n s t ~ d ,viste sig netop at w a r e ti1 de fra Spektrene heregnede Energiforslielle mellem den stationere Tilstand, som Atomet befinder sig i for Stodet. og e n af de s t a t i o n e r e Tilstande, hvori det kan befinde sig efter S a m m e n s t ~ d e t . I det hele tilbyder den omhandlcde Opfattelse e n modsigelsesfri Indordning af Erfaringsmaterialet. men Modsigelsesfriheden e r k u n naaet p a a Bekostning af den n e r m e r e Beskrivelse af de enkelte Overgangsprocesser. Vi er h e r s a a langt fra en Aarsagsbeskrivelse. at et Atom i en s t a t i o n w Tilstand endda i Almindelighed kan siges at v a r e stillet overfor et frit Valg mellem forskellige Overgangsmuligheder ti1 andre stationare Tilstande. F o r Forekomsten af de enkelte Processer k a n der efter Sagens S a t u r kun anstilles Sandsynlighedsbetragtninger, der, som Einstein h a r fremhaevet, udviser e n n0je Lighed med de Forhold, der g d d e r for de spontane radioaktive Smderdelinger. Et for det omhandlede Angreb paa Atombygningsproblemet ejendommeligt T r e k e r den vidtgaaende B r u g af hele Tal, der netop i de empiriske LovmEssigheder for Spektrene spiller e n fremtredende Rolle. Foruden paa Xtomnummeret beror de stationaxe Tilstandes Klassifikation saaledes paa de saakaldte Kvantetal, ti1 hvis Systematik Sommerfeld h a r givet et s a a vigtigt Bidrag. I det hele h a r de omhandlede Synspunkter i betydeligt Omfang tilladt at gore Rede for Grundstoffernes Egenskaber og Slegtskabsforhold p a a Grundlag

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af vore almindelige Forestillinger om Atombygningen. Det kunde maaske undre, at en saadan Redegmelse ti1 Trods for den store Afvigelse fra de sEdvanlige fysiske Forestillinger, som det her drejer sig om, h ar veret mulig, da vort hele Kendskab ti1 Atomernes Byggestene dog liviler paa disse Forestillinger. Uet er jo klart, at enhver Benyttelse af Begreber som Masse og Elektricitetsladning er ensbetydende med Paaberaabelsen af mekaniske og elektrodynamiske Lovmzssigheder. Et Holdepunkt for Xyttiggmelsen af saadanne Begreber ogsaa udenfor de klassiske Teoriers Gyldighedsomraade har vi imidlertid fundet i Fordringen om den kvanteteoretiske Beskrivelses umiddelbare Tilknytning ti1 den szedvanlige Beskrivelsesmaade i det Grsenseomraade, hvor vi kan se bort fra Virkningskvantet. Bestrzbelsen for indenfor Kvanteteorien at gore Brug af ethvert klassisk Begreb i en Omtydning, ber uden at v e r e i Strid med Postulatet om Virkningskvantets Udelelighed imwdekommer denne E’ordring, fandt sit Udtryk i det saakaldte Korrespondensprincip. Gennemforelsen af en streng korrespondensmzssig Beskrivelse h ar dog krevet Overvindelsen af mange Vanskeligheder, og det er f ~ r s t i de seneste -4ar lykkedes at udformc en i sig sammenhsengende Kvantemekanik, der kan opfattes som en naturlig Almindeligg ~ r e l s eaf den lrlassiske Mekanik, hvis sammenhengende Aarsagsbeskrivelse den erstatter med en principielt statistisk Beskrivelsesmaade. Et afgwrende Skridt henimod dette Maal blev taget af den unge tyske Fysiker, W e r n e r Heisenberg, der viste, hvorledea de sEdvanlige Bevzgelsesforestillinger paa konsekvent Maade kan erstattes med en formel Benyttelse af den klassiske Mekaniks Bevegelseslove, ved hvilken Virkningskvantet kun optrzeder i visse Regler for Regningen med de Symboler, der er:tatter de mekaniske Starrelser. Dette sindrige Angreb paa Kvanteteoriens Problem stiller imidlertid store Fordringer ti1 vor Abstraktionsevne, og Fundet af nye Hjzlpemidler, der ti1 Trods for deres formelle Karakter i h ~ j e r eGrad imodekommer vort Krav ti1 Anskuelighed, h a r derfor vaxet af uvurderlig Betydning for Kvantemekanikkens Udformning og Afklaring. Jeg sigter her ti1 de af L o u i s d e B r o g l i e i n d f ~ r t eForestillinger om Materieb~lger,som S c h r o d i n g e r h a r forstaaet at frugtbargwre med saa stort et Held navnlig i Forbindelse med Fore-

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stillingen om stationaere Tilstande, hvis Kvantetal tydes som Antallet af Knuder i de staaende Bslger, hvormed disse Tilstande symboliseres. De Broglies Udgangspunkt var den allerede for Udformningen af den klassiske Mekanik saa betydningsfulde Lighed mellem Lovene for Lysets Forplantning og f o r materielle Legemers Bevaegelser. I Virkeligheden danner Bdgemekanikken et naturligt Modstykke ti1 Einsteins f0rnaevnte Lyskvanteteori. Ligesom ved denne drejer det sig her ikke om en i sig afsluttet Forestillingskreds, men, som isax betonet af Born om et Hjaelpemiddel ti1 Formuleringen af de statistiske Love, der behersker Atomfenomenerne. Vel er den Bekreftelse, som Materiebslgeforestillingen har fundet ved Forssg over Tilbagekastningen af Elektroner f r a Metalkrystaller, paa sin Vis lige s aa afgorende som de eksperimentelle Vidnesbyrd for Bslgeopfattelsen af Lysforplantningen. Dog maa vi betaenke, at Materiebslgernes L4nvendelsesomraade indskraenker sig ti1 de Fenomener, for hvis Beskrivelse en Hensyntagen ti1 Virkningskvantet er vesentlig, og som derfor ligger udenfor det Omraade, hvor der kan vaere Tale om at gennemfsre en Aarsagsbeskrivelse svarende ti1 vore saedvanlige Anskuelsesformer, og hvor vi kan tillaegge Ord som Materiens og Lysets Natur en Mening i saedvanlig Forstand. Ved Kvantemekanikkens Hjaelp behersker vi et udstrakt Erfaringsomraade og kan navnlig gsre Rede for et stort Antal Enkeltheder vedrsrende Stoffernes fysiske og kemiske Egenskaber. 1 den seneste Tid h ar det tilmed veret muligt at opnaa en Tydning af de radioaktive Ssnderdelinger, hvorved de empiriske Sandsynlighedslove, der g d d e r for disse Processer, fremtraeder som en umiddelbar F d g e af den for Kvanteteorien ejendommelige statistiske Beskrivelsesmaade. Denne Forklaring giver et udmaerket Eksempel saavel paa Bdgeforestillingernes Y deevne som paa deres formelle Karakter. P a a den ene Side h a r vi her at g me med en umiddelbar Tilknytning ti1 de saedvanlige Bevaegelsesforestillinger, idet Banerne af de fra Atomkernerne udslyngede Dele paa Grund af disses store Energi kan direkte iagttages. Paa den anden Side lader de mdvanlige mekaniske Forestillinger 0s helt i Stikken ved Beskrivelsen af S~tnderdelingensForleb, da det Kraftfelt, der omgiver Atomkernen, efter disse Forestillinger

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vilde forhindre Partiklerne i at slippe bort fra Kcrnen. Efter Kvantemekanikken stiller Sagen sig imidlertid anderledes, idet liraftfeltet vel danner en Hindring, der i Hovedsagen holder Materiebdgerne tilbage, men dog tillader en ringe Del at sive igennem. Den Del af Bslgerne, der paa denne Maade indenfor en vis Tid undslipper, giver 0s et Maal for Sandsynligheden for Atomkernens Smderdeling i den givne Tid. Vanskeligheden ved at tale om hlateriens Xatur uden at tage det f O r n m n t e Forbehold kunde vel nEppe stilles i grellere Belysning. Ved Lyskvanteforestillingen drejer det sig om et lignende Forhold mellem vore anskuelige Hjdpemidler og Beregningen af Sandsynligheden for Forekomsten af de iagttagelige Lysvirkninger. Svarende ti1 de klassiske elektromagnetiske Forestillinger kan vi imidlertid ikke tilskrive Lyset en egentlig materiel Natur, idet Lysvirkningernes Iagttagelse altid beror paa en Overforelse af Energi og Bevzegelsesmengde ti1 Materiens Smaadele. Lyskvanteforestillingens haandgribelige Indhold er indskrenket ti1 det Regnskab, den hjaelper 0s ti1 at f0re med Energiens og Bevaegelsesmaengdens Bevarelse. Det er overhovedet et af Kvantemekanikkens ejendommeligste Traek, at det uagtet de klassiske mekaniske og elektromagnetiske Forestillingers Begramsning er muligt at opretholde Bevarelsessaetningerne for Energi og Bevaegelsesmaengde. Disse Saetninger danner i visse Henseender et fuldstaendigt Modstykke ti1 den for Atomteorien ti1 Grund liggende Antagelse om de materielle Smaadeles Bestandighed, der ti1 Trods for Opgivelsen af Bevaegelsesforestillingerne strengt opretholdes i Kvanteteorien. I Lighed med den klassiske Mekanik gOr Kvantemekanikken Krav paa at give en udt~mmendeRedeg~lrelsefor de Fenomener, der falder indenfor dens Gyldighedsomraade. Uundgaaeligheden af en principielt statistisk Beskrivelsesmaade for Atomfaenomenernes Vedkommende fremgaar nemlig af en naermere Undersergelse af de Oplysninger, vi ved direkte Maalinger kan skaffe 0s om disse Fznomener, og den Mening vi i denne Forbindelse kan tillaegge Rnvendelsen af de fysiske Grundbegreber. P a a den ene Side maa vi betenke, at disse Begrebers Betydning helt og holdent er knyttet ti1 de szdvanlige fysiske Forestillinger. Saaledes h ar enhver Hentydning ti1 Rum-Tidsforhold Elementarpartiklernes Bestandighed ti1

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Forudsetning, ligesom Setningerne om Energiens og Bevegelsesmendgens Bevarelse ligger ti1 Grund for enhver Brug af Energi- og Bevegelsesmaengdebegrebet. P a a den anden Side betyder Postulatet om Virkningskvantets Udelelighed et for de klassiske Forestillinger fuldstendig fremmedartet Element, der ved Maalinger ikke alene forlanger en endelig Vekselvirkning mellem Genstand og Maalemiddel, men endda krever et aabent Spillerum i vort Hegnskab med denne Vekselvirkning. Som Fslge af denne Sagernes Stilling fordrer enhver Maaling. som tager Sigte paa en lndordning af Elementarpartiklerne i Tid og Rum, en Opgivelse af et n0je Regnskab med Energi- og Bev=gelsesmengdeoms&ningen mellem Partiklerne og de som Henf0relsessystem benyttede Maalestokke og Uhre. Ligeledes fordrer enhver Bestemmelse af Smaadelenes Energi og Bevegelsesmaingde en Opgivelse af deres nsje Forfalgelse i Tid og Rum. I begge Tilfelde er den Paaberaabelse af klassiske Forestillinger, som Maalingens VEesen forlanger, altsaa paa Forhaand ensbetydende med et Afkald paa en streng Aarsagsbeskrivelse. Saadanne Betragtninger fsrer umiddelbart ti1 de af Heisenberg opstillede reciproke Ubestemthedsrelationer, som han h a r lagt ti1 Grund for en indgaaende Undersagelse af Kvantemekanikkens Modsigelsesfrihed. Den principielle Ubestemthed, v i her msder, tar, som Foredragsholderen har paavist, anses for et direkte Udtryk for den absolute Begrensning af anskuelige Forestillingers Anvendelse ved Beskrivelse af Atomfanomenerne, som fremtmder i det tilsyneladende Dilemma, vi stilles overfor ved Sporgsmaalet om Lysets og Materiens Natur. Den Resignation med Hensyn ti1 Ansltuelighed og Aarsagsbeskrivelse, som vi saaledes tvinges ti1 ved Redegarelsen for Atomfenomenerne, kunde maaske opfattes som en Skuffelse af de Forhaabninger, som var Atomforestillingernes Udgangspunkt. Ikke desto mindre maa vi fra Atomteoriens nuvzrende Standpunkt betragte selve denne Resignation som et v z sentligt Led i Fremskridtet af vor Erkendelse. Der er jo ikke Tale om, at de almindelige Grundprincipper for Naturvidenskaben h a r svigtet 0s indenfor det Omraade, hvor vi med Rette kunde regne med deres Stotte. Opdagelsen af Virkningskvantet belerer 0s imidlertid ikke alene om en naturlig Gren s e for den klassiske Fysik, men vi stilles overfor en i Naturvidenskabm

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hidtil ukendt Situation, idet vi i ny Belysning m ~ d e rdet gamle, filosofiske Spargsmaal om Faenomenernes objektiveEksistens uafhaengig af vore Iagttagelser. Som vi har set, k m v e r jo enhver Iagttagelse et Indgreb i Faenomenernes Forlob, der efter sin Art berover 0s Grundlaget for Aarsagsbeskrivelsen. Den Graense, der saaledes af Naturen selv er sat for Muligheden af at tale om selvstaendige Fzenomener, finder efter alt at damme netop Udtryk i Kvantemekanikkens Formulering. Dette maa dog ikke opfattes som en Hindring for videre Fremskridt. v i maa blot vaere forberedt paa Nodvendigheden af stadig videregaaende Abstraktion fra vore tilvante Krav ti1 Naturbeskrivelsens umiddelbare Anskuelighed. Nye Overraskelser kan vi vel i s m vente fra det Omraade, hvor Kvanteteorien mades med Relativitetsteorien og hvor ulaste Vanskeligheder endnu staar hindrende i Vejen for den fuldstendige Sammensmeltning af den Udvidelse af vor Erkendelse og vole Hjaelpemidler ti1 Redegarelse for Xaturfaenomenerne, som disse Teorier har bragt. Selv om det kun er i Foredragets Slutning, er jeg glad for at faa Lejlighed ti1 at fremhaeve den store Betydning, som den af Einstein skabte Relativitetsteori h a r haft for Fysikkens nyere Udvikling netop med Henblik paa vor F r i g ~ r e l s efra -inskuelighedskravet. A€ Relativitetsteorien h a r vi laert, at Hensigtsmsessigheden af den skarpe, af vore Sanser krsevede Xdskillelse mellem Rum og Tid kun beror derpaa, at de saedyanligt forekommende Hastigheder er smaa i Forhold ti1 Lysets Hastighed. I Lighed hermed kan vi sige, at Plancks Opdagelse h a r fart ti1 den Erkendelse, at det kun er Virkningskvantets Lidenhed i Sammenligning med de Virkninger, vi har at gere med ved de saedvanlige Faenomener, der betinger Hensigtsmmsigheden af vor hele af Aarsagskravet praegede Indstilling. Medens vi i Relativitetsteorien paamindes om alle fysiske FEnomeners suhjektive, af Iagttagerens Standpunkt vaesentligt afhengige Karakter, tvinger den af Kvanteteorien klarlagte Sammenksedning mellem Atomfaenomerne og deres Iagttagelse 0s ved Redegmelsen for disse Faenomener ti1 en lignende Forsigtighed i Brugen af vore Udtryksmidler som ved psykologiske Yroblemcr, hvor vi stadig stilles overfor Vanskeligheden ved at afgraense det objektive Indhold. Uden at udsaette mig for den Misforstaaelse, at det var Hensigten at indfme en Mystik, der er uforenelig med Naturvidenskabens Aand, tar jeg maaske G

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i drnnp Forbindelse minde om den ejendommelige Parallel, s o m den fornyede Diskussion om Aarsagssetningens Gyldighed danner ti1 den f r a de seldste Tider fortsatte Diskussion om Viljens Frihed. Ligesom Viljesfriheden er Oplek elsesform for Sjelelivet. t O r Aarsagssammenhengen betragtes som Anskuelsesform f o r Indordningen af Sanseiagttagelserne. Samtidig drejer det sig paa begge Omraader om Idealisnlioiier, hvis naturlige Begrensning kan gares ti1 Genstand for Undersogelse, og som betinger hinanden i den Forstand, at Viljesf~lelseog Aarsagskrav er lige uundvserlige Elementer i Forholdet mellem Subjekt og Objekt, som er Erkendelsesproblemets Kerne. FOr jeg slutter, ligger det nier ved et saadant Fadlesmode af Naturforskere at berare Spargsmaalet om, hvad den beskrevne seneste Udvikling af vort Kendskab ti1 Atomfenomenerne kan l e r e 0s om de Problemer, som de leiende Organismer frembyder. Selv om det vel endnu ikke er muligt at give noget fyldigt S r a r paa dette Spmgsmaal, turde vi allerede skimte en \-is Sammenheng mellem disse Problemer og Tivanteteoriens Forestillingskreds. Et forste Fingerpeg i den Retning finder \ i i den Omstzadighed, at den ti1 Grund for Sanseindtrykkene liggende Vekselyirkning mellem Organismerne og Omverdenen i det mindste under Omstamdigheder kan blive saa ringe, at 1 i iiiermer os 1-irkningskvantet. Som det ofte er hemerket. er saaledes ganske faa Lyskvanter tilstrsekkelige ti1 at frembringe Synsindtryk. Vi ser altsaa, at Organismens Behoi. hvad Selvstasdighed og Fslsomhed angaar, her er tilfredsstillet ti1 den yderste med Naturlovene forenelige Grsense, og vi maa viere forberedt paa at made samme Forhold ogsaa paa andre Punkter af a f g ~ r e n d eBetydning for den biologiske Problemstilling. Men dersom de paagseldende fysiologiske Fsenomener frembyder en ti1 den omhandlede Grrrnse udviklet Forfining, betyder det jo, at vi samtidig n e r m e r 0s Grensen for deres entydige Beskrivelse ved Hjaelp af vore siedvanlige anskuelige Forestillinger. Dette staar ingenlunde i Modstrid ti1 den Kendsgerning, at de levende Organismer i udstrakt Grad stiller 0 s Problenier, der falder indenfor vore Bnskuelsesformers Rakkevidde, og som h a r afgivet saa frugtbart et Anvendelsesomraade for fysiskt! og kemiske Synspunkter. Heller ikke ser vi nogen umiddelbar GrEnse for disse Syns-

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punkters Anvendelse. Lige s a a lidt som vi i Princippet k a n skelne mellem Stromningen i et Vandror og Blodstrommen i Aarerne, lige s a a lidt tor vi p a a Forhaand vente nogen dybere principiel Forskel mellem Sanseindtrykkenes Forplantning i Nerverne og Elektricitetsledningen i e n hletaltraad. Vel gselder det for alle saadanne Fsenomener, at en i Enkeltheder gaaende Redegmelse forer 0s ind paa A4tomteoriens Omraade. ja for Elektricitetsledningens Vedkommende h a r man netop i de sidste A a r erkendt, a t fmst den for Kvanteteorien ejendommelige Begraensning af de anskuelige BevEgelsesforestillinger tillader 0s at begribe, at Elektronerne k a n slippe frem mellem Metalatomerne i Traaden. F o r disse F m o mener g a l d e r det imidlertid, at en saadan Uddybelse af Beskrivelsen ikke e r nodvendig f or R e d e g ~ r e l s e naf de naermest i Betragtning kommende Virkninger. Hvad de dybere biologiske Problemer angaar, hvor det drejer sig om Organismens Frihed og Tilpasningsevne i dens Reaktion overfor ydre P a a virkning. maa vi imidlertid regne med. at Erkendelsen af en storre S a m m e n h s n g k a n krseve en Hensyntagen ti1 de samme Forhold, der betinger Aarsagsbeskrivelsens Begrsensning ved A4tomf~nonienerne.Iovrigt bor vel allerede den Kendsgerning. at Bevidsthed. som vi kender den, e r uadskillelig knyttet ti1 L i v , forberede 0s paa, at selve Grundproblemet om Grsensen mellcm levende og dodt kan unddrage sig en Forstaaelse i dette Ords ssedvanlige Forstand. Som Undskyldning for, at en Fysiker berEirer saadanne Emner, t0r maaske gselde, at den i Fysikken foreliggende, nye Situation p a a s a a eftertrykkelig Maade minder 0 s om den gamle Sandhed, at vi saavel er Tilskuere som Deltagere i Tilvaxelsens store Skuespil. Efter Foredraget, der modtoges med stserkt Bifald af Forsamlingen, bragte Formanden Professor Bohr en varm Tak og sluttede derefter M ~ d e tmed e n Anmodning ti1 Sektionerne om at konstituere sig.

IV The Atomic Theory and the Fundamental Principles underlying the Description of Nature (1929)

Natural phenomena, as experienced through the medium of our senses, often appear to be extremely variable and unstable. To explain this, it has been assumed, since early times, that the phenomena arise from the combined action and interplay of a large number of minute particles, the so-called atoms, which are themselves unchangeable and stable, but which, owing to their smallness, escape an immediate perception. Quite apart from the fundamental question of whether we are justified in demanding visualizable pictures in fields which lie outside the reach of our senses, the atomic theory originally was of necessity of a hypothetical character ; and, since it was believed that a direct insight into the world of atoms would, from the very nature of the matter, never be possible, one had to assume that the atomic theory would always retain this character. However, what has happened in so many other fields has happened also here ; because of the development of observational technique, the limit of possible observations has continually been shifted. We need only think of the insight into the structure of the universe which we have gained by the aid of the telescope and the spectroscope, or of the knowledge of the finer structure of organisms which we owe to the microscope. Similarly, the extraordinary development in the methods of experimental physics has made

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known to us a large number of phenomena which in a direct way inform us of the motions of atoms and of their number. We are aware even of phenomena which with certainty may be assumed to arise from the action of a single atom, or even of a part of an atom. However, at the same time as every doubt regarding the reality of atoms has been removed and as we have gained a detailed knowledge even of the inner structure of atoms, we have been reminded in an instructive manner of the natural limitation of our forms of perception. It is this peculiar situation which I shall attempt to portray here. Time does not permit of my describing in detail the extraordinary extension of our experience, here dealt with, which is characterized by the discoveries of cathode rays, Rontgen rays, and the radioactive substances. I shall merely remind you of the main features of the picture of the atom which we have gained through these discoveries. Negatively charged particles, the socalled electrons, which are held within the atom by the attraction of a much heavier positively charged atomic nucleus, enter as common building stones in all atoms. The mass of the nucleus determines the atomic weight of the element but has otherwise only a slight influence on the properties of the substance, these depending primarily on the electric charge of the nucleus which, apart from the sign, is always an integral multiple of the charge of an electron. Now, this whole number, which determines how many electrons are present in the neutral atom, has turned out to be just the atomic number that gives the place of the element in the so-called natural system, in which the peculiar relationships of the elements as regards their physical and chemical properties are so appropriately expressed. This interpretation of the

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atomic number may be said to signify an important step towards the solution of a problem which for a long time has been one of the boldest dreams of natural science, namely, to build up an understanding of the regularities of nature upon the consideration of pure numbers. The development mentioned above has, to be sure, produced a certain change in the fundamental concepts of the atomic theory. Instead of assuming that the atoms are unchangeable, it is now assumed that it is the parts of the atoms which are constant. I n particular, the great stability of the elements depends upon the fact that the atomic nucleus is not affected by the ordinary physical and chemical influences which produce changes only in the binding of the electrons within the atom. While all our experience strengthens the assumption of the permanence of electrons, we know, however, that the stability of atomic nuclei is of a more limited character. Indeed, the peculiar radiations from the radioactive elements provide us with direct evidence of a disruption of atomic nuclei, whereby electrons or positively charged nuclear particles are ejected with great energy. These disintegrations, so far as we are able to judge from all evidence, take place without any external cause. If we have a given number of radium atoms, we can merely say that there is a definite probability that a certain fraction of them will break down during the next second. We shall return later to this peculiar failure of the causal mode of description which we come upon here and which is closely connected with fundamental features of our description of atomic phenomena. Here, I shall call to mind only the important discovery of Rutherford that a disruption of atomic nuclei may, under certain

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circumstances, be brought about by external influence. As we all know, he succeeded in showing that the nuclei of certain, otherwise stable, elements may be split up when hit by the particles expelled from the radioactive nuclei. This first example of a transmutation of an element, regulated by man, may be said to mark an epoch in the history of natural science and to open up an entirely new field of physics, namely, the exploration of the interior of atomic nuclei. However, I shall not dwell upon the prospects opened up by this new field, but shall confine myself to discussing the general information that we have gained through our endeavours to account for the ordinary physical and chemical properties of the elements on the basis of the conceptions of atomic structure mentioned above. At first glance, it might appear that the solution of the problem considered would be quite simple. The picture of the atom with which we are dealing is that of a small mechanical system which even resembles in certain main features our own solar system, in the description of which mechanics has won such great triumphs and has given us a principal example of the fulfilment of the claim of causality in ordinary physics. Indeed, from a knowledge of the instantaneous positions and motions of the planets, we can calculate, with apparently unlimited accuracy, their positions and motions at any later time. However, the fact that in such a mechanical description an arbitrary initial state may be chosen presents great difficulties when the problem of atomic structure is considered. I n fact, if we must reckon with an infinite number of continuously varying states of motion of the atoms, then we come into obvious contradiction

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with our experimental knowledge that the elements possess definite properties. One might believe perhaps that the properties of the elements do not inform us directly of the behaviour of individual atoms but, rather, that we are always concerned only with statistical regularities holding for the average conditions of a large number of atoms. In the mechanical theory of heat, which not only permits of our accounting for the fundamental laws of thermodynamics, but also gives us an understanding of many of the general properties of matter, we have a wellknown example of the fruitfulness of statistical mechanical considerations in the atomic theory. The elements have, however, other properties which permit of more direct conclusions being drawn with respect to the states of motion of the atomic constituents. Above all, we must assume that the quality of the light which the elements in certain circumstances emit and which is characteristic of each element is essentially determined by what occurs in a single atom. Just as the wireless waves tell us about the nature of the electrical oscillations in the apparatus of the broadcasting station, so should we expect, on the basis of the electromagnetic theory of light, that the frequencies of the individual lines in the characteristic spectra of the elements should give us information as to the motions of the electrons within the atom. However, mechanics does not offer us a sufficient basis for interpreting this information ;indeed, owing to the possibility of a continuous variation of the mechanical states of motion mentioned above, it is not possible even to understand the occurrence of sharp spectral lines. The missing element in our description of nature, evidently required to account for the behaviour of the

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atoms, has been supplied, however, by Planck’s discovery of the so-called quantum of action. This discovery had its origin in his investigation of black body radiation, which, because of its independence of the special properties of the substances, offered a decisive test of the range of validity of the mechanical theory of heat and of the electromagnetic theory of radiation. It was the very inability of these theories to account for the law of black body radiation which led Planck to recognize a general feature of the laws of nature that had hitherto remained unnoticed. This feature, to be sure, is not obvious in the description of ordinary physical phenomena, but it has, nevertheless, caused a complete revolution in our account of such effects which depend on individual atoms. Thus, in contrast with the demand of continuity which characterizes the customary description of nature, the indivisibility of the quantum of action requires an essential element of discontinuity in the description of atomic phenomena. The difficulty of combining the new knowledge with our ordinary scheme of physical ideas became especially apparent through the discussion of the nature of light, which was renewed by Einstein in connection with his explanation of the photo-electric effect, although the question, judging from all earlier experimental results, had found a perfectly satisfactory solution within the frame of the electromagnetic theory. The situation which we meet here is characterized by the fact that we are apparently forced to choose between two mutually contradictory conceptions of the propagation of light : one, the idea of light waves, the other, the corpuscular view of the theory of light quanta, each conception expressing fundamental aspects of our experience. As we

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shall see in the following, this apparent dilemma marks a peculiar limitation of our forms of perception which is bound up with the quantum of action. This limitation is brought to light by a closer analysis of the applicability of the basic physical concepts in describing atomic phenomena. Indeed, only by a conscious resignation of our usual demands for visualization and causality was it possible to make Planck’s discovery fruitful in explaining the properties of the elements on the basis of our knowledge of the building stones of atoms. Taking the indivisibility of the quantum of action as a starting-point, the author suggested that every change in the state of an atom should be regarded as an individual process, incapable of more detailed description, by which the atom goes over from one so-called stationary state into another. According to this view, the spectra of the elements do not give us immediate information about the motions of the atomic parts, but each spectral line is associated with a transition process between two stationary states, the product of the frequency and the quantum of action giving the energy change of the atom in the process. I n this way, it proved possible to obtain a simple interpretation of the general spectroscopic laws which Balmer, Rydberg and Ritz had succeeded in deriving from the experimental data. This view of the origin of spectra was directly supported also by the well-known experiments of Franck and Hertz on collisions between atoms and free electrons. The amounts of energy which can be exchanged in such collisions were found to agree exactly with the energy differences, computed from the spectra, between the stationary state in which the atom was

D E S C R I P T I O N O F NATURE 109 before the collision and one of the stationary states in which it can exist after the collision. On the whole, this point of view offers a consistent way of ordering the experimental data, but the consistency is admittedly only achieved by the renunciation of all attempts to obtain a detailed description of the individual transition processes. We are here so far removed from a causal description that an atom in a stationary state may in general even be said to possess a free choice between various possible transitions to other stationary states. From the very nature of the matter, we can only employ probability considerations to predict the occurrence of the individual processes, which fact, as Einstein has emphasized, exhibits a close similarity to the conditions holding for the spontaneous radioactive transformations. A peculiar feature of this attack on the problem of atomic structure is the extensive use of whole numbers which also play an important r61e in the empirical spectroscopic laws. Thus, the classification of stationary states, besides depending upon the atomic number, also depends on the so-called quantum numbers, to the systematics of which Sommerfeld has contributed so much. On the whole, the views considered have permitted us to account, to a considerable extent, for the properties and relationships of the elements on the basis of our general conceptions of atomic structure. Considering the great departure from our customary physical ideas, one might wonder that such an account has been possible, since, after all, our entire knowledge of the building stones of the atoms rests upon these ideas. Indeed, any use of concepts like mass and electric charge is obviously equivalent to the invocation of mechanical

II0

THE ATOMIC THEORY

and electrodynamical laws. A method of making such concepts useful in other fields than that in which the classical theories are valid has been found, however, in the demand of a direct concurrence of the quantummechanical description with the customary description in the border region where the quantum of action may be neglected. The endeavours to utilize in the quantum theory every classical concept in a reinterpretation which fulfils this demand without being at variance with the postulate of the indivisibility of the quantum of action, found their expression in the so-called correspondence principle. However, there were many difficulties to overcome before a complete description based on the correspondence principle was actually accomplished, and only in recent years has it been possible to formulate a coherent quantum mechanics which may be regarded as a natural generalization of the classical mechanics, and in which the continuous, causal description is replaced by a fundamentally statistical mode of description. A decisive step towards the attainment ofthis goal was made by the young German physicist, Werner Heisenberg, who showed how the ordinary ideas of motion may be replaced in a consistent way by a formal application of the classical laws of motion, the quantum of action appearing only in certain rules of calculation holding for the symbols which replace the mechanical quantities. This ingenious attack upon the problem of the quantum theory makes, however, great demands on our power of abstraction, and the discovery of new artifices which, in spite of their formal character, more closely meet our demands for visualization has, therefore, been of profound significance in the development and clarification of the

DESCRIPTION OF NATURE

I11

quantum mechanics. I am referring to the ideas of matter waves, introduced by Louis de Broglie, which have proved so fruitful in the hands of Schrodinger, especially in connection with the conception of stationary states, the quantum numbers of which are interpreted as the number of nodes of the standingwaves symbolizing these states. De Broglie’s starting-point was the similarity, which had already been so important in the development of classical mechanics, existing between the laws governing the propagation of light and those holding for the motion of material bodies. In fact, the wave mechanics forms a natural counterpart to the abovementioned light quantum theory of Einstein. As in this theory, so also in the wave mechanics, we are not dealing with a self-contained conceptual scheme but, rather, as especially emphasized by Born, with an expedient to formulate the statistical laws which govern atomic phenomena. I t is true that the confirmation of the idea of matter waves, provided by the experiments on the reflection of electrons by metal crystals, is, in its way, just as decisive as the experimental evidence for the wave conception of the propagation of light. However, we must bear in mind that the application of matter waves is limited to those phenomena, in the description of which it is essential that the quantum of action be taken into account and which, therefore, lie outside the domain where it is possible to carry out a causal description corresponding to our customary forms of perception and where we can ascribe to words like “the nature of matter” and “the nature of light” meanings in the ordinary sense. With the help of quantum mechanics, we master an

II2

THE ATOMIC THEORY

extensive range of experience. Especially are we able to account for a large number of details concerning the physical and chemical properties of the elements. Recently, it has been possible even to obtain an interpretation of the radioactive transformations, in which the empirical probability laws holding for these processes appear as an immediate consequence of the peculiar statistical mode of description that characterizes the quantum theory. This interpretation provides an excellent example of the fruitfulness as well as of the formal nature of the wave conceptions. On one hand, we have here a direct connection with the customary ideas of motion, since, owing to the great energy of the fragments expelled by the atomic nuclei, the paths of these particles may be directly observed. On the other hand, the ordinary mechanical conceptions completely fail to provide us with a description of the course of the disintegration process, since the field of force surrounding the atomic nucleus would, according to these ideas, prevent the particles from escaping from the nucleus. On the quantum mechanics, however, the state of affairs is quite different. Though the field of force is still a hindrance which, for the most part, holds the matter waves back, yet it permits a small fraction of them to leak through. The part of the waves which escapes in this way in a certain time gives us a measure of the probability that the disruption of the atomic nucleus takes place during this time. The difficulty of speaking of “the nature of matter ” without the above-mentioned proviso could scarcely be more strikingly brought to light. I n the case of the idea of light quanta, there exists a similar relationship between our conceptual pictures and

DESCRIPTION OF NATURE

1'3

the calculation of the probability of occurrence of the observable light effects. I n accordance with the classical electromagnetic conceptions, we cannot, however, ascribe any proper material nature to light, since observation of light phenomena always depends on a transfer of energy and momentum to material particles. The tangible content of the idea of light quanta is limited, rather, to the account which it enables us to make of the conservation of energy and momentum. It is, after all, one of the most peculiar features of quantum mechanics that, in spite of the limitation of the classical mechanical and electromagnetic conceptions, it is possible to maintain the conservation laws of energy and momentum. In certain respects, these laws form a perfect counterpart to the assumption, basic for the atomic theory, of the permanence of the material particles, which is strictly upheld in the quantum theory even though the conceptions of motion are renounced. As with classical mechanics, so quantum mechanics, too, claims to give an exhaustive account of all phenomena which come within its scope. Indeed, the inevitability of using, for atomic phenomena, a mode of description which is fundamentally statistical arises from a closer investigation of the information which we are able to obtain by direct measurement of these phenomena and of the meaning which we may ascribe, in this connection, to the application of the fundamental physical concepts. On one hand, we must bear in mind that the meaning of these concepts is wholly tied up with customary physical ideas. Thus, any reference to spacetime relationships presupposes the permanence of the elementary particles, just as the laws of the conservation B

8

1 I4

THE ATOMIC THEORY

of energy and momentum form the basis of any application of the concepts of energy and momentum. On the other hand, the postulate of the indivisibility of the quantum of action represents an element which is completely foreign to the classical conceptions ; an element which, in the case of measurements, demands not only a finite interaction between the object and the measuring instrument but even a definite latitude in our account of this mutual action. Because of this state of affairs, any measurement which aims at an ordering of the elementary particles in time and space requires us to forego a strict account of the exchange of energy and momentum between the particles and the measuring rods and clocks used as a reference system. Similarly, any determination of the energy and the momentum of the particles demands that we renounce their exact co-ordination in time and space. I n both cases, the invocation of classical ideas, necessitated by the very nature of measurement, is, beforehand, tantamount to a renunciation of a strictly causal description. Such considerations lead immediately to the reciprocal uncertainty relations set up by Heisenberg and applied by him as the basis of a thorough investigation of the logical consistency of quantum mechanics. The fundamental indeterminacy which we meet here may, as the writer has shown, be considered as a direct expression of the absolute limitation of the applicability of visualizable conceptions in the description of atomic phenomena, a limitation that appears in the apparent dilemma which presents itself in the question of the nature of light and of matter. The resignation as regards visualization and causality, to which we are thus forced in our description of atomic

DESCRIPTION OF NATURE

'I5

phenomena, might well be regarded as a frustration of the hopes which formed the starting-point of the atomic conceptions. Nevertheless, from the present standpoint of the atomic theory, we must consider thisveryrenunciation as an essential advance in our understanding. Indeed, there is no question of a failure of the general fundamental principles of science within the domain where we could justly expect them to apply. The discovery of the quantum of action shows us, in fact, not only the natural limitation of classical physics, but, by throwing a new light upon the old philosophical problem of the objective existence of phenomena independently of our observations, confronts us with a situation hitherto unknown in natural science. As we have seen, any observation necessitates an interference with the course of the phenomena, which is of such a nature that it deprives us of the foundation underlying the causal mode of description. The limit, which nature herself has thus imposed upon us, of the possibility of speaking about phenomena as existing objectively finds its expression, as far as we can judge, justin the formulation of quantum mechanics. However, this should not be regarded as a hindrance to further advance; we must only be prepared for the necessity of an ever extending abstraction from our customary demands for a directly visualizable description of nature. Above all, we may expect new surprises in the domain where the quantum theory meets with the theory of relativity and where unsolved difficulties still stand as a hindrance to a complete fusion of the extension of our knowledge and of the expedients to account for natural phenomena which these theories have given us. Even though it be at the end of the lecture, yet I am

I 16

THE ATOMIC T H E O R Y

glad to have the opportunity of emphasizing the great significance of Einstein’s theory of relativity in the recent development of physics with respect to our emancipation from the demand for visualization. We have learned from the theory of relativity that the expediency of the sharp separation of space and time, required by our senses, depends merely upon the fact that the velocities commonly occurring are small compared with the velocity of light. Similarly, we may say that Planck’s discovery has led us to recognize that the adequacy of our whole customary attitude, which is characterized by the demand for causality, depends solely upon the smallness of the quantum of action in comparison with the actions with which we are concerned in ordinary phenomena. While the theory of relativity reminds us of the subjective character of all physical phenomena, a character which depends essentially upon the state of motion of the observer, so does the linkage of the atomic phenomena and their observation, elucidated by the quantum theory, compel us to exercise a caution in the use of our means of expression similar to that necessary in psychological problems where we continually come upon the difficulty of demarcating the objective content. Hoping that I do not expose myself to the misunderstanding that it is my intention to introduce a mysticism which is incompatible with the spirit of natural science, I may perhaps in this connection remind you of the peculiar parallelism between the renewed discussion of the validity of the principle of causality and the discussion of a free will which has persisted from earliest times. Just as the freedom of the will is an experiential category of our psychic life, causality may be considered as

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a mode of perception by which we reduce our sense impressions to order. At the same time, however, we are concerned in both cases with idealizations whose natural limitations are open to investigation and which depend upon one another in the sense that the feeling of volition and the demand for causality are equally indispensable elements in the relation between subject and object which forms the core of the problem of knowledge. Before I conclude, it would be natural, at such a joint meeting of natural scientists, to touch upon the question as to what light can be thrown upon the problems regarding living organisms by the latest development of our knowledge of atomic phenomena which I have here described. Even though it may not yet be possible to give an exhaustive answer to this question, we can perhaps already catch a glimpse of a certain connection between these problems and the ideas of the quantum theory. A first hint in this direction we find in the circumstance that the mutual action between the organisms and the external world, upon which the sense impressions depend, may, at any rate in certain circumstances, be so small that it approaches the quantum of action. As it has often been remarked, a few light quanta are sufficient to produce a visual impression. We see, therefore, that the needs of the organism for independence and sensibility are here satisfied to the utmost limit permitted by the laws of nature, and we must be prepared to come upon similar conditions aIso at other points of decisive significance for the formulation of biological problems. If, however, the physiological phenomena exhibit a refinement which is developed to the abovementioned limit, then, indeed, this means that we at the

I I8

THE ATOMIC THEORY

same time approach the limit for an unambiguous description of them with the help of our ordinanr visualizable conceptions. This in no way contradicts the-fact that the living organisms to a wide extent present problems to us which lie within the range of our visualizable forms of perception and have formed a fruitful field for the application of physical and chemical points of view. Neither do we see any immediate limit for the applicability of these view-points. Just as we do not need to distinguish, in principle, between the current in a water pipe and the flow of blood in the vessels, no more should we expect, beforehand, any profound fundamental difference between the propagation of sense impressions in the nerves and the conduction of electricity in a metal wire. It is true, for all such phenomena, that a detailed account carries us into the domain of atomic physics ; indeed, so far as the conduction of electricity is concerned, we have just learned, in quite recent years, that only that limitation of our visualizable conceptions of motion, which is characteristic of the quantum theorv, enables us to understand how the electrons can make their way between the metal atoms of the wire. However, in the case of these phenomena, such a refinement in the mode of description is not necessary to account for those effects which first call for our consideration. With regard to the more profound biological problems, however, in which we are concerned with the freedom and power of adaptation of the organism in its reaction to external stimuli, we must expect to find that the recognition of relationships of wider scope will require that the same conditions be taken into consideration which determine the limitation of the causal mode of description in the case of

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atomic phenomena. Besides, the fact that consciousness, as we know it, is inseparably connected with life ought to prepare us for finding that the very problem of the distinction between the living and the dead escapes comprehension in the ordinary sense of the word. That a physicist touches upon such questions may perhaps be excused on the ground that the new situation in physics has so forcibly reminded us of the old truth that we are both onlookers and actors in the great drama of existence.

CAMBRIDGE: PRINTED BY W. LEWIS, M.A., AT THE UNIVERSITY PRESS

IV. INTRODUCTORY SURVEY (1929) WITH ADDENDUM OF 1931 INDLEDENDE OVERSIGT (MED T I L L E G FRA 1931) “Atomteori og Naturbeskrivelse”, Festskrift udgivet af K~benhavnsUniversitet i Anledning af Universitetets Aarsfest 1929, Bianco Lunos Bogtrykkeri, Copenhagen 1929, pp. 5-19 and J.H. Schultz Forlag, Copenhagen 1958, pp. 23-25 INTRODUCTORY SURVEY (WITH ADDENDUM OF 1931) “Atomic Theory and the Description of Nature”, Camb. Univ. Press, 1934 (reprinted 1961), pp. 1-24

See Introduction to Part 11, sect. 2.

P A R T 11: F U R T H E R E L U C I D A T I O N O F T H E C O M P L E M E N T A R I T Y A R G U M E N T

INTRODUCTORY SURVEY (1929) WITH ADDENDUM OF 1931 Versions published in Danish, German and English

Danish :In dledende 0versig t A “Atomteori og Naturbeskrivelse”, Festskrift udgivet af K~benhavns Universitet i Anledning af Universitetets Aarsfest 1929, Bianco Lunos Bogtrykkeri, Copenhagen 1929, pp. 5-19 B “Atomteori og naturbeskrivelse”, J.H. Schultz Forlag, Copenhagen 1958, pp. 9-25 German: Einleitende Ubersicht C “Atomtheorie und Naturbeschreibung” , Julius Springer Verlag, Berlin 1931, pp. 1-15 English: Introductory Survey D “Atomic Theory and the Description of Nature”, Camb. Univ. Press, 1934 (reprinted 1961), pp. 1-24 Apart from a few minor deviations, all of these versions agree with each other. As explained in the Introduction, it was decided at the publication of the German book in 1931 to include Bohr’s address to the 18th Meeting of Scandinavian Scientists in 1929 (document 111). Correspondingly, an Addendum was added, which was then also included in the later English and Danish editions, D and B . To A are added a few words of thanks to Kramers and Klein for their assistance through the years.

FESTSKRIFT UDGIVET AF

K0BENHAVNS UNIVERSITET I ANLEDNING AF

U N IV E R SI T E T E T S AA R SFE S T NOVEMBER 1929

NIELS BOHR

ATOMTEORI OG NATURBESKRIVELSE

*

K 0 R E N H A V N M C MXX I X BIANCO L U N O S ROGTRYKKERI

ATOMTEORI OG NATURBESKRIVELSE 3 ARTIKLER MED EN INDLEDENDE OVERSIGT AF

NIELS BOHR

*

K0BENHAVN

MCMXXIX

BIANCO L U K O S BOGTRYKKERI

v

INDLEDENDE OVERSIGT.

.idenskabens Opgave at forerge og ordne vore Erfaringer frembyder forsltelligartede, ulmeligt forbundne Sider. Kun gennem Erfaringerne selv ltommer de Lovmaessigheder ti1 vor Kundskab, der tillader 0s et Overblik over Frenomenernes Mangfoldighed. Med Erfaringernes Foragelse maa vi derfor stadig vaere forberedt paa, a t d e for deres Ordning bedst egnede Synspunkter vil undergaa Endringer. I denne Sammenhaeng m a a vi ikke mindst betaenke, at alle nye Erfaringer efter Sagens Natur fremtraeder indordnet i den Ramme, som vore tilvante Synspunkter og dnskuelsesformer afgiver. Alt efter Emnets Art kommer de forskellige Sider af vidensliabelig Forskning mere eller mindre i Forgrunden. I Fysikken, hvor det gaelder Ordningen af Erfaringerne o m den ydre Verden, vil vi naturligvis mindre hyppigt v z r e beskreftiget ined Sperrgsmaalet o m vore Ansliuelsesformers Vaesen end i Psykologien, hvor vor egen Tankevirksomhed er selve Unders~gelsensGenstand. Dog kan ti1 Tider netop de fysislie Iagttagelsers ))Objektivitetcc w r e saerlig egnet ti1 at stille a1 Erfarings subjektive Karakter i grel Relysning. Naturvidenskabens Historie opviser mangfoldige Eksempler herpaa. Jeg beherver blot at minde om, hvor meget Udforskningen af Lyd- og Lysfrenomenerne, vore Sansers fysiske Hjaelpemidler, gennem Tiderne h a r betydet for den psykologiske Analyse, eller o m den Betydning som Klarlaeggelsen af Mekanikkens Lovmaessigheder h a r haft for Udviklingen af den alniindelige filosofiske Erkendelsesteori. Ved Fysikkens seneste Cdvikling er det omhandlede T r z k af Videnskabens Vaesen kommet staerkt i Forgrunden. Den store Udvidelse af Erfaringsomraadet h a r bragt de simple mekaniske Forestillingers Utilstraekkelighed for Dagen og dermed rystet Grundvolden for den tilvante Tydning af Iagttagelserne og stillet gamle filosofiske Problemer i ny Belysning. Dette gzelder saavel den Revision af Grundlaget for Tids-Rumbeskrivelsen, som Rela-

6

tivitetsteorien h ar bragt, som den fornyede Diskussion om Aarsagssaetningen , som Kvanteteoriens Udvikling h a r givet Anledning til. Relativitetsteoriens Oprindelse er nsje knyttet ti1 Udviklingen af de elektromagnetiske Forestillinger, der gennem Uddybningen af Kraftbegrebet bragte en saa gennemgribende Omformning af de mekaniske Grundforestillinger. Allerede ved den klassiske Mekaniks Opbygning spillede Erkendelsen af Bevagelsesfznomenernes relative, af Iagttageren afhaengige Karakter, en vzsentlig Rolle og viste sig som et virksomt Hjaelpemiddel ved Udformningen af de almindelige mekaniske Love. Imidlertid fandt de omhandlede Spsrgsmaal en saavel fra fysisk som filosofisk Synspunkt tilsyneladende tilfredsstillende Behandling, og det var forst den Erkendelse af alle Kraftvirkningers endelige Forplantningshastighed, som den elektromagnetiske Teori bragte, der medfclrte, at Sagen blev sat paa Spidsen. Vel viste det sig muligt paa den elektromagnetiske Teoris Grund at opbygge en Aarsagsbeskrivelse med Opretholdelse af Mekanikkens Hovedsaetninger om Energiens og Bevaegelsesmaengdens Bevarelse ved at tillaegge selve Kraftfelterne Energi og Bevaegelsesmaengde. Den for den elektroniagnetiske Teoris Udvikling saa nyttige Forestilling om en Verdenszther fremtraadte imidlertid som et absolut Henfsrelsessystem for Tids-Rumbeskrivelsen, hvis fra filosofisk Synspunkt utilfredsstillende Karakter s tz r k t understregedes ved Strandingen af alle Forssg paa at eftervise Jordens Bevzegelse i Forhold ti1 denne hypotetiske Verdenszther. I denne Henseende aendredes Situationen ikke derved, at alle saadanne Forssgs Stranding kunde vises a t viere i fuld Overensstemmelse med den elektromagnetiske Teori. Fsrst Einsteins Klarlzggelse af den Begraensning, som den endelige Forplantningshastighed af alle Kraftvirkninger, Straalingsvirkningerne iberegnet, paalzegger vore Iagttagelsesmuligheder, og derved Tids-Rumbegrebernes Anvendelsesomraade, indledede en friere Indstilling overfor disse Begreber, som fandt sit mest slaaende Udtryk i Erkendelsen af Samtidighedsbegrebets Relativitet. Som bekendt skulde det lykkes Einstein ud fra denne Indstilling at efterspore be tydningsfulde, nye Sammenhzng ogsaa udenfor h e n elektromagnetiske Teoris egentlige Omraade, og i sin almindelige Relativitetsteori, hvor Gravitationsvirkningerne ikke laengere indtager en Sarstilling blandt de fysiske Fznomener, i

7

uanet Grad a t imsdekomme det Krav om Enhed i Naturbeskrivelsen, der er de klassiske fysiske Teoriers Ideal. Kvanteteoriens Udgangspunkt er Udviklingen af de atomistiske Forestillinger, der i Lsbet af det forrige Aarhundrede i stadig voksende Omfang havde afgivet et frugtbart Anvendelsesomraade for Mekanikken og den elektromagnetiske Teori. I Tiden omkring Aarhundredeskiftet skulde disse Teorier imidlertid i deres Anvendelse paa Atomproblemerne aabenbare en hidtil upaaagtet Begraensning, som fandt Udtryk i Plancks Opdagelse af det saakaldte Virkningskvantum, der paalagger de enkelte Atomprocesser et T r a k af Diskontinuitet, der er ganske fremmedartet for den klassiske Fysiks Grundprincipper, efter hvilke alle Virkninger k a n varieres paa kontinuert Maade. Samtidig med at Virkningskvantet har vist sig mere og mere uundvzrligt for Indordningen af Erfaringerne angaaende Atomernes Egenskaber, er vi Skridt for Skridt blevet tvunget ti1 i stadig videregaaende Grad at give Afkald paa en Beskrivelse af de enkelte Atomers Forhold i Rum og Tid paa Aarsagsszetningens Grund og at regne med frie Valg fra Naturens Side mellem forskellige Muligheder, for hvis Udfald der kun kan anstilles Sandsynlighedsbetragtninger. Bestraebelserne for ved en passende begrxmset Anvendelse af de klassiske Teoriers Begreber at formulere almindelige Love for disse Muligheder og Sandsynligheder har efter en Raekke Udviklingsstadier i de seneste Aar resulteret i Skabelsen af en rationel Kvantemekanik, ved hvis Hja lp det er muligt a t beherske et meget stort Erfaringsomraade, og son1 i enhver Henseende optrader som en Almindeliggsrelse af de klassiske fysiske Teorier. Tilmed er der efterhaanden opnaaet fuld Klarhed over den nsje Sammenhaeng mellem den kvanteteoretiske Beskrivelses Afkald paa Aarsagssammenhzeng og dens Begrzensning i Muligheden af a t skelne imellem et Faenomen og dets Iagttagelse, som Virkningskvantets Udelelighed betinger. Erkendelsen af dette Forhold betyder en vaesentligt aendret Indstilling overfor saavel Aarsagssaetningen som Iagttagelsesbegrebet, Ti1 Trods for a1 Forskel udviser de Problemer, vi msder i Relativitetsteorien og i Kvanteteorien, en dyb indre Lighed. I begge Tilfaelde drejer det sig om Erkendelsen af fysiske Lovnizessigheder, der falder udenfor vort saedvanlige Erfaringsomraade, og som byder vore tilvante Anskuelsesformer Vanskeligheder. Vi belaeres om, a t

8

disse Anskuelsesformer er Idealisationer, hvis Formaalstjenlighed ved Indordningen af de saedvanlige Sanseiagttagelser beror paa Lysets praktisk talt tidlase Forplantning og Virkningskvantets Lidenhed. Ved disse Forholds Bedominelse maa det dog ikke tabes af Syne, at vi ti1 Trods for deres Begransning ikke kan konime bort fra de Anskuelsesformer, under hvilke alle Erfaringer ti1 syvende og sidst optrzeder, og som farver hele Sproget. Det er netop denne Sagernes Stilling, der i forste L i n k betinger de omhandlede Problemers almindelige filosofiske Interesse. Medens den Afrunding af vort Verdensbillede, soni Relativitetsteorien har bragt, allerede er indgaaet i den videnskabelige Almenbevidsthed, er dette dog naeppe i samme Grad Tilfaeldet med de af Kvanteteorien belyste Sider af det almindelige Erkendelsesproblem. Da det faldt i min Lod a t forfatte en Afhandling ti1 Universitetets Festskrift, var det min Hensigt at gplre Rede for de Synspunkter, Kvanteteorien h a r bragt, ved en inuligst let tilgamgelig Fremstilling baseret paa en Analyse af de elementaere Begreber, hvorpaa Naturbeskrirelsen hviler. Optagethed med andre Pligter har imidlertid ikke levnet mig tilstrzekkelig Tid ti1 Fuldfplrelsen af en saadan Fremstilling, hvis Vanskelighed ikke mindst bunder i den stadige Udvikling, hvori de omhandlede Synspunkter befinder sig. Folelsen af denne Vanskelighed forte mig imidlertid ind paa den Tanke i Stedet for en ny Fremstilling at benytte en ti1 denne Lejlighed foretaget dansk Oversaettelse af nogle Artikler, som jeg i Labet af de sidste Aar har offentliggjort i udenlandske Tidsskrifter som Indlaeg i Diskussionen om Kvanteteoriens Problemer, De paagaeldende Artikler er Led i en Raekke Foredrag og Afhandlinger, hvormed jeg fra Tid ti1 anden har splgt at give en sainmenfattende Oversigt over Atomteoriens ojeblikkelige Tilstand. Nogle tidligere paa dansk i Fysisk Tidsskrift udgivne Artikler i denne Rzekke danner i visse Henseender Baggrunden for de tre Artikler, som gengives i det folgende. Dette gzelder navnlig et Foredrag med Titel: ))Atomernes Bygningcc, der blev holdt i Stockholm i December 1922 og samtidig udkoni som saerlig Brochure. De her gengivne Artikler frenitraeder dog som fuldt selvstzendige i deres Form og er indbyrdes naje knyttet sammen, idet de behandler den seneste

9

Fase i Atomteoriens Udvikling, hvor Grundbegrebernes Analyse er kommet saa s tzrk t i Forgrunden. Den Omstaendighed, at Artiklerne fslger Udviklingens Forlsb og derved giver et umiddelbart Indtryk af Begrebernes gradvise Afklaring, turde maaske i nogen Grad bidrage ti1 at gsre Emnet tilgengeligere for Laesere, der ikke tilhsrer Fysikernes snaevrere Kreds. I den nedenstaaende Redegsrelse for de naermere Omstaendigheder ved Artiklernes Fremkomst har jeg endvidere sragt ved nogle orienterende Bemaerkninger a t lette Oversigten over Indholdet og saa vidt rnuligt a t raade Bod paa Fremstillingens Mangler, hvad Vanskeligheder for Tilegnelsen angaar. Den frarste af Artiklerne bringer Udarbejdelsen af et Foredrag holdt ved den skandinaviske Matematikerkongres i Ksbenhavn i August 1925. Den giver i sammentraengt Form en Oversigt over Kvanteteoriens Udvililing indtil det nzevnte Tidspunkt, hvor et nyt Stadium skulde indledes gennem den i Artiklens Slutning nzrmere omtalte Afhaiidling af Heisenberg. Foredraget tager saerlig Sigte paa Anvendelsen af mekaniske Begreber indenfor Atomteorien og viser, hvorledes det nye Udviklingsstadium, der kendetegnes ved Skabelsen af rationelle kvantemekaniske Metoder, var forberedt gennem Ordningen af et stort Erfaringsmateriale ved Hjaelp af Kvanteteorien. Fremfor alt havde denne forudgaaende Udvikling fmt ti1 Erkendelsen af Uigennemfsrligheden for Atomfaxomenernes Vedkoinmende af en sammenhaengende Aarsagsbeskrivelse. E n bevidst Resignation i denne Henseende komnier allerede ti1 Udtryk i den fra de klassiske Teoriers Synspunkt irrationelle Form af de i Artiklen omtalte Postulater, der dannede Forfatterens Udgangspunkt for Anvendelsen af Kvanteteorien paa Atombygningsproblemet. Den Omstaendighed, at alle Wndringer i et Atoms Tilstand i Overensstemmelse med Fordringen om Virkningskvantets Udelelighed beskrives som individuelle Processer, hvorved Atomet fares fra en saakaldt stationax Tilstand ti1 en anden, og for hvis Forekomst der kun ka n aiistilles Sandsynlighedsbetragtninger, maatte paa den ene Side staerkt indskraenke de klassiske Teoriers Anvendelsesomraade. Paa den anden Side gav Nsdvendigheden af desuagtet a t gsre en udstrakt Brug af de klassiske Begreber, hvorpaa ti1 syvende og sidst Tolkningen af alle Erfaringer beror, Anledning ti1 Opstillingen af det saakaldte Korrespondensprincip, der giver Ud tryk for

10

Bestraebelsen paa at udnytte alle klassiske Begreber i passende kvanteteoretisk Omtydning. Den n e r me re Analyse af Erfaringsmaterialet ud fra dette Standpunkt skulde dog g0re det stedse klarere, at m an endnu ikke besad fuldt egnede Hjelpemidler ti1 Gennemfarrelsen af en streng korrespondensmaessig Beskrivelse. Som Farlge af den serlige Lejlighed, ved hvilken Foredraget blev holdt, er der i Artiklen navnlig lagt Vregt paa den Benyttelse af matematiske Hjaelpemidler, der er ejendommelig for den teoretiske Fysik. Matematikens symbolske Udtryksformer er her ikke alene et uundvaerligt Vaerktarj for Beskrivelsen af den kvantitatiye Samrnenhaeng, men tillige et Hovedmiddel ti1 de almindelige kvalitative Synspunkters Afklaring. Det i Artiklens Slutning udtrykte Haab om, at den matematiske Analyse atter denne Gang vilde vise sig i Stand ti1 a t bringe Fysikerne over Vanskelighederne, er i Mellemtiden blevet opfyldt over enhver Forventning. Ikke alene skulde som n e v n t i Artiklen den abstrakte Algebra spille en afgarrende Rolle ved Udformningen af den Heisenbergske Kvantemekanik, men i den n e r me s t paafarlgende Tid skulde ogsaa den klassiske Fysiks Hovedhjaelpemiddel, Differentialligningernes Teori, finde en udstrakt Anvendelse paa Atomproblemerne. Udgangspunktet herfor var den ejendommelige Analogi mellem Mekanik og Optik, hvorpaa allerede Hamiltons betydningsfulde Bidrag ti1 de klassiskTmekaniske Metoders Udvikling hviler. Denne Analogis Betydning for Kvanteteorien blev farrst fremdraget af de Broglie, der i Tilknytning ti1 Einsteins kendte Lyskvanteteori sammenlignede en Partikelbevaegelse med Udbredelsen af Bslgesystemer. Som de Broglie paapegede, gav denne Sammenligning en h'lulighed for en simpel geometrisk Tydning af de i Artiklen naevnte Kvantiseringsregler for Atomernes stationaere Tilstande. Ved en videre Forf~lgelse af disse Betragtninger lykkedes det Schrbdinger at fare det kvantemekaniske Problem tilbage ti1 Larsningen af en vis Differentialligning, den saakaldte Bslgeligning, og dermed a t skaenke 0s et Hjrelpemiddel, der h ar vaeret af afgarrende Betydning for Atomteoriens store Udvikling i de sidste Aar. Den anden Artikel gengiver i udarbejdet Form et Foredrag holdt ved en international Fysikerkongres, der i Anledning af 100-Aarsdagen for Voltas Dard fandt Sted i Como i September 1927.

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Paa dette Tidspunkt havde de ovennzvnte kvanteteoretiske Metoder naaet en stor Fuldkommenhed og bevist deres Frugtbarlmed overfor et stort Antal Anvendelser. Ti1 Gengaeld var der opstaaet en Meningsforskel vedrerrende hletodernes fysiske Tolkning, der havde givet Anledning ti1 megen Diskussion. Kavnlig havde de store Resultater, som den Schrodingerske Bslgemekanik kunde opvise, hos mange Fysikere genvakt Haabet om en Beskrivelse af Atomfaenomenerne efter lignende Retningslinier som de klassiske fysiske Teorier, uden Indfsrelse af ))Irrationalitetercc af den Art son] hidtil havde kendetegnet Kvanteteoriens Anvendelser. I hlodsaetning hertil haevdes det i Artiklen, a t selve Grundpostulatet o m Virkningskvantets Udelelighed fra klassisk Standpunkt netop er et irrationelt Element, der uundgaaeligt kraever en Resignation med Hensyn ti1 Aarsagsbeskrivelsen i Tid og Rum og som F0lge af Sammenkaedningen mellem Faenomen og Iagttagelse henviser 0s ti1 en Beskrivelsesmaade, der betegnes som komplementaer i den Forstand, a t enhver given Anvendelse af klassiske Begreber udelukker den samtidige Benyttelse af andre klassiske Begreber, der i anden Sammenhaeng er lige saa nerdvendige for Fznomenernes Belysning. Det paapeges, hvorledes dette Traek nierder 0s straks ved Spm-gsmaalet o m Lysets og Materiens Vaesen. I den fsrste Artikel var det allerede fremhaevet, at vi ved Beskrivelsen af Straalingsfaenomenerne staar foran et Dilemma ved Valget niellem den elektromagnetiske Teoris Berlgebeskrivelse og Lyskvanteteoriens korpuskulaere Opfattelse af Lysforplantningen. Hvad hlaterien angaar, stiller den Bekraeftelse, sorn de Hroglies Bslgeforestilling i hlellemtiden havde fundet ved de bekendte Forserg over Tilbagekastning af Elektroner fra Metalkrystaller, 0 s foran et ganske tilsvarende Dilemma, idet der ikke kan vaere Tale om a t opgive Forestillingen om de elektriske Elementarpartiklers Individualitet, der paa sin Side udgerr det sikre Grundlag, hvorpaa hele Atomteoriens nyere Udvikling hviler. Det er Artiklens Hovedformaal a t vise, at det omhandlede T r z k af Komplementaritet er afgsrende for den modsigelsesfrie Tydning af de kvanteteoretiske Metoder. Et meget betydningsfuldt Bidrag ti1 denne Diskussion var kort i Forvejen givet af Heisenberg, der havde paapeget den nerje Sammenhaeng mellem de mekaniske Begrebers begrznsede Anvendelsesomraade og den Omstaendighed, at

12

enhver Maaling, der tager Sigte paa en Forfolgelse af de enkelte Individers Revaegelser, paa Grund af det uundgaaelige Indgreb i Fanomenernes Forlsb, indeholder et Element af Usikkerhed, bestemt \Ted Virkningskvantets Stclrrelse. Den Usikkerhed, det her drejer sig om, udviser netop en ejendonimelig, kompletnentaer Karakter, der forhindrer den samtidige Beiiyttelse af Tids-Rumbegreberiie og Bevarelsesszetningerne for Energi og Bevaegelsesmangde, der kendetegner den mekaniske Aarsagsbeskrivelse. For Forstaaelsen af Aarsagsbeskrivelsens Uigennemfsrlighed er det, som det vises i Artiklen, imidlertid vzesentligt at erindre, nt Onifanget af det Iiidgreb, som en Maaling betyder, altid er ukendt, idet den paagzeldende Begrznsning rammer enhver Anvendelse af mekaniske Begreber og derfor lige saa vel galder Iagttagelbesmidleriie som de Fzenomener, der er Unders~lgelsens Genstand. Netop denne Omstaendighed bevirlter, a t enhver Iagttagelse sker paa Bekostning af Sammenhzngen mellem Faenomenernes forudgaaende og fremtidige Forlclb. Overhovedet forhindrer, som allerede nzevnt, \‘irkningskvantets endelige Storrelse den sliarpe Adsliillelse mellem Fzenomen og Iagttagelsesniiddel, der er Forudszetningen for det szdvanlige Iagttagelsesbegreb og derigeiinem for de klassiske Bevzgelsesforestillinger. Med disse Forhold for 0 j e kan det iklie undre os, at de kvantemekanislte Metoders fysiske Indhold indslirzenker sig ti1 e n Formulering af statistiske Lovnizessigheder ~ e d r s rende Sa n i in en haeng e n m ell em d e Maa 1e res ul t at e r, d er k a ra k t eris e rer Fzenomenernes forskellige mulige Forlsb. Det fremhaeves i Artiklen, hvorledes den symbolske Ililaedning, der er ejendommelig for de omhandlede Xletoder, noje svarer ti1 de paagaeldende Problemers principielt uanskuelige Karakter. Et szerlig karakteristisk Eksempel paa den Begraensning af de mekaniske Forestilliiigers Anvendelighed, som det her drejer sig om, moder vi ved Benyttelseii af Begrebet stationaere Tilsiande, der som n a v n t allerede fsr de kvantemekaniske hletoders Udformning indgik som et vasentligt Element i Kvanteteoriens Anvendelse paa Atonibygningsproblemet. Som det eftervises i Artiklen, staar Anvendelsen af dette Begreb i et Udelukkelsesforhold ti1 en Forfslgelse af de enkelte Partiklers Bevaegelser i Atomet. Vi h a r her med et karakteristisk Komplementaritetsforhold at gsre, der er analogt ti1 det, vi traeffer ved Spsrgsmaalet om Lysets og hlateriens Vaesen. Som anfclrt i Artik-

13 len, turde Begrebet stationaere Tilstande indenfor sit Anvendelsesomraade besidde en lige saa stor, eller o m man vil, lige saa ringe nRealitetcc som Elementardelene selv. I begge Tilfaelde drejer det sig o m Hjzlpemidler, der paa modsigelsesfri hlaade tillader at give Udtrgk for vaesentlige Sider a€ Faenomenerne. Ved Brugen af Begrebet stationzere Tilstande stilles vi iervrigt paa lzererig Maade overfor Ncldvendigheden af i Kvanteteorien at have Opmaerksomheden henvendt paa Fzenomenernes Afgraensning, og, som det allerede betones i Artiklens fclrste Paragraf, strengt a t skeine imellem afsluttede og ikkeafsluttede Systemer. For Atomernes Vedkommende medfmer dette, at vi ved Redegerrelsen for Straalingsprocessernes Forekomst stilles overfor en saerlig grel Svigten af Aarsagsbeskrivelsen. hledeiis vi ved Forferlgelsen af de frie Partiklers Bevaegelser lran anskueliggclre Manglen paa Aarsagssammenhaeng yed en Henvisning ti1 vor Mange1 paa samtidigt Kendskab ti1 de Sterrrelser, der indgaar i den ltlassiske mekaniske Beslrriyelse, koinmer ved Redeg~relsen for Atomernes Forhold de klassiske Begrebers begraensede Anvendelighed umiddelbart for Dagen allerede derved, a t Beskrivelsen af det enkelte Atoms Tilstand slet ikke indeholder noget Element, der ruinmer en Henvisning ti1 de frivillige Overgangsprocessers I''orelromst, saaledes a t vi her naeppe undgaar a t tale o m Valg mellem forskellige hluligheder, overfor hvilke Atomet er stillet. I Forbindelse med Spmgsmaalet om Elementarpartiklernes Grundegenskaber k a n det maaske have Interesse a t gclre o p m m k s o m paa et ejendommeligt Komplementaritetsforhold, der nylig er konimet for Dagen. Den Omstaendighed, a t de Erfaringer, der hidtil h a r m x e t forklaret ved at tillaegge Elektronen et magnetisk Moment, finder en utvungen Tydning i den i Artiklens sidste Paragraf kort omtalte Teori af Dirac, er nemlig ensbetydende med, at det iklte er muligt ved Forserg baseret paa en direkte Forfcllgelse nf Elektronens Bevaegelse a t eftervise dens magnetiske Moment. Den Forskel mellem frie Elektroner og Atomer, som vi her m ~ d e r , haenger sammen med, at Maalingen af Atomers magnetiske Monienter i Overensstenimelse nied de almindelige Forhold, der gaelder for Anvendelsen af Begrebet stationaere Tiistande, netop sker under Afkald paa en Forfdgelse af Elementardelenes Bevaegelser. Den vigtige Opgave at sikre Opfyldelsen af det almindelige Relativitetskrav indenfor Kvanteteoriens Rammer, som b e r ~ r e s i Af-

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handlingens Slutning, h a r endnu ikke fundet en fuldt tilfredsstillende Lssning. Diracs nysnavnte Teori, der betsd saa stort et Fremskridt i denne Henseende, h a r nemlig aabenbaret Vanskeligheder, hvis Erkendelse dog samtidig turde aabne nye Udsigter med Hensyn ti1 d e dybtliggende Problemer, som selve Eksistensen af Elementardelene stiller 0s. Medens den hidtidige kvantemekaniske Beskrivelse hviler paa en korrespondensmassig Omtydning af den klassiske Elektronteori, lader de klassiske Teorier 0s ved Spmg9maalet o m en Tydning af saadanne Grundegenskaber hos Elementardelene som deres Masse og elektriske Ladning i endnu hsjere Grad i Stikken. Vi m a a derfor v z r e forberedt paa, at en yderligere Fremtraengen paa dette Omraade vil fordre en endnu videregaaende Abstraktion fra de tilvante Krav ti1 Rum-Tidsbeskrivelseri end Kvanteteoriens hidtidige Angreb paa Atomproblemet og vil kunne byde 0s nye Overraskelser med Hensyn ti1 Anvendeligheden af Bevzgelsesmaengde- og Energibegreberne. Den udstrakte Anvendelse af matematiske Symboler, som er ejendommelig for de kvantemekaniske Metoder, g s r det vanskeligt a t give det rette Indtryk af disse Metoders Sknnhed og Sammenhaeng uden a t gaa ind paa Enkeltheder af matematisk Art. Selr om jeg i den Fremstilling, soin er givet i Artiltlen, h a r bestrzht mig for saa vidt muligt a t nndgaa at gsre Brug af niatematiske Hjzelpemidler, h a r dog Foredragets 0jemed, i en Kreds af Fysiltere a t aabne en Diskussion o m Udviklingens Retningslinier, gjort det nsdvendigt a t ltomme ind paa Enkeltheder, der utvivlsomt vil herede de Laesere Vanskeligheder, som ikke i Forvejen i nogen Grad er fortrolige med Emnet. Ikke mindst i denne Forbindelse vil jeg imidlertid gerne understrege, a t HovedvEgten i Fremstillingen overalt er lagt paa den rent erkendelsesteoretiske Indstilling, der szrlig fremtraeder i Artiklens fsrste Paragraf og i Slutningsbemzerkningerne. I den tredie Ariikel, der e r et Bidrag ti1 et af Tidsskriftet ))Die N at u r w is sen schaft en (( i A nl ed n i n g a f Pla nck s 50-A ar s D o k t o rj u hilaeurn i J u n i 1929 udgivet Festskrift, er der mere udfsrligt gaaet ind paa Kvanteteoriens almindelige filosofiske Side. Ikke mindst med Henblik paa den Beklagelse, som i vide Kredse er kommet ti1 Orde overfor Afkaldet paa en streng Aarsagsbeskrivelse af Atomfaenomenerne, ssger Forfatteren at vise, at de Vanskeligheder for

15

vore Anskuelsesformer, som vi paa Grund af Virkningskvantets Udelelighed msder i Atomteorien, tOr betragtes som en laererig Paamindelse o m de almindelige Vilkaar, der gzlder for menneskelige Begrebsdannelser. Umuligheden af paa tilvant Maade at skelne mellem de fysiske Fznomener og deres Iagttagelse stiller 0s i Virkeligheden overfor en ganske lignende Situation som den, vi kender fra Psykologien, hvor vi stadig paamindes om Vanskeligheden ved at skille mellem Subjekt og Objekt. Det kunde maaske ved ferrste 0jekast se ud, som o m en saadan Indstilling overfor Fysikken giver Plads for en Mystik, der er i Modsaetning ti1 Naturvidenskabens hand. Lige saa lidt som ved andre menneskelige Spargsmaalstillinger turde dog Klarhed paa det omhandlede Omraade kunne naas uden ved at se d e Vanskeligheder i Gjnene, son7 frembyder sig for Begrebsbygningen og Udtryksmidleriies Anvendelse. Efter Forfatterens Opfattelse vilde det saaledes vaere e n Misforstaaelse at mene, at Vanskelighederne paa Atomteoriens Omraade kunde omgaas ved at erstatte den klassiske Fysiks Begreber med eventuelle nye Begrebsdannelser. Som allerede fremhaevet, betyder jo Erkendelsen af vore Anskuelsesformers Begraensning paa ingen Maade, at vi ved Indordningen af Sanseindtrykkene kan undvzre de tilvante Forestillinger eller deres umiddelbare Udtryk i Sproget. Lige saa lidt turde de Grundbegreber, son1 de klassiske fysiske Teorier h a r skzmket os, nogensinde blive overflsdige for Beskrivelsen af de fysiske Erfaringer. Iltke alene beror Erkendelsen af Virkningskvantets Udelelighed og Bestemmelsen af dets Vzrdi paa en Analyse af Maalinger baseret paa klassiske Begreber, men det er stadig udelukkende Anvendelsen af disse Begreber, der betinger Forbindelsen mellem Kvanteteoriens Symbolik og Erfaringernes Indhold. Samtidig m a a vi imidlertid betaenke, a t Muligheden for disse Grundbegrebers entydige Brug alene hviler paa den indre Sammenhaeng hos de klassiske Teorier, hvorfra de er hentede, og at Graenserne for disse Begrebers Anvendelse derfor iferlge Sagens Art er betinget af det Omfang, i hvilket vi ved Redegrarelsen for Faenomenerne kan se bort fra det for de klassiske Teorier fremmede Element, der symboliseres ved Virkningskvantet. Det er netop denne Sagernes Stilling, der holdes 0s for Oje ved det ofte b e r ~ r t e Dilemma vedrsrende Lysets og Materiens Egenskaber. Kun i direkte Tilknytning ti1 den klassiske elektromagne-

16

tiske Teori kan der blive Tale o m a t give Spsrgsmaalet om Lysets og Materiens Vzsen et haandgribeligt Indhold. Vel er Lyskvanter og Materieberlger uvurderlige Hjaelpemidler ved Formuleringen af de statistiske Lovnizssigheder, der behersker saadanne F e n o m e n e r som de fotoelektriske Virkninger og Elektronstraalers Interferens. Men ved disse Faenomer befinder vi 0s jo netop paa et Omraade, hvor en Hensyntagen ti1 Virkningskvantet er uundgaaelig, og hvor en entydig Beskrivelse er uigennemferlig. Den i denne Forstand symbolske Karakter af de naevnte Hjaelpemidler fremtrader ogsaa deri, at en udtermmende Beskrivelse af de elektromagnetiske Berlgefelter ikke levner nogen Plads for Lyskvanter, samt a t der ved Benyttelsen af Forestillingen o m hlateriebalgerne aldrig er Tale om en fuldstaendig Beskrivelse i Lighed med de klassiske Teoriers. Son1 det er frernhaevet i den anden Artikel, kommer den absolute V a r d i af Rslgernes saakaldte Fase jo aldrig i Betragtning ved Tolkningen af Erfaringerne. I denne Forbindelse m a a det ogsaa hetones, a t Betegnelsen ))Sandsynlighedsamplitudercc for Materiebelgernes Amplitudefunktioner er Led i en ofte bekvem Udtryksmaade, der dog iklie ltan gerre Krav paa nogen almindelig Gyldighed. Som n z v n t er det k u n ved H j z l p af de klassiske Forestillinger muligt a t tilskrive Iagttagelsesresultaterne et entydigt lndhold, og ved Sandsynlighedsbetragtninger vil det derfor altid dreje sig o m Udfaldet af Forssg, der kan tydes ved Hjaelp af saadanne Forestillinger. Som Felge lieraf vil den Brug, der gares af de symbolske Hjzlpemidler i hrert enkelt Tilfaelde afhaenge af de naermere Omstaendigheder vedrerende Forssgenes Indretning. Det, der giver den ltvanteteoretiske Beskrivelse dens S a r p r z g , e r n u netop, at vi for a t komme udenom Virkningskvantet m a a benytte forskellige Forssgsindretninger for at opnaa nejagtige Maalinger af de forskellige Stsrrelser, hvis samtidige Kendskab vilde kreves for e n fuldstzndig Beskrivelse baseret paa de klassiske Teorier, saint at disse Maalingsresultater ikke kan suppleres ved gentagne Maalinger. Virkningskvantets Udelelighed forlanger nemlig, a t der ved Tydningen af hvert enkelt Maalingsresultat i Tilknytning ti1 de klassiske Forestillinger tillades et Spillerum i vort Regnskab med Vekselvirkningen mellem Genstand og Maalemiddel, der medfsrer, a t en efterfdgende Maaling i et vist Omfang bererver de Oplysninger, som en forudgaaende Maaling har givet

17 os, deres Betydning for Forudsigelser af Faenomenernes fremtidige Forlnrb. Dette Forhold saetter aabenbart ikke alene en Graense for Omfanget af de Oplysninger, som Maalinger kan give os, men ogsaa for den Mening vi kan tillaegge saadanne Oplysninger. Vi mmder her i ny Belysning den gamle Erkendelse, at der ved Katurbeskrivelsen ikke er Tale om at afdaekke Faenoniernes egentlige Vaesen, men k u n o m i stsrst muligt Omfang a t efterspore Samnienhaeng i vore Erfaringers Mangfoldighed. Paa denne Baggrund m a a Vanskelighederne ved a t give et rigtigt Indtryk af Kvanteteoriens Indhold og dens Forhold ti1 de klassiske Teorier b e d ~ m m e s . Som allerede frenihaevet Ted Omtalen af den anden Artikel, opnaar disse Spnrrgsmaal fnrrst deres fulde Afklaring igennem den matematiske Symbolik, der h a r tilladt at formulere Kvanteteorien som en streng, korrespondensmaessig Omtydning af de klassislie Teorier. Med Henblik paa den reciproke Symmetri, der er ejendommelig for Brugen af klassiske Begreber i denne Symbolik h a r Forfatteren i denne Artikel foretrukket Betegnelsen Reciprocitet €or det i den foregaaende Artikel ved Ordet Komplementaritet kendetegnede for Kvanteteorien ejendonimelige gensidige Udelukkelsesforhold vedrnrrende Anvendelsen af forskellige klassiske Begreber og Forestillinger. Genneni senere Diskussioner er jeg inlidlertid blevet opnisrlisom paa, at den fnrrstnaevnte Betegnelse kan virke vildledende, fordi Ordet Reciprocitet i de klassiske Teorier ofte bruges i ganske anden Mening. Betegiielsen Komplementaritet, der allerede er begyndt a t komme i Brug, turde ogsaa vaere bedre egnet ti1 a t minde om, at det er Samhnrrigheden af de i deli klassiske Beskrivelsesmaade forenede, men i Kvanteteorien adskilt optraedende Traek, der i dybeste Forstand lader denne fremtraede som en naturlig Almindeliggerrelse af de klassiske fysiske Teorier. Inrvrigt er Hensigten med et saadant Kunstord, i videst muligt Omfang at undgaa en Gentagelse af det almindelige Argument og stadig at erindre o m de Vanskeligheder, der, som allerede naevnt, bunder i den Omstaendighed, at alle Sprogets ssdvanlige Ord er prsget af vore tilvante Anskuelsesformer, fra hvis Stade set Eksistensen af et Virkningskvantum er en Irrationalitet. Som Ferlge af denne Situation taber jo selv Ord som v s r e og vide deres entydige Mening. E t interessant Eksempel paa vor Sprogbrugs Tvetydighed i den onihandlede Forbindelse er den Talemaade, hvorefter Aarsagsbeskri2

18

velsens Svigten udtrykkes ved at sige, at det drejer sig om frie Valg fra Naturens Side. Egentlig k rz v e r jo en saadan Talemaade Forestillingen o m en udenforstaaende Valger, hvad der dog allerede modsiges gennem Brugen af Ordet Xatur. Vi stilles her overfor et Grundtraek i det alniindelige Erkendelsesproblem, og vi maa gore 0s klart, at vi efter Sagens Vaesen ti1 syvende og sidst altid er henvist ti1 at udtrykke 0s gennem et Maleri med Ord, der benyttes paa uanalyseret Maade. Som det betones i Artiklen, maa vi jo paa alle Erkendelsesomraader erindre, at vor Bevidstheds Vzsen betinger et Komplementaritetsforhold mellem Analysen af ethvert Begreb og dets umiddelbare Anvendelse. Den Henvisning ti1 visse psykologiske Problemer, som findes i Artiklens senere Del, har et dobbelt Formaal. De Analogier med visse Grundtrzk i Kvanteteorien, som Lovmaessighederne paa det psykiske Omraade udviser, turde ikke alene gore det lettere for 0s at finde 0s ti1 Rette i den for Fysikken nye Situation, hvori vi befinder os, men det vilde maaske ikke v z r e for dristigt a t haabe, at deli Belaering, vi h a r vundet vedrorende de efter Sagens Art saa meget simplere fysiske Problemer, ogsaa vil vise sig behjaelpelig for Bestraebelserne med at opnaa et Overblik over de dybere liggende psykologiske Spmgsmaal. Som freinhzxet i Artiklen, staar det klart for Forfatteren, at det forelobig kun kan dreje sig om mere eller mindre traeffende Analogier. Dog kunde der bag disse ikke alene ligge et Slzgtskab med Hensyn ti1 den erkendelsesteoretiske Side af Sagen, men en dybere Sammenhzng turde ligge skjult bag de biologiske Grnndproblemer, der har direkte Forbindelse ti1 begge Sider. Uden a t Kvanteteorien endnu kan siges paa vaesentlig Maade at have bidraget ti1 Belysningen a € de sidstnzvnte Problemer, er der dog meget, der tyder paa, at vi her msder Spmgsmaal, der staar Kvanteteoriens Forestillingskreds n z r . Det karakteristiske ved de levende Organismer er jo netop Individernes skarpe Afgraensning fra Omverdenen og store Evne ti1 Reaktion paa Indtryk. Det er her tankevzkkende, a t denne Evne i det mindste for Synsindtrykkenes Vedkommende er udviklet ti1 den yderste Graense, som Fysikken tillader, idet, som ofte bemaerket, k u n faa Lyskvanter er tilstraekkelige for a t frembringe Synsfornemmelser. Dog er det selvfolgelig et helt aabent Sporgsmaal, hvorvidt det vundne Kendskab ti1 Atomfaenomenernes Lovmaessig-

19

heder byder 0s tilstraekkeligt Grundlag for a t angribe de levende Organismers Problem, eller om der bag Livets Gaade skjuler sig endnu upaaagtede Sider af Erkendelsesproblemet. Hvad end Udviklingen paa disse Omraader vil bringe, er der, som det betones i Artiklens Slutning, kun Grund ti1 a t glaede sig over, a t vi indenfor Fysikkens relativt objektive Erkendelsesomraade, hvor F~lelsesmomenter i saa hOj Grad trzeder i Baggrunden, har m ~ d tProblemer, som er egnet ti1 paany at minde 0s om almindelige Vilkaar for den menneskelige Erkendelse, som fra de aeldste Tider af har tiltrukket sig Taenkeres Opmaerksomhed.

Ved Afslutningen af denne Oversigt mindes jeg med Taknemmelighed den vzerdifulde Hjaelp, som mine mangeaarige Medarbejdere, Professor H. A. KRAMERS,davaerende Lektor ved Unilrersitetets Institut for teoretisk Fysik og den nuvaerende Lektor ved Instituttet, Dr. 0. KLEIN, har ydet mig ved Artiklernes oprindelige Udarbejdelse; den sidstnzvnte skylder jeg tillige megen T a k for hans Bistand ved den foreliggende Udgivelse.

2'

I n dleden de 0 veiz ig t

TILLEG

(1931)

Den fjerde artikel, som er udarbejdet efter et foredrag holdt ved Det skandinaviske Naturforskermsde i 1929, er nart knyttet til de tre andre afhandlinger og tager sigte p i ud fra samme indstilling at give en oversigt over atomteoriens stilling i naturbeskrivelsen. Iszr betonedes det, at atomteoriens udvikling, trods den store ydeevne af den p i anvendelse af klassiske begreber beroende opdagelse af atomernes byggestene, fremfor alt har ledt ti1 erkendelsen af lovmessigheder, som ikke kan udtrykkes i vore tilvante anskuelsesformer. Som allerede antydet ibner denne belering, som vi har fdet ved opdagelsen af virkningskvantet, nye udsyn der navnlig turde vzre af betydning for diskussionen o m de levende organismers stilling i naturbesrivelsen. N i r vi i overensstemmelse med almindelig sprogbrug betegner en maskine som livlss, betyder det nzppe andet end at vi kan give en for vore formil tilstrakkelig beskrivelse af dens virkemide ved hjalp af de klassiske mekaniske begrebsdamelser. Efter den p i atomteoriens nuverende udviklingstrin klarlagte svigten af klassiske begreber passer dette kriterium imidlertid ikke for atomare fanomeners vedkommende. Dog turde selv kvantemekanikken ikke sti den ti1 vore anskuelsesformer tilpassede klassiske fysiske beskrivelsesmdde tilstrakkeligt fjernt ti1 at vere i stand ti1 at omfatte livets karakteristiske lovmassigheder.

24

Indledende O v e r s i g t

I denne forbindelse n i i det betankes at udforskningen af livsfznomenerne ikke blot, som betonet i artiklen, forer 0 s ind p i et omride af atomfysikken, hvor den for sadvanlig beskrivelse afgsrende skarpe adskillelse mellem faenomen og iagttagelse ikke kan opretholdes, men at der tillige er sat en principiel graense for undersogelsen af disse fanomener ved hjzlp af fysiske begreber, eftersom organismen ville drabes ved det indgreb som en fra et atomfysisk synspunkt fuldstandig beskrivelse fordrer. Med andre ord : den strerige anvendelse af de begrebsdannelser, som er tilpasset beskrivelsen af den livlme natur, turde sti i et udelukkelsesforhold ti1 redegmelsen for lovmmsigheder vedrorende livsfxnoniener. P i samme niidc som det kun p i grundlag af den principielle koniplementaritet mellem anvendelsen af tilstandsbegrebet og atomdelenes rum-tidskoordination er muligt at give en rationel forklaring af den for atomernes egenskaber s i karakteristiske stabilitet, turde livsfznomenernes egenart 06 iser organismernes selvopholdelse vare ulcrseligt forbundet med den principielle umulighed af en indgiende analyse af de fysiske betingelser hvorunder livet udfolder sig. Man kunne miske kort sige at kvantemekanikken drejer sig om den statistiske redegmelse for, hvordan et givet antal atomer opforcr sig under veldefinerede ydre betingelser, medens vi ikke i atomar milestok kan definere et levende vxsens tilstand; p i grund af organismens stofskifte er det ikke engang muligt at afgerre, hvilke atomer der herrer ti1 det levende individ. I denne forstand indtager anvendelsesomridet for den p i korrespondensargumentet opbyggede statistiske kvantemekanik en mellemstilling mellem anvendelsesomridet for den kausale rum-tidsbeskrivelse og det gennem teleologiske betragtningsmider karakteriserede omride af biologien. S k m t denne opfattelse sividt kun angir den fysiske side af sagen, turde den desuden vzre egnet ti1 at skabe en baggrund for indordningen af de ti1 livet knyttede psykiske fznomener. Som diskuteret i den tredie artikel og ogsi berm-t ovenfor, udviser den ved selviagttagelse uundgielige indflydelse p i de af viljesfdelse pragede psykiske oplevelser en sliende lighed med de forhold, som betinger lrsagsforestillingens svigten ved analysen af atomfenomenerne. Fremfor alt turde imidlertid, som antydet,

Indledende O v e r s i g t

25

en vasentlig uddybelse af den oprindelig p i den fysiske irsagsbeskrivelse hvilende psyko-fysiske parallelisine tilbyde sig gennem erkendelsen af den uforudsigelige andring af de psykiske oplevelser, som ethvert forsag p i en objektiv kontrol med de ledsagende fysiske processer i nervesystemet ville medfare. I denne forbindelse m i det dog ikke glemmes, at det ved sammenfatningen af tilvarelsens fysiske og psykiske sider drejer sig om et ejendommeligt komplementaritetsforhold som ikke lader sig fuldt anskueliggare ved ensidige fysiske og psykiske lovmassigheder. I overensstemnielse med atomteoriens alniindelige belaring turde ogsl, som nzrniere forklaret i den fjerde artikel, kun et afkald i denne henseende tillade 0 s at forsti den harnioni, for hvis oplevelse og undersagelse rammerne udgares af viljesfrihed 02 Irsagsbegreb.

Atomic Theory and the Description of Nature I F O U R ESSAYS

With an Introductory Survey BY

NIELS BOHR

CAMBRIDGE A T T H E U N I V E R S I T Y PRESS I934

Of the four articles contained in this volume the first two appeared originally in English in Nature in 1925 and 1927,while the third appeared in German in Die Naturwissenschaften in 1929 and the fourth in Danish in Fysisk Tidsskrift in 1929. The Introductory Survey originally appeared in Danishin the Year Book of Copenhagen University for 1929 together with a Danish translation of the first three articles, the Addendum being first included in the German edition of all four articles published by Jul. Springer, Berlin, in 1931. I am indebted to Prof. Rud Nielsen and Dr Urquhart for their valuable help in preparing the present English translation, and to the Syndics of the Cambridge University Press for their kind interest in this edition, as well as for their courtesy in arranging for this volume to be followed by another containing a number of later essays on the same subject, in which the general point of view is further developed. N. B O H R

COPENHAGEN

February 1934

Introductory Survey (1929) The task of science is both to extend the range of our experience and to reduce it to order, and this task presents various aspects, inseparably connected with each other. Only by experience itself do we come to recognize those laws which grant us a comprehensiveview of the diversity of phenomena. As our knowledge becomes wider, wemust always be prepared, therefore, to expect alterations in the points of view best suited for the ordering of our experience. In this connection we must remember, above all, that, as a matter of course, all new experience makes its appearance within the frame of our customary points of view and forms of perception. The relative prominence accorded to the various aspects of scientific inquiry depends upon the nature of the matter under investigation. I n physics, where our problem consists in the co-ordination of our experience of the external world, the question of the nature of our forms of perception will generally be less acute than it is in psychology where it is our own mental activity which is the object under investigation. Yet occasionally just this " objectivity )' of physical observations becomes particularly suited to emphasize the subjective character of all experience. There are many examples of this in the history of science. I need only mention the great significance that the investigation of acoustical and optical phenomena, the physical media of our senses, has continually had in the development of psychological analysis. As another exB

I

2

T H E ATOMIC THEORY

ample, we may notice the r81e which the elucidation of the laws of mechanics has played in the development of the general theory of knowledge. I n the latest developments of physics, this fundamental feature of science has been particularly prominent. The great extension of our experience in recent years has brought to light the insufficiency of our simple mechanical conceptions and, as a consequence, has shaken the foundation on which the customary interpretation of observations was based, thus throwing new light on old philosophical problems. This is true not only of the revision of the foundations of the space-time mode of description brought about by the theory of relativity, but also of the renewed discussion of the principle of causality which has emerged from the quantum theory. The origin of the theory of relativity is closely bound up with the development of electromagnetic concepts, a development which, by extending the notion of force, has brought about such a profound transformation of the ideas underlying mechanics. The recognition of the relative character of the phenomena of motion, these being dependent upon the observer, already had played an essential part in the development of classical mechanics, where it served as an effective aid in the formulation of general mechanical laws. For the time being, one succeeded in giving an apparently satisfactory treatment of the questions under discussion, both from a physical as well as from a philosophical point of view. It was, in fact, first the recognition, brought about by the electromagnetic theory, of the finite velocity of propagation of all actions of force which brought the matter to a climax. I t is true that it was possible, on the basis of the electro-

3 magnetic theory, to set up a causal mode of description which retained the fundamental mechanical laws of the conservation of energy and momentum, provided one ascribed energy and momentum to the fields of force themselves. However, the conception of a universal ether, which was so useful in the development of the electromagnetic theory, appeared in this theory as an absolute frame of reference for the space-time description. The unsatisfactory character of this conception, from a philosophical point of view, was strongly emphasized by the failure of all attempts to demonstrate the motion of the earth relative to this hypothetical universal ether ;and this situationwas not improved by the recognition that the failure of all such attempts was in complete agreement with the electromagnetic theory. It was Einstein’s elucidation of the limitation which the finite velocity of propagation of all force effects, including those of radiation, imposes upon the possibilities of observation, and, therefore, upon the application of the space-time concepts, that first led us to a more liberal attitude towards these concepts, an attitude which found its most striking expression in the recognition of the relativity of the concept of simultaneity. As we know, Einstein, adopting this attitude, succeeded in tracing significant new relationships also outside the domain to which the electromagnetic theory properly applies, and in his general theory of relativity, in which the effects of gravitation no longer occupy a special position among physical phenomena, he has approached, to a quite unexpected degree, the unity in the description of nature which is the ideal of the classical physical theories. The quantum theory arose out of the development of INTRODUCTORY SURVEY

1-2

4

T H E ATOMIC THEORY

atomic conceptions, which, during the course of the last century, had increasingly provided a fruitful field for the application of mechanics and of the electromagnetic theory. I n the years near the beginning of this century, however, the application of these theories to atomic problems was destined to reveal a hitherto unnoticed limitation that found its expression in Planck’s discovery of the so-called quantum of action, which imposes upon individual atomic processes an element of discontinuity quite foreign to the fundamental principles of classical physics, according to which all actions may vary in a continuous manner. The quantum of action has become increasingly indispensable in the ordering of our experimental knowledge of the properties of atoms. At the same time, however, we have been forced step by step to forego a causal description of the behaviour of individual atoms in space and time, and to reckon with a free choice on the part of nature between various possibilities to which only probability considerations can be applied. The endeavours to formulate general laws for these possibilities and probabilities by a suitably limited application of the concepts of the classical theories have led recently, after a series of phases in its development, to the creation of a rational quantum mechanics by means of which we are able to describe a very wide range of experience, and which may be regarded in every respect as a generalization of the classical physical theories. I n addition, we have gradually reached a complete understanding of the intimate connection between the renunciation of causality in the quantum-mechanical description and the limitation with regard to the possibility of distinguishing between phenomena and their observa-

INTRODUCTORY SURVEY

5

tion, which is conditioned by the indivisibility of the quantum of action. The recognition of this situation implies an essential change in our attitude towards the principle of causality as well as towards the concept of observation. In spite of many points in which they differ, there is a profound inner similarity between the problems met with in the theory of relativity and those which are encountered in the quantum theory. I n both cases we are concerned with the recognition of physical laws which lie outside the domain of our ordinary experience and which present difficulties to our accustomed forms of perception. We learn that these forms of perception are idealizations, the suitability of which for reducing our ordinary sense impressions to order depends upon the practically infinite velocity of light and upon the smallness of the quantum of action. I n appraising this situation, however, we must not forget that, in spite of their limitation, we can by no means dispense with those forms of perception which colour our whole language and in terms of which all experience must ultimately be expressed. It is just this state of affairs which primarily gives to the problems in question their general philosophical interest. While the finish given to our picture of the world by the theory of relativity has already been absorbed into the general scientific consciousness, this has scarcely occurred to the same extent with those aspects of the general problem of knowledge which have been elucidated by the quantum theory. ~

When I was requested to write a paper for the Year Book 1929 of the University of Copenhagen, I first in-

6 THE ATOMIC THEORY tended to give, in the simplest possible form, an account of the new points of view brought about by the quantum theory, starting from an analysis of the elementary concepts on which our description of nature is founded. However, my occupation with other duties did not leave me sufficient time to complete such an account, the difficulty of which arose, not least, from the continuous development of the points of view in question. Sensing this difficulty, I gave up the idea of preparing a new exposition and was led to consider using instead a translation into Danish, made for this occasion, of some articles which, during recent years, I have published in foreign journals as contributions to the discussion of the problems of the quantum theory. These articles belong to a series of lectures and papers in which, from time to time, I have attempted to give a coherent survey of the state of the atomic theory at the moment. Some previous articles of this series form in some respects a background for the three articles which are reproduced here. This is particularly true of a lecture entitled " The Structure of Atoms ", which was given in Stockholm in December 1922,and which was published at the time as a Supplement of Nature. The articles here reproduced appear formally quite independent, however. They are intimately connected with each other, in that they all discuss the latest phase in the development of the atomic theory, a phase in which the analysis of the fundamental concepts has become so prominent. The fact that the articles follow the course of the development, and thus give an immediate impression of the gradual elucidation of the concepts, may perhaps help in some measure to make the subject more easily ac-

I N T R O D U C T O R Y SURVEY

7

cessible to those readers who do not belong to the narrow circle of physicists. I n the following notes on the particular circumstances under which the articles appeared, I have attempted, by the addition of some guiding remarks, to facilitate a general view of the contents and, as far as possible, to make up for such shortcomings of the exposition as might present difficulties to a wider circle of readers. The first artide is an elaboration of a lecture delivered at the Scandinavian Mathematical Congress at Copenhagen in August 1925. It gives in a condensed form a survey of the development of the quantum theory up to that time when a new phase was being ushered in by the paper of Heisenberg which is discussed at the close of the article. The lecture deals with the application of the mechanical concepts within the atomic theory, and shows how the ordering of a vast amount of experimental data with the help of the quantum theory had prepared the way for the new development, which is characterized by the creation of rational quantum-mechanical methods. Above all, the previous development had led to the recognition of the impossibility of carrying out a coherent causal description of atomic phenomena. A conscious resignation in this respect is already implied in the form, irrational from the point of view of the classical theories, of those postulates, mentioned in the article, upon which the author based his application of the quantum theory to the problem of atomic structure. T h e fact that all changes in the state of an atom are described, in agreement with the requirement of the indivisibility of the quantum of action, as individual pro-

8

THE ATOMIC THEORY

cesses by which the atom goes over from one so-called stationary state into another stationary state and for the occurrence of which only probability considerations can be made, must, on one hand, greatly limit the field of application of the classical theories. On the other hand, the necessity of making an extensive use, nevertheless, of the classical concepts, upon which depends ultimately the interpretation of all experience, gave rise to the formulation of the so-called correspondence principle which expresses our endeavours to utilize all the classical concepts by giving them a suitable quantum-theoretical re-interpretation. The detailed analysis of the experimental data from this point of view was, however, destined to show more and more clearly that we did not then possess sufficiently adequate expedients for carrying out a strict description based upon the correspondence principle. Owing to the special occasion on which the lecture was delivered, special emphasis has been placed in the article upon that employment of mathematical aids which is peculiar to theoretical physics. The symbolical forms of expression of mathematics are here not merely indispensable tools for describing quantitative relationships, but they furnish at the same time an essential means for the elucidation of the general qualitative points of view. T h e hope expressed at the conclusion of the article that mathematical analysis would again prove capable of assisting the physicist to surmount his difficulties has in the meantime been fulfilled beyond all expectations. Not only was abstract algebra destined to play a decisive part in the formulation of Heisenberg’s quantum mechanics, as mentioned in the article, but the

9 theory of differential equations-the most important of the expedients of classical physics-was almost immediately afterwards to be extensively applied to atomic problems. The point of departure for this application was the peculiar analogy between mechanics and optics upon which already Hamilton had based his important contribution to the development of the methods of classical mechanics. The significance of this analogy for the quantum theory was first pointed out by de Broglie who, in connection with Einstein’s well-known theory of light quanta, compared the motion of a particle with the propagation of wave systems. As de Broglie pointed out, this comparison made it possible to give a simple geometrical meaning to the quantization rules, mentioned in the article, for the stationary states of the atoms. By a further development of these considerations, Schrodinger succeeded in reducing the quantum-mechanical problem to the solution of a certain differential equation, the socalled Schrodinger wave equation, thus providing us with a method that has played a decisive r61e in the great development which the atomic theory has undergone in the last few years. I N T R O D U C T O R Y SURVEY

The second article is an elaboration of a paper read before an international congress of physicists which took place at Como in September 1927 at the centenary of Volta’s death. By this time the above-mentioned quantum-mechanical methods had reached a high degree of perfection and had demonstrated their fruitfulness in numerous applications. Yet, a divergence of opinion had arisenwith regard to the physical interpretation of the methods, and this had led to much discussion.

I0

THE ATOMIC THEORY

Especially had the great success of Schrodinger’s wave mechanics revived the hopes of many physicists of being able to describe atomic phenomena along lines similar to those of classical physical theories without introducing “irrationalities” of the kind which had thus far been characteristic of the quantum theory. I n opposition to this view, it is maintained in the article that the fundamental postulate of the indivisibility of the quantum of action is itself, from the classical point of view, an irrational element which inevitably requires us to forego a causal mode of description and which, because of the coupling between phenomena and their observation, forces us to adopt a new mode of description designated as complementary in the sense that any given application of classical concepts precludes the simultaneous use of other classical concepts which in a different connection are equally necessary for the elucidation of the phenomena. It is pointed out that we immediately encounter this feature when considering the questions of the nature of light and of matter. It had already been emphasized in the first article that, in our description of radiation phenomena, we are faced with a dilemma as regards the choice between the wave description of the electromagnetic theory and the corpuscular conception of the propagation of light in the theory of light quanta. With regard to matter, the confirmation which, in the meantime, de Broglie’s wave ideas had received by the well-known experiments on the reflection of electrons by metal crystals placed us before a quite similar dilemma, since there can be no question of giving up the idea of the individuality of the elementary particles ;for this individuality forms the secure foundation

INTRODUCTORY SURVEY

I1

on which the whole of the recent development of the atomic theory depends. The main purpose of the article is to show that this feature of complementarity is essential for a consistent interpretation of the quantum-theoretical methods. A very significant contribution to this discussion had been given shortly before by Heisenberg, who had pointed out the close connection between the limited applicability of mechanical concepts and the fact that any measurement which aims at tracing the motions of the elementary particles introduces an unavoidable interference with the course of the phenomena and so includes an element of uncertainty which is determined by the magnitude of the quantum of action. This indeterminacy exhibits, indeed, a peculiar complementary character which prevents the simultaneous use of spacetime concepts and the laws of conservation of energy and momentum, which is characteristic of the mechanical mode of description. To understand why a causal description is impracticable, however, it is essential to remember, as shown in the article, that the magnitude of the disturbance caused by a measurement is always unknown, since the limitation in question applies to any use of mechanical concepts and, hence, applies to the agencies of observation as well as to the phenomena under investigation. This very circumstance carries with it the fact that any observation takes place at the cost of the connection between the past and the future course of phenomena. As already mentioned, the JLinite magnitude of the quantum of action prevents altogether a sharp distinction being made between a phenomenon and the agency by which it is observed, a distinction which

I2

T H E ATOMIC THEORY

underlies the customary concept of observation and, therefore, forms the basis of the classical ideas of motion. With this in view, it is not surprising that the physical content of the quantum-mechanical methods is restricted to a formulation of statistical regularities in the relationships between those results of measurement which characterize the various possible courses of the phenomena. It is emphasized in the article that the symbolical garb of the methods in question closely corresponds to the fundamentally unvisualizable character of the problems concerned. We come across a particularly characteristic example of the limitation imposed upon the possibility of applying mechanical ideas when we employ the concept of stationary states which, as mentioned above, even before the development of the quantum-mechanical methods, entered as an essential element in the application of the quantum theory to problems of atomic structure. As shown in the article, any use of this concept excludes the possibility of tracing the motion of the individual particles within the atom. We are here concerned with a characteristic complementarity analogous to that which we encounter when considering the questions of the nature of light and of matter. As explained in detail the concept of stationary states may indeed be said to possess, within its field of application, just as much, or, if one prefers, just as little " reality" as the elementary particles themselves. I n each case we are concerned with expedients which enable us to express in a consistent manner essential aspects of the phenomena. Besides, when we use the concept of stationary states, the necessity in the quantum theory of paying attention to the delimitation of phenomena and, as emphasized already in the first paragraph of the article, of distinguishing

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strictly between closed and unclosed systems, is brought before us in a very instructive manner. Hence, in the case of atoms, we come upon a particularly glaring failure of the causal mode of description when accounting for the occurrence of radiation processes. While, when following the motions of free particles, we can visualize the lack of causality by considering our lack of simultaneous knowledge of the quantities entering into the classical mechanical description, the limited applicability of classical concepts is immediately evident in our account of the behaviour of atoms, since the description of the state of a single atom contains absolutely no element referring to the occurrence of transition processes, so that in this case we can scarcely avoid speaking of a choice between various possibilities on the part of the atom. In connection with the question of the fundamental properties of the elementary particles, it may perhaps be of interest to call attention to a peculiar complementarity recently disclosed. The fact that the experiments, which so far have been explained by ascribing a magnetic moment to electrons, have been given a natural interpretation by Dirac’s theory, briefly discussed in the last paragraph of the article, is, indeed, equivalent to saying that it is not possible to detect the magnetic moment of an electron by experiments based upon a direct observation of its motion. The difference between free electrons and atoms, which we come upon here, is connected with the fact that measurements of the magnetic moment of atoms involve a renunciation, in accordance with the general conditions holding for the application of the concept of stationary states, of all attempts to trace the motion of the elementary particles. The important task, touched upon at the close of the

I4

T H E ATOMIC THEORY

article, of satisfying the general demand for relativity within the frame of the quantum theory, has not as yet been carried out satisfactorily. Indeed, the abovementioned theory of Dirac, although a great step forward in this respect, has brought to light new difficulties. The recognition of these, however, may lead to the development of new points of view with regard to the profound problems presented by the very existence of elementary particles. While the present quantum-mechanical description depends upon a re-interpretation, based on the correspondence principle, of the classical electron theory, the classical theories offer no guide whatever to the understanding of the existence of the elementary particles themselves and of their specific mass and electrical charge. We must, therefore, be prepared to find that further advance into this region will require a still more extensive renunciation of features which we are accustomed to demand of the space-time mode of description than the quantum theory attack on the atomic problem has required thus far, and we must be prepared to expect new surprises with regard to the applicability of the concepts of momentum and energy. T h e extensive use of mathematical symbols which is peculiar to the methods of quantum mechanics makes it difficult to give a true impression of the beauty and logical consistency of these methods without going into mathematical details. Although in the preparation of this article I have endeavoured, as far as possible, to avoid the use of mathematical artifices, yet the purpose of the lecture, delivered before a group of physicists, to open a discussion on the present tendency in the development of the quantum theory, has made it necessary to go into

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details which will doubtless make it difficult for readers not somewhat acquainted beforehand with the subject. However, I wish to point out that throughout the article the main emphasis has been placed upon that purely epistemological attitude which is particularly apparent in the first section and in the concluding remarks. I n the third article, which is a contribution to a jubilee pamphlet published in June 1929to celebrate the fiftieth anniversary of Planck’s doctorate, I have discussed in more detail the general philosophical aspects of the quantum theory. Partly in view of the regret, so widely expressed, with regard to the renunciation of a strictly causal mode of description for atomic phenomena, the writer attempts to show that the difficulties concerning our forms of perception, which arise in the atomic theory because of the indivisibility of the quantum of action, may be considered as an instructive reminder of the general conditions underlying the creation of man’s concepts. The impossibility of distinguishing in our customary way between physical phenomena and their observation places us, indeed, in a position quite similar to that which is so familiar in psychology where we are continually reminded of the dzficulty of distinguishing between subject and object. It may perhaps appear at first sight that such an attitude towards physics would leave room for a mysticism which is contrary to the spirit of natural science. However, we can no more hope to attain to a clear understanding in physics without facing the difficulties arising in the shaping of concepts and in the use of the medium of expression than we can in other fields of human inquiry. Thus, according

16

T H E A T O M I C THEORY

to the view of the author, it would be a misconception to believe that the difficulties of the atomic theory may be evaded by eventually replacing the concepts of classical physics by new conceptual forms. Indeed, as already emphasized, the recognition of the limitation of our forms of perception by no means implies that we can dispense with our customary ideas or their direct verbal expressions when reducing our sense impressions to order. No more is it likely that the fundamental concepts of the classical theories will ever become superfluous for the description of physical experience. The recognition of the indivisibility of the quantum of action, and the determination of its magnitude, not only depend on an analysis of measurements based on classical concepts, but it continues to be the application of these concepts alone that makes it possible to relate the symbolism of the quantum theory to the data of experience. At the same time, however, we must bear in mind that the possibility of an unambiguous use of these fundamental concepts solely depends upon the self-consistency of the classical theories from which they are derived and that, therefore, the limits imposed upon the application of these concepts are naturally determined by the extent to which we may, in our account of the phenomena, disregard the element which is foreign to classical theories and symbolized by the quantum of action. It is just this state of affairs that is so evident in the frequently discussed dilemma with regard to the properties of light and of matter. Only in terms of the classical electromagnetic theory is it at all possible to give a tangible content to the question of the nature of light and of matter. It is true that light quanta and

17

I N T R O D U C T O R Y SURVEY

matter waves are invaluable expedients in the formulation of the statistical laws governing such phenomena as the photo-electric effect and the interference of electron rays. However, these phenomena belong, indeed, to a domain in which it is essential to take into account the quantum of action and where an unambiguous description is impossible. The symbolical character, in this sense, of the artifices mentioned also becomes apparent in that an exhaustive description of the electromagnetic wave fields leaves no room for light quanta and in that, in using the conception of matter waves, there is never any question of a complete description similar to that of the classical theories. Indeed, as emphasized in the second article, the absolute value of the so-called phase of the waves never comes into consideration when interpreting the experimental results. I n this connection, it should also be emphasized that the term “probability amplitude’’ for the amplitude functions of the matter waves is part of a mode of expression which, although often convenient, can, nevertheless, make no claim to possessing general validity. As mentioned above, only with the help of classical ideas is it possible to ascribe an unambiguous meaning to the results of observation. We shall, therefore, always be concerned with applying probability considerations to the outcome of experiments which may be interpreted in terms of such conceptions. Consequently, the use made of the symbolic expedients will in each individual case depend upon the particular circumstances pertaining to the experimental arrangement. Now, what gives to the quantum-theoretical description its peculiar characteristic is just this, that in order to evade the quantum of action we must use separate B

2

18

THE ATOMIC THEORY

experimental arrangements to obtain accurate measurements of the different quantities, the simultaneous knowledge of which would be required for a complete description based upon the classical theories, and, further, that these experimental results cannot be supplemented by repeated measurements. I n fact, the indivisibility of the quantum of action demands that, when any individual result of measurement is interpreted in terms of classical conceptions, a certain amount of latitude be allowed in our account of the mutual action between the object and the means of observation. This implies that a subsequent measurement to a certain degree deprives the information given by a previous measurement of its significance for predicting the future course of the phenomena. Obviously, these facts not only set a limit to the extent of the information obtainable by measurements, but they also set a limit to the meaning which we may attribute to such information. We meet here in a new light the old truth that in our description of nature the purpose is not to disclose the real essence of the phenomena but only to track down, so far as it is possible, relations between the manifold aspects of our experience. It is against this background that we must judge the difficulties which we come upon if we attempt to give a correct impression of the content of the quantum theory and of its relation to the classical theories. As already emphasized when discussing the. second article, these questions can be fully elucidated only in terms of the mathematical symbolism which has made it possible to formulate the quantum theory as a rigorous re-interpretation, based upon the idea of correspondence, of the classical theories. In view of the reciprocal symmetry

INTRODUCTORY SURVEY

I9 peculiar to the use of the classical concepts in this symbolism, the writer in this article has preferred the term " reciprocity to the word " complementarity '),used in the preceding article to denote the relation of mutual exclusion characteristic of the quantum theory with regard to the application of the various classical concepts and ideas. Meanwhile, as the result of further discussion, it has come to my notice that the former term may be misleading because the word " reciprocity ')is frequently used in the classical theories with a quite different meaning. The term " complementarity", which is already coming into use, may perhaps be more suited also to remind us of the fact that it is the combination of features which are united in the classical mode of description but appear separated in the quantum theory that ultimately allows us to consider the latter as a natural generalization of the classical physical theories. Moreover, the purpose of such a technical term is to avoid, so far as possible, a repetition of the general argument as well as constantly to remind us of the difficulties which, as already mentioned, arise from the fact that all our ordinary verbal expressions bear the stamp of our customary forms of perception, from the point of view of which the existence of the quantum of action is an irrationality. Indeed, in consequence of this state of affairs, even words like " to be ') and " to know lose their unambiguous meaning. I n this connection, an interesting example of ambiguity in our use of language is provided by the phrase used to express the failure of the causal mode of description, namely, that one speaks of a free choice on the part of nature. Indeed, properly speaking, such a phrase requires the idea of an external chooser, the existence of ))

))

2-2

20

T H E A T O M I C THEORY

which, however, is denied already by the use of the word nature. We here come upon a fundamental feature in the general problem of knowledge, and we must realize that, by the very nature of the matter, we shall always have last recourse to a word picture, in which the words themselves are not further analyzed. As emphasized in the article, we must, indeed, remember that the nature of our consciousness brings about a complementary relationship, in all domains of knowledge, between the analysis of a concept and its immediate application. The reference to certain psychological problems in the latter part of the article has a twofold purpose. The analogies with some fundamental features of the quantum theory, exhibited by the laws of psychology, may not merely make it easier for us to adjust ourselves to the new situation in physics, but it is perhaps not too ambitious to hope that the lessons we have learned from the very much simpler physical problems will also prove of value in our endeavours to obtain a comprehensive survey of the more subtle psychological questions. As stressed in the article, it is clear to the writer that for the time being we must be content with more or less appropriate analogies. Yet it may well be that behind these analogies there lies not only a kinship with regard to the epistemological aspects, but that a more profound relationship is hidden behind the fundamental biological problems which are directly connected to both sides. Although it cannot yet be said that the quantum theory has contributed essentially to the elucidation of the latter problems, still there is much which indicates that we are concerned here with questions which closely approach the circle of ideas of the quantum theory. Indeed, living

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organisms are first of all characterized by the sharp separation of the individuals from the outside world and their great ability to react to external stimuli. It is very suggestive that this ability, at least so far as sight impressions are concerned, is developed to the utmost limit permitted by physics; for, as has often been remarked, only a few light quanta are sufficient to produce a visual sensation. Nevertheless, it is obviously a quite open question whether the information we have acquired of the laws describing atomic phenomena provides us with a sufficient basis for tackling the problem of living organisms, or whether, hidden behind the riddle of life, there lie yet unexplored aspects of epistemology. Whatever the development in this domain may be, we have, as emphasized at the close of the article, every reason to rejoice that, within the relatively objective domain of physics, where emotional elements are so largely relegated to the background, we have encountered problems capable of reminding us anew of the general conditions underlying all human understanding, which, from time immemorial, have attracted the attention of philosophers.

Addendum (1931). The fourth article, which is an elaboration of a lecture delivered before the Scandinavian Meeting of Natural Scientists in 1929,is closely related to the other three articles, since it attempts to give a survey, against the same background, of the place of the atomic theory in the description of nature. In particular, it was my desire to emphasize that, despite the great success attending the discovery of the building stones of atoms-a discovery depending on the applica-

22

THE ATOMIC THEORY

tion of classical concepts-the development of the atomic theory has, nevertheless, first of all given us a recognition of laws which cannot be included within the frame formed by our accustomed modes of perception. As already indicated above, the lessons we have learned by the discovery of the quantum of action open up to us new prospects which may perhaps be of decisive importance, particularly in the discussion of the position of living organisms in our picture of the world. If, according to the ordinary usage, we speak of a machine as dead, this only means that we can give a description, sufficient for our purpose, of its working in terms of the conceptual forms of classical mechanics. However, in view of the present recognition of the insufficiency of classical concepts in the domain of atomic theory this criterion of the inanimate is no longer suitable so far as atomic phenomena are concerned. Nevertheless, even the quantum mechanics may hardly depart sufficiently from the mode of description of classical mechanics to be capable of mastering the characteristic laws of life. I n this connection, however, we must remember that the investigation of the phenomena of life not only leads us, as emphasized in the article, into that domain of atomic theory where the usual idealization of a sharp distinction between phenomena and their observation breaks down, but that, in addition, there is set a fundamental limit to the analysis of the phenomena of life in terms of physical concepts, since the interference necessitated by an observation which would be as complete as possible from the point of view of the atomic theory would cause the death of the organism. I n other words : the strict application of those concepts which are

23 adapted to our description of inanimate nature might stand in a relationship of exclusion to the consideration of the laws of the phenomena of life. In exactly the same way as it is only possible on the basis of the fundamental complementarity between the applicability of the concept of atomic states and the coordination of the atomic particles in space and time to account, in a rational manner, for the characteristic stability of the properties of atoms, so might the peculiarity of the phenomena of life, and in particular the selfstabilizing power of organisms, be inseparably connected with the fundamental impossibility of a detailed analysis of the physical conditions under which life takes place. To put it briefly, one might perhaps say that quantum mechanics is concerned with the statistical behaviour of a given number of atoms under well-defined external conditions, while we are unable to define the state of a living being in terms of atomic measures ; in fact, owing to the metabolism of the organism, it is not even possible to ascertain what atoms actually belong to the living individual. In this respect, the domain of the statistical quantum mechanics, which is based on the correspondence argument, occupies an intermediate position between the domain of applicability of the ideal of causal space-time mode of description and the domain of biology which is characterized by teleological arguments. Although, put in the above way, this idea concerns only the physical aspect of the problem, it may perhaps also be suited to form a background for the ordering of the psychical aspects of life. As explained in the third article, and also touched upon above, the unavoidable influencing by introspection of all psychical experience, INTRODUCTORY SURVEY

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THE ATOMIC THEORY

that is characterized by the feeling of volition, shows a striking similarity to the conditions responsible for the failure of causality in the analysis of atomic phenomena. Above all, as indicated there, an essential refinement of our interpretation, originally based on physical causality, of the psycho-physical parallelism ought to result from our taking into consideration the unpredictable modification of psychical experience produced by any attempt at an objective tracing of the accompanying physical processes in the central nervous system. With regard to this, however, it must not be forgotten that, in associating the psychical and physical aspects of existence, we are concerned with a special relationship of complementarity which it is not possible thoroughly to understand by one-sided application either of physical or of psychological laws. I n consideration of the general lessons we have learned from the atomic theory, it would also seem likely that only a renunciation in this respect will enable us to comprehend, in the sense explained more fully in the fourth article, that harmony which is experienced as free will and analyzed in terms of causality.

w

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The participants at the Copenhagen Conference, 1929.

INTRODUCTION bY

J0RGEN KALCKAR

1.

AN INTERLUDE: THE MAGNETIC ELECTRON

O n Monday, April 8, 1929, a conference, which was to become the first of the celebrated series of annual “Copenhagen Conferences”, opened at the Institute. It was quite a family gathering which included among others Darwin, Ehrenfest, Kramers, Pauli, Goudsmit, Kronig, Mott (on the last day of the week) and the young Leon Rosenfeld, who has left us a vivid account of the meeting’. The introductory lecture was given by Bohr, and Rosenfeld depicts it thus: “I am not sure whether Bohr’s introductory talk at the conference was really worse than the average; perhaps he had not prepared it so thoroughly, since the idea was to have quite informal discussions: no programme had been set up in advance - Bohr took in turn each of the participants aside and asked him what topic he wished to bring up. At any rate, here is the impression this talk has left in my memory, as I described it (with some hindsight) in 1945: ‘He had begun with a few general considerations calculated, no doubt, to convey to the audience that peculiar sensation of having the ground suddenly removed from under their feet, which is so effective in promoting receptiveness for complementary thinking. This preliminary result being readily achieved, he had eagerly hastened to his main subject and stunned us all (except Pauli) with the non-observability of the electron spin. I spent the afternoon with Heitler pondering on the scanty fragments of the hidden wisdom which we had been able to jot down in our note books.’ It was comforting to hear from Klein, when I told him some time ago of our failure to understand what Bohr meant by the impossibility of measuring

’

L. Rosenfeld, Quantum Theory in 1929. Recollections from the first Copenhagen Conference in the jubilee publication, Institute f o r Theoretical Physics - The Niels Bohr Institute 1921-71, Rhodos, Copenhagen 1971. Also Nordita publication, 1971. Cf. also, My Initiation, Journal of Jocular Physics 2 , 1945.

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

Pauli and Ehrenfest on the ferry boat on their way to the Copenhagen Conference, 1929 (Courtesy AIP, Goudsmit Collection).

Discussions at the blackboard (Courtesy AIP, Goudsmit Collection).

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

Bohr with Ehrenfest, Casimir and Klein.

the spin of the electron, that he had had the same difficulty when Bohr first discussed the matter with him in the autumn of 1928. Guided by the general correspondence idea, Bohr argued that such a purely quanta1 concept as the electron spin, vanishing from the theory in the classical limit, could not possibly be brought in direct relation with classical quantities like angular momentum or magnetic moment2. It was not immediately clear to Klein, however, how this correspondence argument could be reconciled with the Stern-Gerlach effect, which clearly exhibited a contribution to the magnetic moment of an atom from an electron bound in a 2S state; but what Bohr demonstrated was precisely that with a free electron a Stern-Gerlach experiVide infra, a manuscript on the magnetic electron, reproduced on p. [331], as well as the discussion by Pauli at the Solvay Meeting 1930, reproduced on p. [337]. Cf. also the correspondence between Bohr and Joffe: Bohr to JoffC, 25 October 1928 and 27 December 1928, and Joffe to Bohr, 18 January 1929. BSC, microfilm no. 12.

En Fysiker-Kongres i Kjsbenhavn. Professor Niels Bohr hrr paa sit InstItut ramlet en Raekke berrmte fy. eikere ti1 Kmgree, der aabnedes her i Byen i O a r .

Profersor Nicls Bohr aabnede i Gaar

ode paa rit Inrtituf paa Blegdamsvejen an Kongrea, hvori mange beremte Fgrikere tager Del. Man ma den hollandder en Ovcrrke Professor C'mmcrr den gang var knyttet ti1 Inrtitutet engelske Professor Dancin, Tyrkeren, Professor Ehtenfess, den ruasiske Dr. Cumon, de tyske Doktorer KZein, Pauli, Beiden og A'ordheim og K i n w r e n YeiPien-C'liou. Mellern de danske Tilherere vnr i Gnar Profemorerne HamM Bohr, Jacob Nielscn og Bierrum, Docent Wcrnet og Forf8tteren H c l g e B d s t . Da Profeseor Nieh Bohr pa. Tysk bavde budt vclkommen, redegjorde han for Eongreeseas FOrm881: Gennem Foredrag og Diskuesioner vil man rege a t uddrbe de mange Atomberimer, eom bar met Dagenr LY'Rde ridste Aor, og 198 muligt sege a t rkabe en Sammenhleng imellem dem. 1 et to Timer langt Foredrag, soom hilstee med begejsttct Bifald, holdt Profesmr Bohr rine Tilherere i Aande om dette Glmrgsniaal.

Professor Niels Bohr:

Dr. Pauli talte derefter om Einskins 6tilling ti1 de Dohreke Probleincr, og derpled var en ivrig Diskuuion atrake i Gang. Man blev red a t diskutere lierotn i flcrc Timer, og man fortsaetter i Formiddag,

Kongressen varer Ugen ud. Fernte ingene Dirkussioner blev ud over 64. ran8 livlige, 888 man i videnskabelige Krese venter 8ig meget of Fortssettelren.

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Newspaper report of the Copenhagen Conference, 1929 (Polifiken, 9 April 1929).

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

ment could not succeed, because the effect of the Lorentz force would inevitably blur any Stern-Gerlach pattern. This is the point he ineffectually tried to make in his talk. Fortunately, Mott, during his stay at the Institute, had been engaged in the problem of electron polarization, and in the paper3 in which he brilliantly showed how this property could in principle be ascertained by a double scattering experiment, he gave a very clear account of the whole situation. He finished writing this paper shortly after the conference (it was sent off by Bohr on the 25th of April) and we were thus soon able to appreciate at leisure the full force of Bohr’s famous argument.” In spite of the completion of Mott’s paper3 Bohr continued to ponder the problem of the measurability of the magnetic moment of a free electron, and three months later (July 1st) he sent Pauli a small manuscript (together with another draft concerning P-decay4): “In order that you may see that it was not altogether untruthful to promise the other notes, I enclose two fragments. One is the beginning of a note on the magnetic electron, which I had to put aside because of the Planck issue. The other is a little piece about the P-ray spectra, which I have had in mind for a long time, and which has been typed in the last few days, but I have not yet made up my mind to send it off, since it yields so few positive results and has been written so sketchily. I shall be happy to hear your opinion about all of this, no matter how severe or how mild the expressions which you feel it appropriate to use.”

Bohr to Paul], 1 Jul) 29 Danish text on p. 14411 Translation o n p . [4411

As always Bohr could reckon upon a prompt and direct reaction from Pauli. Whereas the draft on P-decay gave him “very little satisfaction”, he comments favourably on the note about the magnetic electron: “I liked the little note on the magnetic electron so much that I deeply regretted that it has not been completed and sent off for printing. If I may give you a piece of advice, then it would be to send off this note w o r e anything else, and to do it as soon as possible!”

Bohr’s manuscript note on the magnetic electron is reproduced on p. [331]. We may surmise that during Pauli’s visit to Tisvilde in August (or on later occasions), he and Bohr subjected the matter to further scrutiny, since we find N.F. Mott, The Scattering of Fast Electrons by Atomic Nuclei, Proc. Roy. SOC.A124 (1929) 425-442. The discussion of the measurability of the magnetic moment of a free electron is added in an appendix. For a discussion of Bohr’s ideas on a possible violation of energy conservation in 8-decay, cf. the Introduction to Vol. 9. Cf. the part of Pauli’s letter of July 17, quoted in the Introduction to Part I1 (p. [194]).

Pauli to Bohr, 17 Jul) 29 German text on p , [444] Translation on p. 14461

P A R T 111: GENERAL ASPECTS O F PHYSICAL DESCRIPTION

that at the Solvay Meeting of 1930 on Magnetism in Brussels (October 20-25), Pauli includes in his report a thorough discussion of Bohr’s proof that the magnetic moment of a free electron cannot be measured either by magnetometers or by a Stern-Gerlach experiment. Since Pauli explicitly refers to Bohr as the source for his argumentation, and since we are fortunate enough to possess in the Archive a copy of Pauli’s German manuscript, we reproduce the relevant section of this in the original language together with the corresponding part of the subsequent discussion (vide p. [3371). Bohr’s demonstration is in itself quite clear, but the question remains as to what are the general implications of the arguments6. Thus, Darwin conceived the idea that he could demonstrate the impossibility of measuring the charge of a free electron. On his return to Edinburgh he sent Bohr a note to that effect with an accompanying letter The note has not been found and from the few hints in Darwin’s letter it does not seem possible to reconstruct the line of his argumentation. The aim was to show, as he puts it in the letter, “that just as one could not measure the moment without loading the electron with an atom, so one could not measure the charge without loading it with an oil drop.” Darwin’s note and letter prompted Bohr not only to point out the flaw in Darwin’s argument, but also to comment on further aspects of the magnetic measurements. Some of these points were later to come to the fore in connection with his elaborate investigation (together with Rosenfeld) of the measurability of electromagnetic fields8 (to which we return in Vol. 7), and for that reason we quote the following two letters in full:

’.

[Copenhagen,] November 15th, [19]30 Dear Charles, I have read your kind letter and note with great interest, but I can not agree with your conclusions regarding the inobservability of the charge of a free electron. Vaguely I remember that, as you write, you made once here a question, whether there might not be an analogy between the possibilities of measuring the magnetic moment and the electric charge in that sense that in both cases it should be necessary that the electron was coupled to a heavy par-

Bohr to D a w i n , 15 Nov 30 English

In the later discussions of this issue one has not always with sufficient clarity distinguished between measurements of the electron spin and of its magnetic moment. In fact, any single component of the electron spin can be measured with arbitrary accuracy by relying on the conservation law for angular momentum. In contrast, the determination of the magnetic moment depends upon specific assumptions regarding the electromagnetic properties of the electron, in particular the gyromagnetic factor. Letter from Darwin to Bohr, 10 November [1930]. BSC, microfilm no. 18. N. Bohr and L. Rosenfeld, Zur Frage der Messbarkeit der elektromagnetischen Feldgrossen, Mat.Fys. Medd. Dan. Vidensk. Selsk. 12, no. 8 (1933). Reproduced in Vol. 7 .

’

’

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

ticle. I do not remember, however, if I gave an answer, but from my point of view the question itself involves a contradiction, since it should in no case be possible to “measure” the magnetic moment of the electron. The point about the Stern-Gerlach effect is just that by classical concepts we cannot distinguish between an intrinsic electronic magnetism and a magnetism originating from intra-atomic motions. In Millikan’s measurements of the electronic charge the situation is quite different as the classical interpretation of the phenomenon is not affected by the mechanism by which the electron is attached to the oil drop. This drop forms of course a convenient tool of handling the electron, but although in practice it may be very difficult, there are no principal difficulties involved in measuring the electric and magnetic forces exerted by a free electron. Consider for instance an ordinary deflection experiment where electrons are passed separately between certain electrified or magnetized bodies, and assume that these bodies are free to move, so that we can measure the momentum delivered to them during the passage of each electron. Due to the conservation of momentum the limitation imposed by the uncertainty relation on such measurements corresponds exactly to that imposed by this relation on the deflection measurements themselves. It is thus only the detection of spin effects which is excluded, and from the argument it follows at once, that this will just represent the limit of the correlation between simultaneous measurements of the magnetic force exerted by the electron on different test bodies used as magnetometers. Of course the measuring instruments must be properly chosen, and I am sorry that in our interrupted conversation with Richardson in Brussels I did not get opportunity to enter more fully on this point. Thus, with my remark as to the importance of taking the time necessary for the measurements into account, if one tries to use the Zeeman effect on spectral lines as magnetometer, I had a calculation in mind by which some time before we had convinced ourselves by our discussions here that the Zeeman effect is wholly unsuited to measure the magnetic force from a single free electron. Indeed this may be considered as a special case of the general theorem very easily proved, that it is never possible to measure the magnetic effects exerted by two electrons on one another, because these effects are always shadowed by the uncertainty in the motion. Principally, however, there is no difficulty in measuring the electric repulsion for instance by following the path of two free electrons during a collision. As a simple argument shows, the condition for such a measurement is only that the relative velocities of the electrons are small compared with the “velocity” in the “normal orbit” of the hydrogen atom. In this connection it is interesting to note that your demonstration of the unsuitability of the Stark effect as the measuring instrument for the electric force from an electron depends entirely

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O X

on the assumption, that the hydrogen atom concerned finds itself in the normal state. If you had considered stationary states of high quantum numbers you would have found quite different results. The marked difference between the Zeeman and the Stark effects is just that the latter in contrast to the first differs widely for different spectral lines. Since we returned from Brussels I have been busy in preparing an account of my views on the general problems of atomic theory. In a few weeks I hope also to complete the manuscript of the address I delivered in Edinburgh in May, the work on which to my great regret has been delayed by a sequence of unforeseen disturbances. I shall be thankful if you will kindly explain matters to Professor Samfrson, to whom I shall write very soon. With kindest regards to Katherine and yourself from us all, yours [Niels Bohr] P.S. Calculation of 2 electron case (normal hydrogen velocity u o = e 2 / h ) electric interaction

magnetic interaction

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Bohr to Daruin. 18 h o \ 30 English

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[Copenhagen,] November 18th, [19]30 Dear Darwin, Since I wrote to you I have considered the problem of measuring electric and magnetic fields more closely, and to avoid misunderstanding of my letter I should like to add a few remarks. In measuring fields we must distinguish between static, or quasi stationary, fields on one hand and electromagnetic wave fields which in no system of reference are stationary on the other hand. In the first case, which corresponds to the considerations in my letter, the time of propagation of the fields is unessential, and in discussing the limit of accuracy obtainable we need neither consider the reaction of the measuring instruments on the source of the field nor the interaction between these instruments on one another. Also in the second case the reaction of the measurements on the source is in general only of secondary importance, but

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O K

Charles G . Darwin.

it is often essential not to forget the interaction of the different measuring instruments on one another. We have here to do with the problem considered by Heisenberg in his new book9, where he has formulated a reciprocal uncertainty relation between the accuracy with which electric and magnetic fields can be measured simultaneously. These considerations are very illuminating, but in the book it is hardly sufficiently emphasized that the uncertainty in question is only material for electromagnetic wave fields. The sharp distinction between wave fields and static fields may perhaps look artifical from the point of view of relativity theory. Nevertheless it is only the possibility of a widegoing distinction of this kind, originating in the small value of e2/hc on which the whole correspondence attack on atomic problems rests. Of course it is also this situation, which makes the developing of a proper relativistic quantum theory so difficult a task. Yours, [Niels Bohr] W. Heisenberg, Die physikalischen Prinzipien der Quantentheorie, Hirzel Verlag, Leipzig 1930. English edition: The Physical Principles of the Quantum Theory, Chicago University, 1930. Reprinted by Dover Publ.

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Darwin readily recognized1° that his argument for the non-measurability of the electron charge did not hold water, and he proposed to publish a revised version of his note, in which he also discussed an extension of Bohr’s magnetic experiments to a situation involving several magnetometers”. Within three days Bohr drafted an answer to Darwin, which was however discarded (the manuscript, dated December 16, is preserved in the Archive collection). The final version of the letter (December 21) is interesting because Bohr here - in his mild way, but nevertheless quite emphatically - suggests a difference in point of view between Darwin and himself as regards the r6le of the uncertainty relations. In the introductory paragraph to his paper “Examples of the Uncertainty Principle”, Darwin had written:

“In the general synthesis of classical dynamics with the quantum theory, the Uncertainty Principle* plays a most useful part. It is of course only one aspect of the new mechanics, but is is a very helpful one since by its means it becomes easy to see where the old classical ideas broke down. The state of affairs in the quantum theory is not unlike that of the early days of relativity, when most of those who studied the subject felt the need of supporting the formal theory by seeing how the old ideas failed in specific cases. Here the formal theory is very abstract and is not easy to follow intuitively, and the Uncertainty Principle plays much the same r61e as did the examples of clocks and rods in relativity theory. For this reason it is more appropriate for illustrative examples, than for any extreme generality, and though a number of examples have been already given by Heisenberg and Bohr**, it may not be amiss to have some more. There are probably some who will have shared my experience that it is often by no means easy to detect how the uncertainty enters into a given experiment, though once detected the arguments are usually very simple.” Bohr’s letter runs as follows: [Copenhagen,] December 21, [19]30 Dear Charles, It was a great pleasure to me from your new note to see, how completely you have succeeded in clearing up all those points which puzzled you hitherto. Your analysis of the reaction accompanying field measurements by means of Stark- and Zeeman-effects brings out in a very beautiful way indeed the lack * “Heisenberg, ‘Z. Physik’, vol. 43, p. 172 (1927).” * * “See for example, Heisenberg, ‘The Physical Principles

of the Quantum Theory’, chap, 2.”

Letter from Darwin to Bohr, 13 December 1930. BSC, microfilm no. 18. C.G. Darwin, Examples of the bncertainty Principle, Proc. Roy. SOC.A130 (1931) 632-639. Cf. also, The Diamagnetism of the Free Electron, Proc. Camb. Phil. SOC.27 (1930-1931) 86-90. lo

‘I

Bohr IO Daruin, 21 Dec 30 English

P A R T 111: G E N E R A L ASPECTS OF PHYSICAL DESCRIPTIOY

of correlation required for the consistency of quantum theory. From my own experience I appreciate fully the trouble you have had in getting this question straight. The salient point, the impossibility of fixing the time of the reaction, reminded me of the puzzle dealt with in my old Nature note: why the knowledge of the position of a particle is spoiled by the measurement of its velocity by Doppler effect. In your letter I recognize my own feelings when after a good deal of trouble I got hold of this point; it was a lesson about the shortcoming of our ordinary ideas in accounting for the behaviour of the measuring tools; which is of course the bottom of the uncertainty. For readers’ orientation it may perhaps be helpful to mention this close analogy between velocity measurements by Doppler-effect and measurements by means of Zeeman- or Stark-effects. In location by a microscope, the time coordination comes only in secondary, and the essence of the uncertainty argument is, as you know, easily confused by the complication of focusing, not met with in the other cases. Another point in your note, on which I should like to comment, is your reference to our conversations on the magnetic electron, which might give readers the impression that I have been sceptical as regards your present endeavours, while I have only been on the defensive of the views you attacked at the London meeting, and about the innocence of which I take it that we now agree entirely. In saying “agree” I shall confess, however, that sometimes I feel a little puzzled by the way you like to express yourself regarding the tendencies symbolized in the uncertainty principle: as if this symbol of our common earnest striving for an objective description of atomic phenomena was but a padagogical trick to appease ourselves with the caprices of nature in her artful play with particle and wave toys. Of course, this is a matter of style in which you are the best judge yourself, and I need hardly add, that all my remarks are just meant as background for my hearty appreciation of your successful endeavours. With kindest regards from us all and best wishes for Christmas and the new year for yourself and Katherine and the children, yours, [Niels Bohr]

2.

THE MAXWELL AND FARADAY LECTURES. BOHR’S VIEWS ON THERMODYNAMICS A N D STATISTICAL MECHANICS

The first half of 1930 saw Bohr even more than usually busy lecturing. Thus, in February and March he held three lectures in Fysisk Forening’* and appeared

’’

Manuscript, Kvunfefeorien og de klussiske fysiske Teorier [Quantum Theory and the Classical Physical Theories] (lectures given o n 10 February, 24 February and 24 March 1930). Bohr MSS, microfilm no. 12.

P A R T I l l : G E N E R A L A S P E C T S OF P H Y S I C A L D E S C R I P T I O N

in Cambridge as the first “Scott Lecturer” on three occasions at the end of April and beginning of May13. The Faraday Lecture was delivered before the Chemical Society in London on May 8, towards the end of the month Bohr addressed the Royal Society of Edinburgh14, and in June he lectured in Berlin”. Various manuscript notes and shorthand reports of the lectures (rather incomprehensible except in the case of the Faraday Lecture) are preserved in the Bohr Archive, but only the Faraday Lecture was published in elaborated form - “delayed owing to unforeseen circumstances” - in February 1932 in the Journal of the Chemical Society, LondonI6. During the almost two years between the delivery and publication of the Faraday Lecture, we have from Bohr’s hand summaries of lectures at the Danish Academy and at the Centenary Meeting of the British Association for the Advancement of ScienceI8. Furthermore, there is the Maxwell Lecturelg from October 1931. The formulations in the printed version of the Maxwell Lecture are strikingly less careful than in Bohr’s other papers. This circumstance, together with the fact that the lecture was published only three weeks after its delivery, may suggest that it was printed from a shorthand version which was not corrected by Bohr himself. However that may be, it is of interest to notice how forcefully Bohr here emphasizes his view on the purely symbolic character of the photon concept. On October 5 , 1931, Bohr lectured at the University of Bristol on “SpaceTime-Continuity and Atomic Physics”20. The lecture was never published, but a fairly complete, typewritten manuscript is preserved in the Archive. We have l 3 Manuscript, Cambridge Lectures, The Principles of Atomic Theory (lectures given at the Cavendish Laboratory on 28 April, 29 April and 2 May 1930). Bohr MSS, microfilm no. 12. l 4 Manuscript, Philosophical Aspects of Atomic Theory (lecture given on 26 May 1930, on the award of the James Scott Prize). Bohr MSS, microfilm no. 12. Summary published in Nature 125 (1930) 958. This summary is reproduced on p. [351].

Manuscript, Das Wirkungsquantum [The Quantum of Action] (lecture given before the Deutsche Physikalische Gesellschaft on 20 June 1930, on the award of the Planck Medal). Bohr MSS, microfilm no. 12. l 6 N. Bohr, Chemistry and the Quantum Theory of Atomic Constitution, J . Chem. SOC.London, 1932, pp. 349-384. Reproduced on p. [371]. N . Bohr, Om Benyttelsen af Begreberne Rum og Tid i Atomteorien, Overs. Dan. Vidensk. Selsk. Forh. Juni 1930 - Maj 1931, p. 26. Translation: The use of the concepts of space and time in atomic theory, Nature 127 (1931) 43. Reproduced on p. [353]. N. Bohr, On Atomic Stability, Brit. Ass. Adv. Sci., Report of the Centenary Meeting, London - 1931, September 23-30, London 1932, p. 333. Reproduced on p. [355]. l9 N. Bohr, Maxwell and Modern Theoretical Physics. Address delivered on the occasion of the Maxwell Centenary Celebrations at Cambridge on October 1, 1931. Published in Nature 128 (1931) 691-692. Reproduced o n p. [357]. 2o Manuscript, Bristol Lecture, 1931. Reproduced on p. [361].

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P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

not been able to decide whether the manuscript was prepared by Bohr before the lecture or whether it was written up afterwards, perhaps on the basis of a stenographic report. The formulations are evidently of a preliminary character, perhaps suited to oral delivery, but very far from the polished style of Bohr’s writings. Of special interest in this manuscript is the discussion of the weighing experiment, proposed by Einstein during discussions in connection with the Solvay Meeting in 1930 with the intention of circumventing the uncertainty relation between energy and time. This is the earliest reference we have from Bohr’s hand to this experiment and not least for this reason the manuscript is included in the present volume. An improved version of Bohr’s refutation of Einstein’s argument is found in his article on the occasion of Einstein’s 70th birthday in 1949 (cf. the Introduction to Part I, ref. 41), to which we return in Volume 7. Later in the same month, October 1931, Bohr participated in a congress in Rome arranged by the Fondazione Alessandro Volta. Bohr elaborated some remarks he made during the discussions into a small paper that was published in 193221. Since Bohr here discusses in greater detail the problem of “intranuclear electrons”, this paper is reprinted in Volume 9, among his contributions to nuclear physics. As far as the Faraday Lecture is concerned, what appears to be a typed shorthand copy is preservedz2. It is of considerably less interest than the printed version, and runs only to 12 typewritten pages whereas the published paper covers 45 pages. The major part of the published lecture is taken up with a survey of the development of atomic theory. However, the paper is of particular interest because it contains the only published hint of Bohr’s views on the problem of irreversibility (vide pp. 376-377). Since Bohr was throughout his life deeply interested in statistical mechanics and pondered much upon the relation between mechanical and thermodynamical descriptions, it may be appropriate here to digress for a while in order to illustrate his line of thought.

*** From the time when he worked on his doctoral thesis, on the electron theory of metals (1911), we already find scattered remarks that testify to Bohr’s occupation with the foundations of statistical mechanics. As we shall see, his critical at21 N.

Bohr, Atomic Stability and Conservation Laws, Atti del Convegno di Fisica Nucleare della “Fondazione Alessandro Volta”, Oct. 1931. Reale Accademia d’Italia, Rome 1932, pp. 119-130. Cf. VOl. 9. 22 Manuscript, Chemistry and the Quantum Theory, in folder: Faraday Lecture, 1930. Bohr MSS, microfilm no. 12.

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titude towards Boltzmann as well as his deep admiration for Gibbs date from these years. Many of us remember how Bohr in later years used to acknowledge Gibbs as “the one who really understood the meaning of the probability concept”. It was the idea of the “ensemble” that Bohr would often return to as providing the very basis for the physical application of the probability calculus23. A curious fragment is found among Bohr’s lecture notes from 1912 on the Mechanical Foundation of thermodynamic^^^. It indicates the mood of Bohr’s early reflections (cf. also the facsimile on p. 13211).

“Suppose you have a very large country, quite flat, but with a few sharply peaked mountains (see the figure) and this country is full of people perpetually wandering. If I know a man who finds himself in the flat land, then I may say that in a year I will also expect to find him there, because it would be a strange coincidence if, on his way, he should just hit on the one sharply peaked mountain. If I know a man who finds himself on the side of the mountain, I do not know whether he at this moment is walking up or down, but I may surely expect that in a year he will be in the flat land. In my opinion this is all one can say about the entropy theorem, and the theorem understood in the sense that in a year there will be fewer people than now on the mountain is in my opinion not correct2s.”

Undated manuscript by Bohr Danish

In the autumn of 1915 Bohr lectured in Manchester on the kinetic theory of matter26. In his lecture of November 19, he discussed the H-theorem and in this In this respect there is a sharp contrast between Bohr’s views and those of Rosenfeld, who was a strong adherent of Boltzmann’s view and censured Gibbs for his “timid and ambiguous attitude” towards the problem of the atomistic foundation of statistical mechanics. Gibbs is further rebuked for shunning “the whole problem of the ergodic hypothesis” and taking refuge in “the crudest platonistic conception of mechanical ‘analogies’ of the thermodynamical laws”. Even the discussion of the famous stirring experiment is not considered with favour by Rosenfeld who writes: “This simile so aptly illustrates the ‘mixing process’ in phase space that even Gibbs did not forsake it in the obscure pages he grudgingly devoted to the problem in his book” (L. Rosenfeld, On the Foundutions of Statistical Thermodynamics, Acta Physica Polonica 14 (1955) 3-39). 24 Manuscript, Optegnelser ti1 Forelzsninger om Det mekaniske Grundlag f o r Termodynamikken [Notes for Lectures on The Mechanical Foundations of Thermodynamics], 1912. Bohr MSS, microfilm no. 4. 25 In this argument some premise concerning the initial conditions seems to be tacitly implied. 26 Manuscript, Lectures on the Kinetic Theory of Matter I , 1915. Bohr MSS, microfilm no. 5 .

23

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

connection he returned to this picture of the wandering people in the flat country with only a single mountain peak. Already in a draft of a letter to Oseen from 1911, Bohr had expressed serious reservations as regards Boltzmann’s H-theorem: “While I am on this point, I would also like to say a few words on the paradox concerning Boltzmann’s minimum theorem. Because I think definitely (I believe I said something similar last summer) that the solution is27that Boltzmann has not at all proved that an arbitrary distribution will always (on the average) approach the Maxwell distribution (this would of course also not be true), but only that a distribution in which the molecules within the various velocity domains at the given moment are distributed in space in such a way that the collisions within the next small time interval can be evaluated in the manner used by Boltzmann, in the first instance will approach the Maxwellian. However, this has nothing to do with the entropy theorem, but has merely to be regarded as a very limited mechanical theorem. Boltzmann’s calculations cannot be justified by assuming the ‘molecular chaos’, since this itself presupposes the Maxwell distribution. As far as the entropy theorem itself is concerned (I mean the theorem that the entropy always increases), I do not think that it can be justified by probability arguments (yet, I dare not express myself too definitely tonight, since I only really thought about this long ago, and it is one of those problems that almost always slip away between one’s fingers just at the moment when one believes one really has caught them; while I am writing this, is thus strikes me that it may depend on the way the theorem is defined), but only by considering the way in which deviations from what is called the molecular chaos are brought about. ”

Bohr to Oseen, Draft of I Dec 11 Danish text in Vol. 1, p. [422] Translation in Vol. I , p. [426]

In his later years Bohr was fond of revealing to his younger pupils what he considered the essential insights in the foundations of physics. Many of us have been taught about Gibbs by Bohr, but nothing could better convey the trend of Bohr’s teaching than the following extract of one of Thomas Kuhn’s interviews with Heisenberg: A H Q P , Intenleu with Heisenberg, 12 July 63 Transcript, p 15 English

H:

“You know he loved Gibbs, as I mentioned today, yes.

TSK: Incidentally, I’d be very glad to have this on this record also because this statement about his strong preference for Gibbs - 1 take it that he felt that a slip of the pen. The translation of this entire quotation is slightly more literal than the translation of the whole letter given in Vol. 1. Actually the passage quoted is deleted in the draft [the deletion in pencil is invisible in the reproduction) and omitted in the final letter dated December 4, 1911. ” The original has “namely” in place of “is”, but it seems to be

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

Gibbs had seen what this was all about in some sense in which Boltzmann had totally missed the point. H:

Yes, that was Bohr’s definite meaning. Well, he asked me, ‘Have you studied thermodynamics?’ And that was very early, I think in one of the first weeks I came here and I said, ‘Well, I have studied my lectures with Sommerfeld.’ And it turned out that he had followed more or less Boltzmann’s line. Then he said, ‘Well, you know that is all really not understood. That’s not the real point. You read Gibbs and there are these chapters in Gibbs’ book and that is really everything that can be said about thermodynamics.’ And this view was very different from the view I had learned in Gottingen or in Munich. Because all these other physicists always felt that the canonical ensemble was something dreadful, I mean an ensemble of things which didn’t exist, of which one example existed and all the rest was just imagination. That was, for most physicists of that time, something which was extremely disagreeable, but I have in the course of time very well understood why Bohr had such an emphasis on this point. As I say today, Bohr emphasized the complementarity between temperature and energy to the extreme. He said, ‘As soon as I know the temperature, then the concept of energy has no meaning. I mean this is just one example of a canonical ensemble which means that I do not know the energy. So either I can know the energy or I can know the temperature, but I can never know the energy and the temperature.’ ...

TSK: Well, clearly in the earliest discussions that you had with him of this, there was no notion of complementarity in anything like its later form. Can you remember at all the sort of thing he would have said before he had? H:

That’s very difficult. That’s more or less imagination what I say, but still, when he spoke about Gibbs, he felt that it was such a reasonable step to take to introduce some paradox to the very beginning. Simply to say, ‘Let us, to begin with, not start with statistical averaging in time or such things but let’s say we start from pure imagination, that this thing here is only one piece of an infinite number of similar things. Then let us study how this thing will behave if we stick to that notion and say we define a modulus of canonical distribution. Then, of course there comes these famous arguments of Gibbs about what happens if I have two such objects and they touched each other’. That Bohr always liked to say was real physics, that one finds out that if one once had started with this concept

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

of the canonical distribution, then everything comes out as it should be or as it experimentally does.” Throughout his life Bohr kept alive his old interest in the statistical interpretation of the second law, which we have seen him ponder as a young man. Thus, in 1947 we find him engaged in a vivid discussion with Pauli about the entropy increase associated with observation. The matter had apparently been debated on earlier occasions, and Bohr returns in January 1947 to the subject in a letter to Pauli2*: “The idea that any observation must necessarily involve an increase in entropy has been much discussed and I remember that already in the discussions with Stern and you in Hamburg, when you helped me with the ‘proofs’ of the old paper on complementarity, I stressed the principal irreversibility of the concept of observation. More specifically, any observation must make use of some registering device, whether through a photographic plate or directly by the retina of the eye, which involves processes of amplification by which free energy is spent. I know that also Teller is interested in these problems which were discussed in Los Alamos. I shall be glad if either you or Stern would write to me what has come out of your discussions in Zurich.”

Bohr to Pauli, 15 Jan 47 Tull text on p [449]

Pauli answers promptly: “The discussions which I had here with Stern (he left Zurich a few days ago) concerned the quantitative side of the connection of the concepts of entropy and of observation, a connection which, as we all agree, is of a very fundamental character. The problem arises whether there is a well defined minimum of the increase in entropy, independent of the particular experimental arrangement in use, if a certain quantity (‘observable’) is measured. Our discussions seemed to indicate that this is actually the case, although we did not reach yet any final conclusion. The increase in entropy can easily [be] computed if one starts with a ‘mixture’ as initial states and changes it into a ‘pure case’ by constatation of the value of a certain quantity. If, however, before and after the measurement the observed system is in pure cases the situation seems to be more difficult to judge. (Example: a single particle in a closed box. Before the measurement it is supposed to be in a certain eigenstate with a sharply fixed value of the energy. Then one observes the place of the particle in space, perhaps by constatation that it is in a certain partial volume. What one can [can one] say about the amount of the increase

Paul1 10 Bohr, 28 Jan 41 Full text on p I4501

28

The following exchange of letters between Bohr and Pauli is in English.

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in entropy through this measurement, independent of a particular experimental arrangement?) We discussed different experimental arrangements, but we are not sure how general our preliminary results are. Stern and I are both trying to continue the considerations of the problems, possibly by correspondence. Needless to say how anxious I am to hear what you know yourself on this quantitative side of the increase in entropy by observations and how grateful I would be if you could write to me your views on this problem. I feel that you might know the answer already or at least that you may find it out quicker than we.” In May Bohr takes up the problem anew in two letters to Pauli: “AS regards the problem of the entropy increase connected with observations I have in the last days been thinking anew over the situation and have also revived my reminiscence of earlier discussions on this point. You will remember our talks with Stern in Hamburg at the time you so kindly assisted me in gradually working my first article on complementarity from proof back to manuscript. The question was then of Boltzmann’s ideas regarding the direction of time and my point was that the very concept of observation entails an irreversibility in principle. In the following years, this problem came into the foreground in connection with the question of the consistency of the interpretation of the quantum mechanical formalism and in continuation of our discussions in Warsaw I had during the war some talks in America with v. Neumann who still felt some uneasiness about the apparent arbitrariness in the distinction between the objects and the measuring instruments, and with Teller who just like Stern, as I gather from you, was endeavouring to look for the elucidation of the paradoxes in a more quantitative connection between thermodynamics and the observational problem. After reconsidering the question, I feel myself that the whole question is purely epistemological and therefore of qualitative rather than quantitative character. On the one hand, it is evident that any practical observational arrangements, making use of photographic plates, cloud chambers or direct sensual impressions, involve a mechanism of amplification in the working of which free energy is spent in amounts out of all proportion with the energy exchanges characterizing the individual atomic processes under investigation. On the other hand, it is equally clear that, for the interpretation of [the] quantum mechanical formalism and the elucidation of the paradoxes involved, the problem is how to account consistently for the phenomena defined by means of measuring agencies and recording devices which serve to fix the external conditions and register the experimental results and which, for this purpose, are to be treated as ideal classical instruments.

Bohr paul,, 16 Ma) 4rexr on ,451,

P A R T II!:

GENERAL ASPECTS OF PHYSICAL DESCRIPTlON

Of course, it is true that the constitution and operation of the instruments is ultimately subject to the laws of atomic mechanics and that a consideration of this point may perhaps eventually prove a guide for the overcoming of the still unsolved difficulties in quantum theory, but I am sure we agree that this point has as little to do with the questions for which Einstein feels such uneasiness as the final clarification of the still unsolved problems of cosmology has to do with futile criticism of the foundation of relativity theory. The irreversibility in any observational problem has its root in a certain degree of complication of the interaction of the object with the measuring agencies and, trying to make the situation more clear to me, I have considered experimental arrangements where the critical element of irreversibility may be arbitrarily far removed from the final macroscopic recording. For instance, we may, for the localization of a particle, instead of catching it directly on a photographic plate, allow it to enter through a small hole in a box from which the probability of escaping is vanishingly small and where, therefore, the presence of the particle can be ascertained in some suitable way at a later time. The degree of irreversibility here depends on the complicated character of the state of motion of the particle in the box, and the problem presents a certain analogy with the entropy increase accompanying irreversible expansion of a gas from a smaller volume u to a larger volume V which pro gas molecule is given by k log V / u . From such considerations it follows that there will be a close correspondence between the degree of complexity required and the degree of irreversibility practically demanded and that, under optimal circumstances, the unavoidable entropy increase may be brought down to the order of k , representing the limit for the unambiguous use of the very concept of entropy. Here, I have in mind such considerations about the complementary relationships between thermodynamical and mechanical concepts as I tried to indicate in my old Faraday lecture. Just as such considerations offer a consistent attitude to the well-known paradoxes of irreversibility in thermal phenomena, so it appears to me that, notwithstanding the obvious qualitative relationship between such phenomena and the irreversibility of observations, we may more adequately regard thermodynamical considerations and the essence of the observational problem as different complementary aspects of the description. I do not know how far such remarks meet the problems in your mind and I am afraid that you may feel that I have got complementarity on the brain, but I know we both consider it a task for physicists to develop a way of talking which is suited not to hinder, but to stimulate the search for ordering experience. Of course, there must be a division of labour corresponding to the

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

different temperaments which, after all, are themselves complementary in the deepest sense. ” Copenhagen, May 20, 1947 Dear Pauli, I hope you have received my letter of May 16th and are not too unsatisfied with the cursory character of my remarks. When writing you I was unfortunately not able to find your letter of January 28th in which you told me about your discussions with Stern and I was, therefore, not quite sure what details you were thinking about. After a renewed search, the missing letter has, however, reappeared and I wish, therefore, to add that we have also here been aware of the various points you mentioned. As regards the distinction between pure and mixed cases, one must remember that, in any experimental arrangement aiming at a test of the paradoxical consequences of the quantum mechanical formalism, one has to do with a pure case while, in mixed cases, we have to do with a closer analogon to ordinary statistical mechanics. The element of irreversibility inherent in observations is common to both cases but, as regards entropy considerations, we are in the latter case already in the investigated system itself presented with thermodynamical analogies while, in the former case, such concepts apply primarily to the interaction between the objects and the measuring instruments. It is for such reasons that I feel it dubious to look for more explicit connections between thermodynamics and the special observational problems in quantum mechanics. I am, of course, prepared that you may have something far more positive in mind and, as already said in my letter, I shall most eagerly await your reactions. With kindest regards, Yours ever, Niels Bohr

Bohr to Pauli, 20 Ma) 47 English

In his reply Pauli inclines to accept Bohr’s doubt as to the fruitfulness of looking for more explicit connections between thermodynamics and the observational problem in quantum mechanics. However, he still maintains that it ought to be possible to estimate quantitatively the minimal entropy increase associated with an observation: “The most difficult topic of your letters is the problem of the connection between the entropy increase with every observation and the paradoxes of the observational problem in quantum mechanics. I am very grateful that you discussed so many details of these thorny questions in your letter and I am inclined to accept your point of view that the two mentioned sides of the problem should be considered as different complementary aspects of the descrip-

Paul)

to

Bohr,

~ , ~ , ~ :,3551 ~ ~ n

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O X

tion. I have, however, some hope that it may be possible to say about the unavoidable entropy increase, under optimal circumstances, in case of an observation something more quantitative than the statement that it may be brought down to the order of k (which is certainly true). I have not yet a definite opinion about this point and I hope to come back to it in another letter in a few days. Meanwhile I have sent to Stern a copy of the passages of your letters which are dealing with this observational problem.” In August Otto Stern writes to Bohr on the same problems and again in October. On this occasion he also refers to the familiar machinery discussed by Szilard in connection with his analysis of the entropy balance in observation processes29. From the last months of 1947 there is an exchange of letters on this topic between Bohr and Stern. These letters are included among the correspondence in this volume (p. [467]) together with some preparatory notes by Bohr in Aage Bohr’s handwriting.

*** After this digression we return to the Faraday Lecture. Towards the end of the talk Bohr touches on the current questions of the relativistic electron theory and comments on the problems of nuclear constitution. As regards this latter part, the reader should of course bear in mind that the article was completed just before Chadwick’s discovery of the neutron, which finally solved the puzzle of the “intra-nuclear electrons” and their contribution to the statistics of the nuclei. This story as well as that about Bohr’s renewed doubts as regards energy conservation (in P-decay), which was laid to rest through Pauli’s neutrino hypothesis, is beyond the scope of the present volume and is told by Rudolf Peierls in Volume 9, which deals with Bohr’s contributions to nuclear physics.

L. Szilard, Uber die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen, Z . Phys. 53 (1929) 840-856. English translation, On the Decrease ofEntropy in a Thermodynamic System by the Intervention of Intelligent Beings, Behavioral Science 9 (1964) 301-310. Both are reprinted in The Collected Works of Leo Szilard: Scientific Papers (eds. B.T. Feld and G. Szilard), The M.I.T. Press, 1972, pp. 103-129. 29

I. THE MAGNETIC ELECTRON [ I ] UNPUBLISHED MANUSCRIPT FROM FOLDER LABELLED MAGNETIC ELECTRON, 1929

See Introduction to Part 111, sect. 1

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

The folder “Magnetic Electron”, 1929, contains two versions of a typewritten manuscript in English, entitled “The Magnetic Electron”, each of 5 pages, and two sheets of notes written in ink, one in Bohr’s and one in Casimir’s handwriting. There is a carbon copy of one version. On top of what apparently is the earlier version Bohr has written in pencil, “Tisvilde Sommer 1929”. In this version the symbols and formulae have been entered in ink. There are a few corrections in pencil and ink in Bohr’s handwriting. In the second version these corrections have been included. In the carbon copy of this the symbols and formulae have been entered in ink, possibly by Casimir. This is the version reproduced here. The manuscript is on microfilm Bohr MSS no. 12.

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

The Magnetic Electron As well known the idea of ascribing t o the electron, besides its electric charge and inertial mass, a magnetic moment, equal t o a quantum theory magneton, has proved very successful in the interpretation of atomic spectra and the periodic table of the elements. Still the desirability of a direct experimental verification of this idea and a determination of the electronic magnetic moment, on similar lines as the determination of atomic magnetic moments in the beautiful experiments of Stern and Gerlach, has been often expressed. It is clear that the presence of the electronic charge makes any experiment devised for this purpose very difficult, and from a careful examination of various experimental arrangements L. Brillouin concludes (Proc. Nat. Acad. Washington 1928) that such experiments will have only little hope of success. It does not seem to be generally recognised, however, that the possibility of a direct observation of the magnetic moment of the electron would be inconsistent with the fundamental principles of quantum theory. This conclusion is suggested by the very expression for the magneton p=-

eh 4rcrnc

where -e and rn are the charge and mass of the electron, c is the velocity of light and h is Planck’s constant. Indeed the fact that p is proportional to h leads one to suspect that any effect of the magnetic moment ascribed to the electron will disappear in the region of legitimate application of classical mechanics, which involves the neglect of the quantum of action. This suggestion is also confirmed by the ingenious theory of Dirac in which the effects hitherto pictured by the magnetic electron appear as inherent features of relativistic quantum mechanics. To this aspect of the problem we shall return below. It may be of interest however, first to show how the truth of the above statement can be made evident from a direct examination of the extent to which the analysis of experiments on free electrons are affected by the limitation of classical concepts. As well known the region of unambiguous application of these concepts is limited by the condition that all dimensions entering into the description of our apparatus are large compared with the wavelength

A=-- h

mv

associated with electrons of velocity u according to the original ideas of de

~ s ,2

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

Broglie which have proved themselves able to account in a complete way for the remarkable experiments on reflection and diffraction of electron beams from crystals. Let us consider the deflection of a beam of electrons in a magnetic field and assume for the moment that we can speak of an orientation of the magnetic axis of the electron parallel and opposite to the magnetic force, just like the orientation of the silver atoms in Stern and Gerlach’s experiments. Let the main direction of the magnetic force acting on the beam be parallel to the z-axis of a Cartesian coordinate system. The force acting on the electrons due to their magnetic moment will therefore be parallel to this axis and be given by:

where the two signs correspond to the two orientations of the magnetic axis assumed. Due to the solenoidal character of the field any inhomogeneity in the intensity of the magnetic force will be accompanied, however, by an inhomogeneity in direction. To fix our ideas let us assume that the magnetic force is always perpendicular to the x-axis and that it is symmetrical with respect to the u:plane. In the neighbourhood of this plane the variation in the magnetic force in the direction of the y-axis will thus be governed by the relations:*

MS, p. 3

Let us now to begin with assume that the initial direction of the electron beam is parallel to the x-axis, corresponding to the arrangement in the Stern-Gerlach experiment. The Lorentz force on the moving electronic charge has in this case a component in the z-direction equal to:

and at two points within the electron beam of distance A y this force will differ by the amount:

nz,=

eu aH,

- - -dy

c

aY

* [Evidently there is a slip of the pen in the first equation. With the symmetry of the arrangement described in the text, aH,/az should here be replaced by 8 H Y / 8 z . ]

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

Due to (l), (2), (3) and (4) we have thus

Now the divisor on the right side, is a measure for the angular diffraction in the xy plane to be expected on quantum theory for an electron beam of width d y . We see therefore that it is impossible to reconcile the idea of a well-defined direction of the beam, with the condition that the variation of the Lorentz-force within the beam shall not exceed the deflecting force due to the electron magnetization. In other words if we would try to meet this condition, necessary for the measurement of the magnetic moment, by gradually narrowing the width of the beam in the direction of the y-axis, the diffraction would before our aim is reached have grown into a complete diffusion of the beam excluding an unambiguous application of the classical concept of force. We may thus conclude that the Stern-Gerlach arrangement is unsuited for observation of a magnetization of free electrons. Since, however, for the above argument the consideration of the component of the Lorentz-force in the direction of the deflection due to the magnetization is essential, it might perhaps be thought that another orientation of the direction of the beam relative to the magnetic field would be more suitable for the purpose. We shall therefore consider also the cases where the beam is directed parallel to the y and z-axis respectively. In the case the electron beam is parallel to the y-axis the Lorentz-force is parallel to the x-axis and will not directly influence the deflection in the direction of z. It can be shown, however, that it will produce a diffusion of the beam so large as to prevent any such deflection to be measured. If we consider two points within the beam of distance A z the difference of the Lorentz-force is obviously:

Comparing again with (l), (2), and (3) we have consequently:

Here the divisor on the right side gives a measure for the angular diffraction in the xz-plane of the beam which will result from a finite width A z in the direction of the z-axis. The possibility, however, of observing an angular deflection due to the magnetization depends obviously on its being larger than this diffraction.

hlS, p 4

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

MS, p. 5

But the expression on the left side of ( 6 ) gives just the ratio of the angular diffusion in the xy-plane, suffered by the beam on account of the Lorentz-force, to the orbital deflection due to the magnetization of the electrons when passing the same part of the field. Equation ( 6 ) therefore means that this latter diffusion will be complete before the electrons have penetrated far enough in the field to make a deflection due to their magnetization observable. This argument is really quite general and will be seen immediately also to apply to the above Stern-Gerlach arrangement. We shall see that in substance it will also apply to the third case named above where the electron beam is parallel to the z-axis. The case of an electron beam moving parallel to the magnetic force corresponds to an arrangement for measuring the magnetic moment of the electron proposed by Brillouin and discussed in the paper cited above. The idea is that such an arrangement may allow a separation of the electron beam in two fractions corresponding to an orientation of the magnetic axis parallel and opposite to the magnetic force respectively. While the first fraction of the electrons, the motion of which is accelerated in the field, are supposed to penetrate to a receiver suitable for their observation, the second fraction of electrons, which are retarded in the field, are supposed to return before they reach this receiver. We shall see, however, that such a separation is prevented by diffusion of the beam due to the component of the Lorentz-force perpendicular to the motion. In fact, assuming that the beam of electrons is directed parallel to the z-axis we get with the same notation as above

Taking dy as the width of the beam in the direction of the y-axis the divisor on the right side is here the angular diffraction of the beam in the yz-plane and must for a well defined beam be small compared with unity. The expression on the left side gives a measure of the angular diffusion of the beam in the xz-plane produced by the Lorentz-force within the time interval necessary for the force arising from the magnetization to effect a considerable change of moment[um] in direction of the z-axis. We see therefore that a complete diffusion will take place before the acceleration and retardation of the electrons will be sufficient for the separation looked for.

[3361

11. THE MAGNETIC ELECTRON [2] W. PAUL1

DISKUSSION EINIGER VERSUCHSANORDNUNGEN ZUR BESTIMMUNG DES SPINMOMENTES AN FREIEN ELEKTRONEN EXTRACT OF GERMAN MANUSCRIPT OF CONTRIBUTION TO THE REPORT OF THE SOLVAY MEETING 1930 and Discussion Remarks (Translation)

See Introduction to Part 111, sect 1.

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

This manuscript consists of 30 cyclostyled foolscap pages, viz. a title page: “Congres Solvay 1930. Rapport de M. Pauli”, 2 numbered pages of Pauli’s Introduction, the whole of Part I1 of Pauli’s report, comprising 25 numbered pages, and 2 pages of references. Apart from the title page and the main title, which are in French, the manuscript is in German. On the title page, along with a few formulae and a sketch, is written in pencil in Bohr’s handwriting: “Magnetisk Moments [?I med Hastighed. Stern GerfachF o r s ~ g ”(The [?] of the magnetic moment with velocity. Stern Gerlach experiment). We have here reproduced the first section of Part I1 of Pauli’s report, dealing with the measurement of the magnetic moment of a free electron (which, according to the first footnote, is based on a personal communication from Bohr), together with the relevant references. This is followed by the relevant part of the subsequent discussion, translated from the Solvay Report. There are no further discussion remarks by Bohr reported in the proceedings of this Solvay Conference. The manuscript is in the Niels Bohr Archive, in the collection of Manuscripts by Other Authors. At the time of publication of the present volume this collection has not been microfilmed. The printed French version of the report is in the proceedings of the Solvay Conference: “Le magnetisme. Rapports et discussions du sixieme Conseil de physique tenu a Bruxelles du 20 au 25 octobre 1930”, Gauthier-Villars, Paris 1932, pp. 217-225. The discussion of Part I1 of the report is on pp. 275-280.

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

II. Die relativistische Quantenmechanik des Elektrons. 1. Diskussion einiger Versuchsanordnungen zur Bestimmung des Spinmomentes an freien Elektronen.

Bevor wir auf die relativistische Dirac’sche Theorie des Elektrons naher eingehen ( § 2 ) , mogen hier an Hand einiger moglicher Versuchsanordnungen die BOHR’sche These diskutiert werden, dass bei freien Elektronen (oder freien Protonen) die innerhalb des Anwendungsbereiches der anschaulichen Vorstellung der Bewegung einer Partikel (klassisch mechanische Bahn, geometrisch-optischer Strahlbegriff) liegen, ihr Spinmoment niemals nachgewiesen werden kann. Es zeigt sich namlich, dass infolge der Grosse

eh

iG& des magnetischen Momentes freier Elektronen (vgl. I, (4)) die Bedingung dafiir, dass die Wirkung der das Spinmoment angreifenden Krafte durch die Wirkung der Lorentzkraft nicht verdeckt wird, gerade zur Folge hat, dass die erstgenannten Wirkungen sich infolge des Auftretens von Beugungseffekten der Beobachtung entziehen. a) Bremsung von Elektronen bestimmter Spinorientierung durch ein Gegenfeld. Es mogen langsame Elektronen parallel der z-Achse gegen ein zur z-Achse entgegengerichtetes Magnetfeld anlaufen. 1st dieses Magnetfeld inhomogen und aH,/az> 0, so werden die Elektronen mit einem Spin, der zur z-Achse entgegengesetzt gerichtet ist, nach einer durch die Gleichung

‘

aHZ t mv, = ,uo-

az

gegebenen Zeit t zur Ruhe gebracht und nachher zum Umkehr gezwungen werden. Die Anzahl der Elektronen, die Strecken grosser als u,t in der z-Richtung Diese Versuchsanordnung wurde von BRILLOUIN (32) angegeben. Sie enthalt iibrigens noch technische Verfeinerungen, die es gestatten, statt eine sehr kleine Gesarntgeschwindigkeit der Elektronen die Kleinheit einer gewissen Geschwindigkeitskornponente auszunutzen. Diese Verfeinerung wurde im Text nicht erwahnt, da sie fur die Erorterung der rnehr prinzipiellen Resultate nicht wichtig ist. Die Kenntnis der Ueberlegung des Textes verdanke ich rnundlichen Diskussionen rnit Herrn N . BOHR.

us, p . 2

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

zurucklegen, wird also nur halb so gross sein als wenn kein Spin vorhanden ware. Es muss jedoch wesentlich berucksichtigt werden, dass es bei einem inhomogenen Feld unmoglich ist, im ganzen Raum eine Feldstarke parallel zur z-Achse zu realisieren. Denken wir uns die Feldrichtung stets in der xz-Ebene gelegen, so muss vielmehr

aHx-

aH,

az

ax

sein. Nehmen wir weiter an, dass fur x = 0 die Feldstarke streng parallel haben wir in einem Abstand Ax von der x-Achse

z ist, so

Diese bewirkt innerhalb der Larmorperiode 1/a ein Umklappen der Geschwindigkeit u, (vgl. I, §3), wobei a gegeben ist durch

a=---eHx - POHX 4nmoc h Sol1 die Umkehr der Geschwindigkeitskomponente uz sicher vom Spinmoment der Elektronen herruhren, so muss also ta-

h POHX

sein oder

MS, p. 3

also nach (2)

mu,. Ax4 h

(4)

Die letztere Bedingung ist nun (obwohl formal von der Unsicherheitsrelation verschieden) wegen der Wellennatur des Elektrons nicht wahrend der ganzen Dauer des Versuches erfullbar, da die Wellenlange A gerade h/mu, betragt und Dimensionen Ax von Strahlquerschnitten, fur welche A x e A ist, nicht realisierbar sind.

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

Wurde man umgekehrt versuchen, einen solchen Strahlquerschnitt durch eine Blende zur erzwingen, so ware

Av,>

h mAx

-

also nach (4)

erforderlich und der Ausfall des Experimentes liesse sich nicht mehr mittels der klassischen Mechanik vorhersagen. Dieses Resultat beruht wesentlich darauf, dass das magnetische Moment des Elektrons durch (1) gegeben ist. Wurden wir den Wert des Spinmomentes noch unbestimmt lassen (er sei p ) , so ware an Stelle von (2) zu setzen

m v , = p -aHZ t

az

und der Vergleich mit (3) wurde ergeben P mu,. A x e h PO

Dies ware mit der Bedingung dx9A = h / m v , vertraglich, wenn p 9 p 0 ware. Entsprechendes gilt bei allen im folgenden diskutierten Versuchsanordnungen. b) Das Stern-Gerlach 'sche Experiment fur freie Elektronen2 Wir denken uns die Elektronen parallel zur y-Richtung bewegt, die Richtung des Magnetfeldes stets in der xz-Ebene gelegen und fur x=O auch H,= 0 . In der yz-Ebene wirkt also in der z-Richtung die Kraft

auf die Elektronen, die Lorentzkraft wirkt in der x-Richtung und hat den Betrag

* Vgl.

hierzu eine Arbeit von MOTT (33). - Das gewohnliche Stern-Gerlachexperiment liefert dagegen die Bestimmung des magnetischen Momentes eines ganzen Atoms.

MS, p . 4

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

e H . Fur Elektronen etwas ausserhalb der yz-Ebene hat man eine Feldstarke c y

-u

H =-AX= aHx

--Ax aHz

az

ax

und eine Lorentzkraft in der z-Richtung

e aHz -u,-Ax c az Sol1 der Versuch gelingen, so muss

e u, -Ax aHZ c

az

Q

aHz

-

az

oder

das heisst

MS, p . 5

sein, das heisst man befindet sich ausserhalb des Anwendungsbereiches der klassischen Mechanik. Es bleibt noch der Fall zu diskutieren, dass die Bewegungsrichtung der Elektronen parallel zur x-Achse ist, wahrend die Feldrichtung wie friiher stets senkrecht steht auf der y-Achse; in der x-Richtung konnte die Spalte jetzt beliebig lang sein. Die Lorentzkraft ist jetzt parallel zur y-Achse und betragt

ihre Differenz an zwei verschiedenen Punkten mit dem Abstand dz betragt dK,=-

eu, aHz

c

-d~

az

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

wahrend die auf den Spin wirkende Kraft

betragt. Also wird

AK, - -= 2nAz K, hmou,

--

A z =4n A a,

471-

wenn a = A / A z den Beugungswinkel (in der xz-Ebene) bedeutet. Damit ein beobachtbarer Effekt resultiert, muss die Strecke die das Elektron im Feld durchlauft so gross sein, dass die durch die Kraft K, bewirkte Abbeugung gross gegen a, ist. Die Gleichung ( 5 ) zeigt, dass dann die durch die Lorentzkraft AKy bedingte Unbestimmtheit der Geschwindigkeitsrichtung in der xy-Ebene gross gegen 2n ist. Man konnte vielleicht zunachst meinen, dass dieser Umstand noch nicht ausreichend sein wurde, um das Bild der beiden abgelenkten Strahlen in einer Ebene senkrecht zur x-Achse zu verwischen. Indessen bewirkt eine Geschwindigkeit u, eine zusatzliche Lorentzkraft K; in der z-Richtung von der Grosse e c y

K;=-u H

*

Selbst wenn am Beginn der Bewegung im Feld H,=O, so ist wegen aH,/ax= - aHz/az und der Unbestimmtheit Ax> h / m u , sicher

Da nun wahrend des Durchganges der Teilchen durch das Feld wie wir gesehen haben,

wird, so wird auch AKiSK, was die vollige Verwischung des Aufspaltungsbildes zur Folge hat.

~

5 P ,6

P A R T 111: G E K E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O K

c) Kompensation der Lorentzkraft durch eiektrische Felder3. Wir denken uns die Elektronen mit der Geschwindigkeit u parallel zur z-Achse bewegt und unter dem gleichzeitigen Einfluss eines elektrischen sowie eines magnetischen Feldes stehend, die beide parallel zur xy-Ebene gerichtet sind und einen von z unabhangigen Betrag haben. Mit Einfuhrung des elektrischen Potentials @ und des magnetischen Potentials A parallel zu z (beide nur von x , y abhangig) werden die Feldstarken bestimmt durch MS, p . 7

( A @ = A A = 0) und die Kraft auf das Elektron mit der Ladung -e wird

e a K,= -eE,x+-uHy=e-(@c ax

U

-A) C

Fur eine bestimmte Geschwindigkeit u tritt also im ganzen Raum eine Kompensation der Krafte ein, wenn U

@=-A c

(6)

gewahlt wird. Auf den ersten Anblick scheint es also, dass die ubrigbleibenden das Spinmoment angreifenden Krafte in diesem Fall bequem zur Beobachtung gelangen konnten, da die genaue Festlegung des Elektronenortes senkrecht zum Strahl hier nicht erforderlich ist. Indessen erweist es sich als unmoglich, Elektronen mit vom Ort (x, y ) unabhangiger Geschwindigkeit u, im Feld ohne Benutzung von Blenden herzustellen, wenn zugleich noch o x = uy = 0 sein soll. Denn es ist im Feld nicht u,, sondern

e pz=muz-- A c

Dieser Versuch wurde von KNAUER Besprechungen mit Herrn N. BOHR, sowie the letters from Knauer to Bohr, 29 August Knauer, 4 September [1929] (PWB I , letter

(7)

(34) vorgeschlagen. Seine Diskussion verdanke ich einer brieflichen Mitteilung von Herrn W. LENZ. [Cf. 1929, Bohr to Knauer, 3 September 1929, and Pauli to [236]) (all on microfilm BSC no. 13).]

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

ein Integral der Bewegungsgleichungen. Denkt man sich ferner zunachst bei Abwesenheit der Felder Elektronen mit einem bestimmten u, und dann die Felder adiabatisch eingeschaltet, so bleibt ebenfalls nach der klassischen Mechanik die Grosse p z und nicht die Grosse u, hierbei erhalten4. Wir denken uns also p , = const. und ein Biindel mit kreisformigem Querschnitt vom Durchmesser d daraus ausgeblendet. Es sei

Das Feld Hy = a bewirkt die Richtungsquantelung und

wird die ablenkende Spinkraft. Die unkompensierte Lorentzkraft A K , auf die Rander des Biindels wird e

e

C

C

A K , = - A U, H,= - A U, bd und wegen

e

e mc

Au, = -AA mc

= -bd2

also muss

e

e

Pob >> -* - b 2 d 3 mc c

Dass es sich hier um etwas prinzipielles handelt, geht daraus hervor, dass im Magnetfeld nicht mehr u, und u z , sondern u, und p z miteinander vertauschbar sind. Fur dux und A u z gilt die Unbestimmtheitsrelation Av,Au,--

1 eh -Hy-

rnz c

1 ,PQH,

& I S , p.

x

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

oder po% MS, p. 9

e - a

e

- bd3

mc c

sein. Damit der Strahlbegriff anwendbar bleibt, muss ferner langs des ganzen Weges I durch das Feld die Querdimension des Biindels kleiner als d bleiben, was wegen der Beugung urn den Winkel A/d (A = h/mv,)

A -bed d

oder

AI

1%-

d2

(9)

ergibt. Dies mit (8) multipliziert gibt

und mit Beriicksichtigung des Wertes von p o g%pob-!-(;)2 d

Hier steht nun links der Beugungswinkel, rechts der Ablenkungswinkel infolge der Spinkraft und das Resultat bedeutet, dass der letztere unbeobachtbar bleibt. (Er wurde es nicht sein, wenn das magnetische Moment des Elektrons betrachtlich grosser als p 0 ware.) d) Messungen des von den Elektronen erzeugten Magnetfeldes'. Man konnte endlich noch daran denken, durch Messungen des raumlichen Verlaufes der magnetischen Feldstarke zwischen dem von der Bewegung und dem vom Spin herruhrenden Teil des Feldes zu unterscheiden. Betrachten wir nur die Betrage (nicht die Richtungen) und die Grossenordnungen dieser Feldstarken, so haben wir MS, p. 10

Vergl. hierzu MOTT (33).

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

also bei festem Ort und A r t r

1 e A->o e h AH - _ +-e h 2 - r2 c cr2 mAr mc r3 also

Die beiden Teile der Feldstarke sind nicht trennbar. Die Ergebnisse dieses § sind nicht so zu verstehen, dass bei freien Elektronen vom Spin herruhrende Effekte uberhaupt nicht beobachtbar waren; dies gilt vielmehr nur von solchen Effekten, die mittels anschaulicher Bewegungsvorstellungen (klassische Mechanik) deutbar sind. Im $3 werden wir dagegen sehen, dass die Herstellung polarisierter Wellen von freien Elektronen moglich ist .

*..

Literatur

(32) L . Brillouin, Comptes rendus, Acad. Sci. Paris, 184, 82, 1927. Proc. Nat. Ac. 14, 775, 1928. (33) N.F. Mott, Proc. Roy. SOC. 124, 425, 1929*. (34) F. Knauer, Zs. f. Phys. 59, 807, 1930.

...

Discussion Remarks Pertaining to the Preceding Part of Pauli’s Report. TRANSLATION

...

Bohr - One can see in a very general way that it is impossible experimentally to detect the intrinsic magnetic moment of the electron. Any attempt in this direction would actually be based on an application of the classical concepts, in particular that of the trajectory of the electron, which essentially amounts to ignoring the elementary quantum of action. Now, in Dirac’s theory, all effects due to the intrinsic magnetic moment of the electron automatically disappear if Planck’s constant is put equal to zero. This theory is actually based on an expres-

* [Cf. also N.F. Mott and H A W . Massey, The Theory ofAtomic Collisions (2nd ed.), Oxford Univ. Press, 1949 (reprinted 1952), ch. IV, 52.1

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

sion for the Hamiltonian derived from the classical theory of electrons in which, apart from the velocity of light, only the mass and the charge of the electron enter. It follows that these quantities are the only ones which can be determined from considerations based on an analysis of the classical concept of trajectory.

Dirac - I agree with Bohr in a general way,

...

Kapitza - One cannot measure the magnetic moment of an isolated electron; one can only measure the moment when the electron is bound in an atom. But where is the limit? What is the dimension of the outermost orbit in which one can no longer measure it? Bohr - It is perfectly correct that the magnetic moment of an atom in a stationary state can be determined by a Stern-Gerlach experiment. Only, in this case it is absolutely impossible to obtain separately the contribution arising from the orbital motion of the electron and that due to its intrinsic magnetism. We have here to do with an impossibility in principle, due to the fact that the concept of stationary state of an atom ips0 facto excludes a space-time description of the electron in this state. It follows that if the angular momentum of the atom is quantized, which implies that the atom is in a stationary state, the azimuth conjugated to the momentum must be completely undetermined. Furthermore, an indeterminacy of the azimuth smaller than 2n entails an indeterminacy of the conjugated angular momentum larger than h / 2 n . In any experiment carried out with free electrons, the indeterminacy of the angular momentum with respect to any given direction is precisely larger than h/277. On the other hand, the magnetic field created by a moving electron is given by the classical expression e I -.me r3’ where r is the absolute value of the distance from the electron to the point where one observes the field and I the orbital angular momentum vector of the electron with respect to the same point. From this one sees immediately that the indeterminacy of the field due to the motion of the electron is always larger than the field produced by a magnetic dipole of moment he/4;rcrncI . In the following discussion Darwin asked whether the previous argument is absolutely compelling. Since the direction of the classical magnetic field produced by a moving electron does not in general at each point coincide with the direction of the field produced by the magnetic dipole associated with

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

Kapitza - However, we can measure the magnetic moment and experimentally determine its ratio to the spin in magnetic materials by means of saturation magnetization or by the gyromagnetic phenomenon. Bohr - Neither saturation magnetization nor the gyromagnetic phenomenon allows an experimental determination of the intrinsic angular momentum of the electron (spin) or of its intrinsic magnetic moment. In both cases, what one determines is the total angular momentum and the total magnetic moment of the body. It is appropriate here to repeat the remark already made in connection with the Stern-Gerlach experiment. Richardson - I would like to ask, what are strictly speaking the objections against denying the reality of the existence of the spin of a free electron? One says that even if it exists, we are nevertheless unable to know it. Obviously, we know that various spectroscopic and magnetic phenomena force us to admit the existence of the electron spin in atomic and material systems. It occurs to me that the spin is something which manifests itself as a consequence of the interaction between the electron and another material system, e.g., a nucleus, to take a simple case. I think that the spin must be confined to this domain of cases of a rather complicated character where it has a real existence. Van Vleck - By observing the paramagnetism of a free electron, to which Pauli’s calculations apply, we recognize the usefulness of the spin, even outside atoms. We must always take it into consideration, even if the electrons are free. Richardson - But we know that in metallic conductors the conduction electrons are not free. We know it from the Fermi statistics and from Sommerfeld’s theory of metallic conduction. I myself do not consider them free electrons. Bohr - I would like immediately to remark that the impossibility of a direct observation of the intrinsic magnetic moment of the electron does not at all imply that the concept of spin has lost its significance as a means of explaining the fine structure of the spectral terms and the polarization phenomena of electron waves. Only, the way in which the concept of spin enters in the formalism of quantum mechanics is such that it does not lend itself to any independent interpretation, based on classical concepts. the electron, one might imagine, by making observations at several points suitably chosen, a possible compensation of the fields due to the motion of the electron, which would thus make the dipole field apparent. But Darwin has himself shown (Proc. Roy. SOC.A130 (1931) 632), in a particularly suggestive way, that such a procedure for measuring the field necessarily produces, by the reaction of the measuring apparatus on the electron, a perturbation of its motion which escapes our control and makes any correlation between the observations made at different points impossible.

111. PHILOSOPHICAL ASPECTS OF ATOMIC THEORY Nature 125 (1930) 958 Address to the Royal Society of Edinburgh on 26 May 1930 on the Award of the James Scott Prize ABSTRACT

EDINBURGH.

Royal

Society, May 26. - Niels B o h r : Philoso-

phical aspects of atomic theory : Recent experimental and theoretical studies of physical phenomena have revealed a limitation in our ordinary concepts of natural philosophy as regards the description of the behaviour of single atoms. This limitation is an immediate consequence of the discovery of the elementary quantum of action, which excludes a simple distinction between the atomic phenomena and the observation, since any observation necessarily involves a finite change in the course of the phenomena. This circumstance prevents a pictorial description of atomic phenomena and allows us to apply physical concepts only in connexion with probability considerations. The new situation in physics with which we are thus confronted presents a remarkable analogy with situations with which we are familiar from studies in biology and psychology.

IV. THE USE OF THE CONCEPTS OF SPACE AND TIME IN ATOMIC THEORY OM BENYTTELSEN AF BEGREBERNE RUM OG TID I ATOMTEORIEN Overs. Dan. Vidensk. Selsk. Forh. Juni 1930 - Maj 1931, p. 26 THE USE OF THE CONCEPTS OF SPACE AND TIME IN ATOMIC THEORY Nature 127 (1931) 43 Communication to the Royal Danish Academy on 17 October 1930 ABSTRACT

NIELS BOHRgav en Meddelelse: Om Benyttelsen af Begrebenie Rum og Tid i Atomteorien. Atomteoriens seneste Udvikling h a r aabenharet en principiel G r e n s e for Beragelsesforestillingernes Anrendelighed. Ti1 Belysning af dette Forhold underkastes Grundlaget for hfaalinger af a t o m a r e Partiklers Rum-Tidskoordinater en n a r m e r e Prsvelse.

COPENHAGEN. Royal Danish Academy of Science a n d Letters, Oct. 17.-Niels B o h r : The use of the concepts of space and time in atomic theory. The recent development of atomic theory has disclosed a principal limitation of our ideas of motion. I n this connexion a closer analysis of the foundation of space-time measurements of atomic particles is attempted.

V. ON ATOMIC STABILITY Brit. Ass. Adv. Sci., Report of the Centenary Meeting, London - 1931, September 23-30, London 1932, p. 333 Address to the British Association for the Advancement of Science on 28 September 1931 ABSTRACT

Prof. N. BoHR.-on

Atomic h'tabdity.

The characteristic stability of atoms presents us, as well known, with a fundamental limitation in the applicability of the concepts on which the so-called classical physical theories rest. An analysis of this limitation will be attempted in connection with a discussion of the difficulties disclosed by the recent development of the theory of the electron, especially as reqards t,he problems of nuclear cnnntitution.

VI. MAXWELL AND MODERN THEORETICAL PHYSICS Nature (Suppl.) 128 (1931) 691-692 Address Delivered on the Occasion of the Maxwell Centenary Celebrations at Cambridge on 1 October 1931

See Introduction to Part 111, sect. 2 .

OCTOBER24, 19311

NATURE

691

Maxwell and Modern Theoretical Physics.* By Prof. NIELSBOHR,F0r.Mem.R.S.

PEEL greatly honoured in being given this opporImemory tunity of paying a tribute of reverence t o the of James Clerk Maxwell, the creator of the

electromagnetic theory, which is of such fundamental importance to the work of every physicist. I n this celebration we have heard the Master of Trinity and Sir Joseph Larmor speak, with the greatest authority and charm, of Maxwell’s wonderful discoveries and personality, and of the unbroken tradition upheld here in Cambridge connecting his life and his work with our time. Although I have had the great privilege, in the years of my early studies, of coming under the spell of Cambridge and the inspiration of the great English physicists, I fear that it may not be possible for me t o add anything of sufficient interest in this respect, but it gives me very great pleasure indeed to be invited to say a few words about the relation between Maxwell’s work and the subsequent development of atomic physics. I shall not speak of Maxwell’s fundamental contributions t o the development of statistical mechanics and of the kinetic theory of gases, which Prof. Planck has already discussed, especially as regards Maxwell’s fruitful co-operation with Boltzmann. It is only my intention t o make a few remarks about the application of the electromagnetic theory to the problem of atomic constitution, where Maxwell’s theory, besides being extremely fruitful in the interpretation of the phenomena, has yielded the utmosf any theory can do, namely, t o be instrumental in suggesting and guiding new developments beyond its original scope. I must, of course, be very brief in commenting upon the application of Maxwell’s ideas to atomic theory, which in itself constitutes a whole chapter of physics. I shall just recall how successfully the idea of the atomic nature of electricity was incorporated into Maxwell’s theory by Lorentz and Larmor, and especially how it furnished an explanation of the dispersion phenomena, including the remarkable features of the Zeeman effect. I would also like t o allude to the important contribution t o the electron theory of magnetism made by Prof. Langevin, whom we much regret not t o be able to hear to-day. But above all, I think in this connexion of the inspiration given by Maxwell’s ideas to Sir Joseph Thomson in his pioneer work on the electronic constitution of matter, from his early introduction of the fundamental idea of the electromagnetic mass of the electron, t o his famous method, valid t o this day, of counting the electrons in the atom by means of the scattering of Rontgen rays. The developments of the atomic theory brought us soon, as everybody knows, beyond the limit of direct andconsistent application of Maxwell’s theory. I wish to emphasise, however, t h a t it was just the possibility of analysing the radiation phenomena * Address delivered on the occasion of the Maxwell Centenary Celebrations a t Cambridge on Oct. 1.

No. 3234, VOL. 1281

provided by the electromagnetic theory of light which led t o the recognition of an essentially new fea,ture of the laws of Nature. Planck’s fundamental discovery of the quantum of action has necessitated, indeed, a radical revision of all our concepts in natural philosophy, Still, in this situation, Maxwell’s theory continued t o provide indispensable guidance. Thus the relation between energy and momentum of radiation, which follows from the electromagnetic theory, has found application even in the explanation of the Compton effect, for which Einstein’s idea of the photon has been so appropriate a means of accounting for the marked departure from the classical ideas. The use of Maxwell’s theory as a guide did not fail either in the later stage of atomic theory. Although Lord Rutherford’s fundamental discovery of the atomic nucleus, which brought our picture of the atom to such wonderful completion, showed most strikingly the limitation of ordinary mechanics and electrodynamics, the only way t o progress in this field has been to maintain as close contact as possible with the classical ideas of Newton and Maxwell. At first sight it might perhaps look as if some essential modification of Maxwell’s theory was needed here, and it has even been suggested t h a t new terms should be added to his famous equations for electromagnetic fields in free space. B u t Maxwell’s theory has proved far too consistent, far too beautiful, to admit of a modification of this kind. There could only be a question, indeed, of a generalisation of the whole theory, or rather of a translation of it into a new physical language, suited t o take into account the essential indivisibility of the elementary processes in such a way that every feature of Maxwell’s theory finds a corresponding feature in the new formalism. In the last few years, this aim has actually been attained to a large extent by the wonderful development of the new quantum mechanics or quantum electrodynamics, connected with the names of de Broglie, Heisenberg, Schrodinger, and Dirac. When one hears physicists talk nowadays about ‘ electron waves ’ and ‘ photons ’, it might perhaps appear t h a t we have completely left the gronnd on which Newton and Maxwell built ; but we all agree, I think, that such concepts, however fruitful, can never be more than a convenient means of stating characteristic consequences of the quantum theory which cannot be visualised in the ordinary sense. It must not be forgotten that only the classical ideas of material particles and electromagnetic waves have a field of unambiguous application, whereas the concepts of photons and electron waves have not. Their applicability is essentially limited t o cases in which, on account of the existence of the quantum of action, i t is not possible t o consider the phenomena observed as independent of the apparatus utilised for their observation. I would like to mention, as a n example, the most conspicuous application of Maxwell’s ideas, namely,

692

XATURE

the electromagnetic waves in wireless transmission. It is a purely formal matter to say that these waves consist of photons, since the conditions under which we control the emission and the reception of the radio waves preclude the possibility of determining the number of photons they should contain. I n such a case we may say that all trace of the photon idea, which is essentially one of enumeration of elementary processes, has completely disappeared. For the sake of illustration, let us imagine for a moment that the recent experimental discoveries of electron diffraction and photonic effects, which fall in so well with the quantum mechanical symbolism, were made before the work of Faraday and Maxwell. Of course, such a situation is unthinkable, since the interpretation of the experiments in question is essentially based on the concepts created by this work. But let us, nevertheless, take such a fanciful view and ask ourselves what the state of science would then be. I think it is not too much to say that we should be farther away from a consistent view of the properties of matter and light than Kewton and Huygens were. We must, in fact,

[OCTOBER. 24,

1931

realise that the unambiguous interpretation of any measurement must be essentially framed in terms of t h e classical physical theories, and v e may say that in this sense the language of Kenton and Maxwell will remain the language of physicists for all time. I do not think that this is a proper occdsion t o enter into further details regarding these problems, and to bring new views under discussion In conclusion, however, I am glad to give expression t o the great expectation with m hich the whole scientific world follows the exploration of an entirely new field of experimental physics, namely, the internal constitution of the nucleus, which is now carried on in Rlaxwell’s laboratory, under the great leadership of the present Caventiish professor. In the fact that nobody here in Cambridge is likely to forget Newton’s and Naxwell’s work, we see perhaps the very best auguries for the continued success of these endeavours. Even if we must be prepared for a still further renunciation of ordinary visualisation, the basic concepts of physics which we o u e to the great masters will certainly prove indispensable in this new field as well.

VII. SPACE-TIME-CONTINUITY AND ATOMIC PHYSICS H.H. Wills Memorial Lecture, given at the University of Bristol on 5 October 1931 UNPUBLISHED MANUSCRIPT

See Introduction to Part 111, sect. 2.

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

This manuscript consists of 14 typewritten pages with handwritten figures, formulae and corrections. It is in English. In this document Bohr seems to have used the symbol T for the wave number. To avoid confusion, however, we have, as in some earlier documents, replaced it by 0,the symbol used in the printed versions of the Como Lecture. We have corrected a slip of the pen in the formulae. The manuscript is on microfilm Bohr MSS no. 12.

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

SPACE-TIME-CONTINUITY A N D ATOMIC PHYSICS I hope that I shall not disappoint you too much with the subject of the lecture I have chosen. I shall not speak about any striking new results or try to tell you anything which you have not heard, but I should like to bring forward some simple considerations regarding space-time co-ordination in atomic physics, a subject which has given rise to much discussion in the last years. I shall directly go into the matter: How do we introduce the idea of space at all. Suppose we want to know the place where an electron is, then we have to let it pass through a diaphragm But in passing through the diaphragm there comes in the phenomenon of diffraction which is accounted for by a wave in much the same way as the diffraction of a light wave falling on such a diaphragm. This diffraction is determined by the momentum P of the electron by means of the fundamental relation P = ho which connects up the impulse with the wave number. (It is more convenient to use the wave number than the wave length because one can consider o as a vector and consider its components in different directions.) This formuia is in full analogy to Planck’s formula E=hv which Einstein applied to light quanta. On the basis of this wave picture of the electron we will now try to work out the diffraction of it when it passes through a hole or diaphragm. Let us first consider a finite train of waves. As you all know, such a finite train can be analysed according to harmonic components. In order that the wave amplitude outside the wave train shall cancel out there must be components, the wave numbers of which are spread over a range so that there is an uncertainty of one wave over the length of the wave train x , so that we have the relation

To what diffraction will such an uncertainty in the wave number lead? The diaphragm will cut out from the on-coming electron wave the finite range in the

MS. P 2

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O K

x-direction of the length Ax and so makes the x-component o , ~of the wave number uncertain by l/Ax. From this results an uncertainty in the direction of Aa, 1 the diffracted wave w -- -. This means simply that the angle of diffraca aAx tion is equal to the reciprocal of the number of wave periods contained in the opening of the diaphragm. Now we ask: what is the uncertainty in the momentum of the scattered beam? It is AP,= hAa,

-

or us, p . 3

MS, p. 4

AxAP,=h

The result is therefore that by forcing the electron through the hole we lose the knowledge of its momentum. This is the principle of indeterminacy introduced by Heisenberg in order to show the consistency of the new methods of the quantum mechanics. We shall have to repeat this argument several times. Is this argument quite conclusive, or can one avoid its consequences? One could suggest, for instance, that the diaphragm during the interaction with the electron gets momentum and as the momentum is conserved also in quantum mechanics we should be able by measuring the momentum of the diaphragm before and after the interaction of the electron to draw additional information, additional to the information about the co-ordinate of the electron. Now to begin with we cannot measure the momentum when the diaphragm is fixed. We must therefore have a loose diaphragm and measure its momentum relative to another fixed one before and after the passage of the electron. Let us do this by passing light quanta through a second hole in the loose diaphragm on to a photographic plate attached to the fixed one. But these light quanta again will be diffracted and give momentum to the loose diaphragm. If we aim, on the other hand, at great accuracy in the measurement of the momentum by this method we find we lose accuracy in our knowledge of the position of the diaphragm. The general result is that the idea of space and time involve the use of measuring rods which can be considered to be outside the system under investigation so that the interaction with these rods is not traceable in the experiment. This interaction is always finite according to the quantum theory so that the application of the space-time concepts involve a renunciation with respect to the knowledge of energy and momentum. SPRING (?> A diaphragm hanging on a spring. The diaphragm has to be brought to rest. The temperature motion prevents that.

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

Proposition: to cool down to absolute zero. But then absolute zero point motion, which is confirmed by X-ray measurements. Let us turn now to more refined experiment which involves interference of electronic waves. We have two diaphragms I and I1 fixed to a frame. (See Fig:)

Electrons passing the first diaphragm may go through either of the two holes of I1 and finally fall on a screen S. Here they will produce interference fringes. This means that there will be black lines on the screen on which the electrons cannot fall. Now this seems very odd. If it is assumed for instance that a certain electron passes through the upper hole of diaphragm I1 then the mere fact that the lower hole in this diaphragm is open shall prevent the electron from falling on one of the dark lines. This apparent paradox can be solved most satisfactorily in the following way: Suppose we want to find out through which hole the electron has passed. We could do so by measuring the impulse given up to the screen. According to whether the electron has passed the upper or lower hole, the impulse given up will be directed downwards or upwards. But the point is that, if the measurement of the impulse of the screen before and after the electron has reached it should be done with sufficient accuracy to decide the origin of the oncoming electron, then this would again force us to leave an uncertainty in the position of the screen and therewith deprive us of the possibility of settling the question where the black lines on the screen ought to lie. In fact the impulse to be measured is

MS.

5

VS,

6

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

The uncertainty in the position of the diaphragm becomes therefore Ax>

h ~

P; - P.;

On the other hand the number of interference fringes per unit length is

, n = 0,- a, I,

or

n = ( P l - P;)

1 h

-

their distance therefore

q.e.d. MS,p , 7

We may state the result of the foregoing considerations in the following way: We are free to make the arrangement either so that we get interference fringes on the screen (fixed screen) or so that we can decide through which hole each electron comes (loose screen), but in the case of interference we have no hope to ascertain anything about the path of the electron. This is the solution of the paradox. The arrangement of the interference experiment makes it senseless to talk about the electron going through one or the other hole, in the sense that the whole phenomenon then cannot be further analysed. In the foregoing examples the analysis was particularly simple because the time was not explicitly involved and we shall now proceed to see what alterations in the argument are necessary. Our apparatus shall consist of a diaphragm fixed to a support with an opening width A x and a shutter behind it which passes the opening with the velocity V at a certain time To. The diaphragm will therefore be open during a time d T = d x / V . During this time a finite train passes the diaphragm which will contain harmonic components of such a range that dT.Av= 1

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

or

ATAE=h

The energy of the electron which may have been known accurately before the passing of the diaphragm is therefore uncertain to the extent A E . The question arises, where does this energy go to? The case is very similar to the previously discussed one of diffraction. Just as before the diaphragm swallowed the untraceable part of the momentum it is now the shutter which takes up the surplus energy. Again if we try to make a subsidiary arrangement to measure the energy of the ensemble of shutter and clock we lose the knowledge of its position and therewith of the accurate time at which it opens the diaphragm. In fact, we require an accuracy

his, p.

x

MS.

9

h AE< AT On the other hand

A E = u,A P, = 0-,

h h - h = -A x Ax/u, A T

The determination of the energy precludes therefore the measuring of the time. Summing up, therefore, we will say that we have traced down the indeterminacy to a first principle which is inherent in the finite interaction between subject and object. CONCAVE GRATING (?) These new principles, introduced by quantum mechanics, are of similar character as those of Einstein’s theory of relativity and the question arises whether they are independent from each other, or one superseding the other, or contradictory to each other, or reconcilable. To my mind there is not the least difficulty, as the fundamental relations are relativistic[ally] invariant and can be adapted to each Lorentz-frame of coordinates, but I shall enlarge on this subject in the published account of this lecture*.

* [At this place in the manuscript an unknown hand has added the encouraging remark “ H m t ! Hart!” (hear, hear!) in the margin. It is all the more regrettable that the lecture, as far as we know, was never published.]

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

I shall now proceed to discuss a problem designed by Einstein which seemed to show that the foregoing considerations are only valid as long as the view point of theory of general relativity is left out of the discussion, i.e. as long as gravitation is left out. Einstein proposed the following arrangement with which he thought it possible to determine the energy change of an object accurately and also the time at which this change took place accurately: The box contains a clock which, by some arrangement, can move a shutter, so as to open the box ---- 3 for a very short time A T suffi'-.J cient to let pass out of the box 5n just one light-quantum. We /' know, therefore, the time at wqbd.~ which the light-quantum escaped with accuracy A T which can be made indefinitely small. In addition to this we will weigh the box in a gravitational field before and after the process to which there is also no limit of accuracy, so that time and energy of the emission of the light-quantum would both be known accurately. The solution of this paradox can be found by taking into account the principles of general relativity to a fuller extent, and by going into greater detail about the actual process of weighing. We will do the weighing by hanging up the box on a spring and reading the extension of the spring by means of a pointer attached to the box and running over a vertical scale. Now the reading of the pointer cannot be done with infinite accuracy because this would involve infinite inaccuracy in our knowledge of the momentum of the whole box which would mean that the whole process of weighing was illusory. We must therefore be content with a finite accuracy in the determination of the mass. Let us say, for instance, that we do the experiment of reading the pointer by some arrangement which takes the time T. Then the uncertainty in mass A M will be connected with the gravitational force @/ax and the uncertainty in the momentum by the equation:

1 7 3 0

MS, p. 10

7

t + -J -

--

-

IMS, p. 11

or

AM A@ T = h

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

Such an arrangement would therefore involve also an uncertainty about the gravitational potential at which the whole box and the clock inside of it was placed. But this fact, according to a fundamental principle of general relativity would implicate uncertainty in the time scale indicated by the clock:

A T = A @ .T / C ~ or, according to (1)

h AT= c2 AM Finally, we must remember that, also according to the relativity theory, mass and energy are in every case connected by the relation

Mc'

=

E

and this brings (2) back to the same form of uncertainty relation which we found valid in all those cases in which gravitation was not involved. The foregoing analysis of the paradox brought forward by Einstein shows that the phenomenon of gravitation which, up till now, in its mathematical treatment stands so far apart from the rest of modern physics does so by no means with respect to the primary physical concepts evolved by Einstein in his theory of general relativity in order to remove the obstacles which until then seemed to prevent the reconciliation of these two groups of physical phenomena.

MS, P . 12

COMPLEMENTARITY We have thus either space-time description or description where we can use the laws of conservation of energy and momentum. They are complementary to one another. We cannot use them both at the same time. If we want to use the idea of space-time we must have watches and rods which are outside and independent of the object under consideration, in that sense that we have to disregard the interaction between the object and the measuring rod used. We must refrain from determining the amount of momentum that passes into the instrument in order to be able to apply the space-time point of view. This is something unavoidable. We cannot hope to overcome this limitation of applicability because it lies absolutely embodied in the process of the definition of space and time. There are some types of experiment in which it appears to be possible to define energy and time simultaneously but a closer analysis shows that in all those cases

MS, P , 13

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O N

%IS, p. 14

we have actually to do with another use of the words space and time. The point is that in such cases the information about the time of a certain event cannot be put into relation to an independent time scale and, therefore, cannot be used to compare the time of this event with the time of another event, whereas the ordinary idea of time implies that the clock indicating it can be read twice without being disturbed. One must be very careful, therefore, in analysing which concepts actually underlie limitations. With the foregoing analysis we have described the new point of view brought forward by the quantum theory. Sometimes one has described it as leaving aside the idea of causality. I think we should rather say that in the quantum theory we try to express some laws of nature which lie so deep that they cannot be visualised, or, which cannot be accounted for by the usual description in terms of motion. This state of affairs brings about the fact that we must use to a great extent statistical methods and speak of nature making choices between possibilities*. We might for a moment think what is the characteristic of classical Physics. Here we deal with an idealisation. We have to do with phenomena which are not influenced by observation. Our instruments have a negligible interaction with the object and we speak of the phenomena, therefore, as independent systems in themselves and consequently of causality. But it is interesting to note that also in the classical description there is a certain freedom which amounts to a choice of nature. We often say that the object of physics is to control the phenomena of nature, and to do this we must arrange the initial conditions of simple phenomena according to simple laws in an adequate way. In atomic physics we cannot in this simple way control the initial conditions. We cannot even find them out in the same way and, therefore, this choice which, in classical theory, we keep for ourselves has (in atomic physics) gone to nature. The new situation is, therefore, that choice and laws combine in a different way in the description although we are using the same concepts in both cases.

* [It is thought-provoking that an anonymous question mark has been added in the margin here, because this is the very formulation which Bohr repudiates in the article on his discussions with Einstein (Introduction t o Part I , ref. 41).]

VIII. FARADAY LECTURE CHEMISTRY AND THE QUANTUM THEORY OF ATOMIC CONSTITUTION J. Chem. SOC.London, 1932, pp. 349-384 Address Delivered to the Fellows of the Chemical Society at Salters’ Hall on 8 May 1930 on the Award of the Faraday Medal

See Introduction to Part 111, sect. 2.

P A R T 111: G E N E R A L A S P E C T S O F P H Y S I C A L D E S C R I P T I O X

A report of the original address was given in Nature 125 (1930) 788-789.

Reprinted from the Journal of the Chemical Society, 1932.

Faraday Lecture. (DELIVEREDBEFORE THE FELLOWS OF THE CHEMICAL SOCIETYAT THE SALTERS' HALLO X R h Y 8TH, 1930.) By NIELS BOHR.

Chemistry and the Quantum Theory of Atomic Constitution." IT is with a feeling of deep reverence t h a t I accept the kind invita-

tion of the Chemical Society t o deliver this lecture in commemoration of the great genius t o whom we owe so large a part of the common foundation on which chemists and physicists build to-day. Indeed, Faraday's work may be taken as a symbol of the intimate relationship of our sciences, between which all sharp distinction is now disappearing on account of the rapid growth of our insight into the atomic constitution of matter. The peculiar feature of the great recent advance in this field is not only the intense mutual fertilisation of chemical and physical investigations ; but i t will even appear that a fusion of the attitudes of mind by which the study of the laws of nature has been approached by physicists and chemists is essential for the proper appreciation of the situation with which the recent development of atomic theory has faced us. Certainly, a leading idea in physical theory has been t o seek the ultimate cause of all natural phenomena in the relative displacements of material bodies ; Q-hilethe proper field of chemistry may be said, I think, t o be the study of those transmutations of the substances which defy a simple visualisation in terms of displacements. As is well known, atomic ideas originated just in the endeavour t o bridge the gap between these different lines of research. Also, I need not remind you that, in the present epoch of science, the existence of atoms is more than a fruitful hypothesis. A large number of physical and chemical discoveries have given us direct evidence of effects of individual atoms, and we now possess several methods of counting the molecules in a body v i t h great precision. Thanks, above all, to the great English pioneers in intra-atomic research, Sir Joseph Thomson and Lord Rutherford, we have even gained detailed information regarding the structure of atoms, which t o a wide extent allows us t o interpret the properties of the chemical elements as consequences of general physical laws. At the same time, however, we have in this new field met with a peculiar insufficiency of the ordinary ideas of natural philosophy, for the first disclosure of which

* This article, the publication of which has been delayed owing t o unforeseen circumstances, is an elaboration of the author's Faraday Lecture, the plan and substance of which are maintained, while a number of details, omitted a t the verbal delivery, are added. 349

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we are indebted t o the venerated master of the great German school of theoretical physics, Max Plancli. I n the lecture I have the honour and pleasure t o deliver, I shall attempt t o show, in a retrospective survey, how the fundamental idcas regarding the constitution of atoms have logically developed, and how it has gradually been recognised t h a t the very stability of atomic structures, nhich is essential for our analysis of natural phenomena, imposes an unavoidable limitation of the use of space-time pictures in accounting for atomic reactions. Indeed, we meet here with an illustration of the old truth t h a t our power of analysing a harmony and the width of its perception will always exhibit rt mutually esclujive, complementary relationship. I n the scientific literature of the centuries folloxing Sewton’s great work one often meets the expression “ mechanical system of natural philosophy,” with reference not only t o the masterly explanation of astronomical facts, but also to the kinetic theory of matter which permitted an interpretation on atomic ideas of the laws of thermodynamics so fruitful in the study of chemical reactions. If nowadays a similar comprehensive expression were t o be used, we should surely speak of a n “ electromagnetic description of the world ” and should thereby think not only of the imposing structure built on the discoveries of Volta, Prsted, Faraday, and AIasirell, which has been essential for modern technical development, but just as much of the revolution in our ideas of atomic processes brought about by the creation of the electric theory of matter, for which the discovery of the elementary quantum of electricity mas fundamental. As was pointed out by Stone\- in his British Association Address of 1874, and especially emphasisrd by Helmholtz in his famous Faraday Lecture of 1881, this discovery may be regarded, from the standpoint of Dalton’s atomic theory of chemical combination, as an immediate consequence of Faraday’s fundamental work on electrolytic equivalence. Time will not allow me to enter here on the great importance of this discoverx for the whole science of electrochemistry and especially for the theory of electrolytic dissociation, on the development of which Arrhenius lectured before this Society in 1914. I n electrolysis we follow the motions of the ions through the transport of chemical substance, but a still closer examination of ionic properties has been afforded by discharges in rarefied gases, t o the study of nhich Crookes and Lenard contributed so materially. I n fact, the deflection of electric rays in discharge tubes allows us to measure the ratio between the mass and the charge of individual ions, and, as is well known, such measurements led at the close of the last century to the epoch-

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making discovery of the electron as a universal constituent of matter. This electron carries a negative charge equal to the elementary quantum and exhibits a mass-charge ratio very small compared with that of chemical ions in electrolysis. The incorporation of the idea of the atomic nature of electricity in the general electro-magnetic theory of Maxwell was most successfully performed in those years by Lorentz and Larmor. It is above all Thomson, however, who took a leading part, not only in the establishment of the fundamental experimental evidence but also in the attack on the problem of the electric constitution of matter. His ingenious methods of estimating the number of electrons in atoms, based on the scattering of Rontgen rays and the effects accompanying the penetration of swiftly-moving ions through matter, led him t o approximately correct values for this electron number in the different chemical elements. Surely, few achievements have made a greater impression than the attempt t o interpret the general relationship between the elements, which Thomson outlined in 1904 on the basis of these results. Indeed, it brought in a most suggestive way t o the notice of physicists the wonderful outlooks into the central problems of atomic constitution which Tvere opened u p by the recognition of the peculiar periodicity in the chemical properties of the elements when arranged according t o atomic weights, of which RIendeleiev spoke with such great enthusiasm and foresight in his Faraday Lecture of 1889. A more detailed insight into the problem of atomic constitution was at that time hindered by our ignorance of the forces by which the negatively-charged electrons are held in the atoms, or, in other words, of the distribution of intra-atomic positive electri$cation. Decisive progress in this direction, however, was soon made possible through the startling discovery of the radioactivity of certain elements, in the history of which the isolation of radium by Madame Curie forms a conspicuous landmark. This phenomenon, which contrasts markedly with the ordinary physical and chemical properties of matter and at first even threatened t o overthrow the general principle of the conservation of energy, found, as is well known, a lucid and complete interpretation in the disintegration theory of Rutherford and Soddy. According t o this theory, the radioactivity of the substances in question is due t o a spontaneous disintegration of the atoms which follows a simple probability law quite independent of the physical and chemical conditions t o which the atoms are subjected. I n this audience, I need not spend many words in reminding you of the unparalleled success with which Rutherford and his collaborators pursued this new clue t o the problem of atomic constitution. Within the first decade of this century he built up a whole new branch of chemical and physical science which embraces

*

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t h e remarkable changes in the properties of the radioactive substances accompanying the disintegration of their atoms as well as the nature of the rays emitted by these disintegrations and consisting, as is well known, partly of electromagnetic radiation (y-rays) and partly of high-speed electrons (P-rays) and positively-charged ions (a-rays). Above all, however, the results of these studies proved t o be, in Rutherford’s hands, powerful tools for the exploration of intra-atomic structures. Thus, the careful study of the remarkable phenomenon of large-angle scattering of x-rays in passing through matter led him in 1911 t o the fundamental discovery t h a t the positive electrification in any atom is confined within the so-called nucleus, the dimensions of m hich are exceedingly small compared with ordinary atomic dimensions, and which is a t the same time the seat of practically the whole mass of the atom. Not least from the chemical point of view Rutherford’s discovery was of decisive importance, offering us for the first time an unambiguous distinction between atoms and molecules. I n fact, while a n atom possesses only one nucleus, a molecule is a structure in which two or more nuclei enter as separate parts. We thus recognise at once the origin of the remarkable stability of the natural elements in contrast t o chemical compounds. While a separation and a displacement of t h e various atomic constituents are sufficient for chemical substitutions, we learned t h a t a realisation of the old aim of the alchemists, namely the transmutation of elements, implies a radical change of the atomic nucleus itself. Now, i t is just a n explosion of the nucleus which m-e witness in the spontaneous disintegration of radioactive elements. I n fact, after the expulsion from the nucleus of a n CT- or a P-particle, a new atomic nucleus remains, which corresponds to an element of quite different physical and chemical properties. I n this connexion, it was also most instructive t h a t Rutherford mas able t o prove that the helium generated by radium, first observed by Ramsay and Soddy, is a direct product of the neutralisation of the emitted x-rays by capture of t u o electronb, a-particles thus being identified with helium nuclei. As everyone knows, the first artificial transmutation of elements was achieved about ten years later by Rutherford ,when hediscovered that the penetration of a-rays through matter in certain cases is accompanied by the generation of high-speed, singly-charged, positive ions which proved t o be identical with hydrogen nuclei. These are ejected from the nuclei of the atoms bombarded, this process leading t o the formation of new nuclei consisting of the remainder of the original nuclei, with which, in some cases, the impinging a-particle may associate itself. At the end of this lecture me shall discuss the outlook created by

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these last achievements which have inaugurated a new epoch in science. To keep t o our subject, however, we shall for the moment return t o the time when the fundamental ideas of the electric constitution of atoms were taking shape. To everyone who, like myself, had the good fortune t o visit the physical laboratories in Cambridge and Rlanchester about twenty years ago and Tvork under the inspiration of the great leaders it was a n unforgettable experience t o p-itness almost every day the disclosure of hitherto hidden features of nature. I remember, as if it were yesterday, the enthusiasm with which the new prospects for the whole of physical and chemical science, opened by the discovery of the atomic nucleus, were discussed in the spring of 1912 among the pupils of Rutherford. Above all, we realised that the localisation of the positive electrification of the atom nithin a region of practically infinitesimal extension allowed a great simplification in thc classijcation of the properties of matter. I n fact, it permitted a far-reaching distinction between such atomic properties as are wholly determined by the total charge and mass of the nucleus and those a hich depend directly on its internal constitution. Radioactivity, which according t o all experience is independent of the physical and chemical conditions, is typical of the last class of properties. The ordinary physical and chemical properties of matter, on the other hand, depeiid in the first place on the total charge and mass of the atom as aell as on the electronic configuration round the nucleus, which is responsible for the reaction of the atom t o external influences. Moreover, in an isolated atom this electronic configuration must be expected t o depend almost entirely on the nuclear charge and very little on its mass, this being so large compared with the electronic mass t h a t the nuclear motion t o a first approximation can be neglected in comparison with t h a t of the electrons. These simple deductions from the nuclear atomic model offered, indeed, a n immediate explanation of the fact th a t two elements of different atomic weights and with quite different radioactive properties may be so alike as regards other properties t h a t they are inseparable by chemical methods. The first evidence of such a case had been obtained a few years earlier by Boltwood’s discovery of ionium, which is chemically inseparable from thorium and even possesses a n optical spectrum indistinguishable from the thorium spectrum, as was proved just a t that time in Rutherford’s laboratory by experiments of Russell and Rossi. Two such elements, which evidently have equal nuclear charge, occupy the same place in the periodic table and are appropriately called isotopes, according to the proposal of Soddy, through whose extensive investigation of the chemical properties of the radioactive elements in t h e preceding years the general importance

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of isotopy was first recognised. The intimate connexion between the periodic table and the nuclear charge, of uhich we shall soon speak, led to further expectations regarding the connexion between radioactivity and chemical properties which were confirmed by investigations in Manchester b y Hevesy and Russell. A complete coordination of the experimental evidence on this question was soon obtained, as is -ell known, in the formulation of the so-called displacement laws, according t o which any u-ray disintegration is accompanied by a descent of the element of two steps in the periodic table, and any p-ray disintegration by an ascent of one step. I n accordance with this law a n especially instructive case of isotopy is exhibited by two members of a nadioac tive family between T+hicli one u-ray and two p-ray disintegrations take place. I n fact, the identity of the nuclear charge of two such elements follolis a t once, if we realise t h a t in the triple process the nucleus lows two negativelycharged electrons besides the a-particle with its double positive charge. This confirmation of the views in question is all the more interesting because the final empirical establishment of the generalrlisplacement law by Fajans and Soddy in 1913 was quite independent of the development of the ideas on atomic structure here discussed. As we now know from Aston’s ingenious refinement of the analysis of ionic rays originated by Thomson, the existence of isotopes is not confined t o the radioactive elements, but almost all ordinary chemical elements consist of a mixture of isotopes of different atomic masses. The usual atomic weights are thus mean values of secondary importance as regards ordinary chemical properties. Moreover, Aston’s discovery t h a t all atomic masses are very close to simple multiples of the atomic mass of hydrogen made it clear t h a t the nucleus of any atom is built u p of electrons and hydrogen nuclei. Indeed, we iind here a n interesting revival of the ideas of Prout, which a hundred years ago caused so much discussion among chemists. The recognition t h a t the electron and the hydrogen nucleus, generally termed ‘‘ proton,” form the ultimate units of atomic structures places before us the prospect of a purely electric constitution of matter. Still, as we have seen, the interpretation of the bulk of chemical and physical experience is independent of the problem of the internal constitution of atomic nuclei, TT hich presents peculiar aspects t o be discussed later. For this interpretation, it is sufficient to consider the nucleus as a charged mass point, and we are solely concerned with the problem of the configuration of the extra-nuclear electrons, the number of which in a neutral atom is, of course, determined b y the nuclear charge. Now, the &-st element i n the periodic table, namely hydrogen, contains one electron in the atom, and the second element, helium, contains two extra-

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nuclear electrons. From the general ideas of Thomson concerning the relation between electron number and periodic table, it was therefore a n unavoidable generalisation t h a t for any element the number of extra-nuclear electrons in the neutral atom is given by the integer, the so-called atomic number, which fixes its position in the table, often termed the ‘( natural system of the elements.” This view was in obvious conformity with the radioactive displacement law and agreed, within the limits of experimental error, with Rutherford’s original estimate of the nuclear charge, derived from the measurements of cc-ray scattering by Geiger and Rlarsden. It has since been directly confirmed by the refined measurements of Chadn ick on this phenomenon, as well as by renewed investigations, interpreted by Thomson’s famous formula, of the scattering of Riintgen rays by matter. Above all, the experimental evidence regart ling this fundamental point has obtained a most extraordinary amplification, as we shall see, through Moseley’s brilliant researches on the characteristic Rontgen spectra of the elements. Summarising the situation, we may say t h a t , as regards the co-ordination of all ordinary properties of matter, Rutherford’s model of the atom puts before us a task reminiscent of the old dream of philosophers : to reduce the interpretation of the laws of nature to the consideration of pure numbers. On setting out to work on this attractive programme, however, one was a t once confronted with difficulties of a most serious character, which would a t f i s t sight even appear t o be fatal t o the wholc conception of the electric constitution of atoms. Indeed, on clawical theories no system of charged material points will exhibit a stability of the kind w-hich must be attributed t o atomic structures in order t o account for the chemical and physical properties of matter. Such systems will not possess statical states of stable equilibrium in the ordinary mechanical sense, nor will any dynamical state fulfil the conditions required. Even in the simplest case of an atom consisting of a positive nucleus and one electron, this is quite evident. I t is true that, according t o Kewtonian mechanics, two particles attracting each other with a force, ruled by Coulomb’s law, will revolve in Keplerian ellipses round their common centre of gravity. But this solution, which accounts satisfactorily for the stability of planetary motions, does not enable us t o understand why, by the combination of a n electron and a proton, an atom is formed q i t h properties corresponding t o the chemical behaviour and characteristic linespectrum of hydrogen. Without entering into any speculation regarding the origin of the solar system, it is clear t h a t the dimensions of the earth’s orbit and the length of the year are essentially deter-

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mined by the initial conditions and may any day be permanently modified by a collision with a meteor. On the other hand, the definiteness of the hydrogen atom under most varied conditions is impressively exhibited by the identscation of spectral lines from distant stars with the hydrogen spectrum obtained in ordinary discharge tubes. If we look closer into the origin of this spectrum, the situation becomes even worse. I n fact, the very emission of radiative energy from the atom will be accompanied, according t o ordinary electromagnetic ideas, by a gradual decrease of the sue of the electron orbit and the period of revolution, which process will prevent the appearance of sharp monochromatic spectral lines and ultimately cause the electron to combine with the proton into a neutral system of linear dimensions exceedingly small compared with those of actual atoms. Similar remarks obviously hold for any atomic system of the type under consideration. Indeed, from ordinary mechanics and electrodynamics no argument can he derived which allows us to explain why the electric constituents of the atom do not neutralise each other in a way catastrophic to the stability of material bodies. Clearly, an entirely new idea was needed before the discoveries regarding the ultimate electric particles could be properly utilised in the interpretation of the general properties of matter. I n the search for such an idea, however, one had not to look far : A clue to the solution of the difficulties was offered by Planck’s fundamental discovery of the elementary quantum of action, which, especially in the hands of Einstein, had already proved so fruitful in co-ordinating physical experience of most varied kinds. Indeed, this discovery had disclosed a novel feature of atomicity in the laws of nature, quite foreign t o the classical ideas of physics, and in a certain sense even more so than the atomic nature of electricity. Of course, no explanation based on general electromagnetic theory can be given of the existence of the elementary quantum of electricity and of the specific values of the masses of the electron and of the proton, but i t must be remembered that the measurements of the charge and the mass of these particles rest on experimental evidence vhich permits of an unambiguous interpretation by means of classical ideas. No account of the existence of the quantum of action can be given, however, which does not involve a radical departure from ordinary physical principles. Of course, the determination of Planck’s universal constant is also based on measurements classically defined, but in contrast to the case of the electronic charge and mass, no rational interpretation in electromagnetic terms can be given of the derivation of the action quantum from these measurements. The field of unambiguous applicability of classical concepts

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is limited to processes where the mechanical action involved is large compared with the quantum, as in the deflection experiments with electric rays; and the insufficiency of these ideas to account for atomic reactions is due just t o the fact that a detailed analysis of the intra-atomic motions would involve the consideration of elements of electron paths for which the action is of the same order of magnitude as, and even smaller than, the quantum. Certainly, the two fundamental aspects of atomicity, symbolised by the elementary quanta of electricity and action, are intimately connected, and. when we come t o the problem of the constitution of atomic nuclei, we shall see t h a t it is no longer possible to use the ideas of electron charge and mass without ambiguity. But, as regards the extranuclear electronic configuration, a great simplification arises from the fact t h a t the dimensions of the constituent particles, defined in a classical sense, can be considered as negligibly small compared with those of the whole atom. Indeed, this idealisation, on which the simple classification of atomic properties rests, allows us in the region outside the nucleus to regard the specific properties of the electrons as independent of the quantum of action. Already in the years preceding the establishing of the nuclear model of the atom, the question of the bearing of Planck’s discovery on the problem of atomic constitution had been discussed from various sides, and approximate relationships between atomic constants had been suggested. Previous atomic models, however, constructed from the point of view of mechanical stability, were obviously unsuited t o a satisfactory interpretation of the specific properties of the elements, and, as they were in themselves fully determined as regards dimensions and frequencies, the introduction of the action quantum meant no decisive improvement in this respect. The situation was completely changed by Rutherford’s discovery. Indeed, the evident insufficiency of simple mechanical ideas as regards the interpretation of atomic stability not only made n radical departure from classical principles inevitable but left a t the same time sufficient freedom for the utilisation of the guidance offered by the direct evidence concerning the physical and chemical properties of the elements. A suitable basis for the use of this evidence I found in two simple -pothetical constituents named ortho- and para-helium. Wliile the normal state of the helium atom belongs to the para-system, the first term of the ortho-system corresponds t o the so-called metastable state of the helium atom, the remarkable propertie, of XT hich were first pointed out by Franck. A very interesting recent contribution to the problem of the constitution of the helium atom i h due to Xylleraas who, on the basis of wave mechanics, developed a precision method for the numerical evaluation of wave-functions and derived a ralue for thc ionisation potential of helium mhich agree-. uithin the limits of experimental error, M ith the spectral evidence. Indeed, this result represents the first quantitative derivation of a constant depending on the constitution of a n atom with more than one electron. Recently, Hylleraas' calculation has obtained further striking confirmation by E d l h ' s analysis of the spectra emitted by singly. doubly, and trebly charged ions of beryllium, boron, anti carbon, respectively. For atoms and ions with more than two electrons, no exact calculations of spectral terms have hitherto been performed ; still, for such atoms also, the wave-functions derived by the approximation method of Hartree have proved helpful, especially in accounting for the spatial electronic distribution in atoms determining the dispersion of Rontgen rays. The wonclerful tool of quantum mechanics, with the incorporation of the exclusion principle, has been not only essential for the detailed treatment of properties of isolated atoms, but also indispensablr as regards the problem of molecular constitution. For this problem the study of the so-called bund spectra plays its fundamental a part as that of series spectra for the constitution of atoms. Just a i the latter spectra inform us about the states of binding of the electrons in the atoms, the analysis of band spectra tells us about the electron binding in the molecules, and, in addition, about the vibrations of the nuclei relative to each other and the states of rotation of the whole molecule. The gradual elucidation of this problem gives an interesting illustration of the general development of theoretical spectroscopy. The infra-red absorption bands

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of polar molecules had already been satisfactorily accounted for on classical electromagnetic theory, as resulting from vibrations of the constituent ions relative to each other ; and, in contrast t o the case of spectra resulting from changes in the electron binding, this explanation needsonlyslight modification in the quantum theory, since the ionic masses are so large that vibrations involving several quanta may still be considered as small harmonic oscillations about equilibrium positions. The expectations of classical theory regarcling the influence on spectra of translatory and rotatory motions of an atomic system as a whole were originally discussed by Rayleigh. While his deductions regarding the breadth of spectral lines resulting from the translatory thermal agitation of gas molecules are still valid in investigations of the masses of the emitting systems, the expectations regarding the influence of thermal rotation constituted a fundamental difficulty for the understanding of the observed sharpness of the lines of atomic spectra until the advent of the nuclear model, according to which the essential part of the mass of the atom does not contribute to its moment of inertia. I n the case of molecules, however, thermal rotation is essential for the shape of the infra-red absorption bands, as was first recognired by Bjerrum in 1912. His considerations were quite independent of special ideas on atomic constitution, and, in accordance with Einstein’s quantum theory of specific heats, he made the important prediction that the bands under consideration should show a fine structure in which each component should correspond to a different number of rotation quanta. On the present view of the origin of spectral lines, this interpretation of the fine structure has t o be modified in the sense t h a t each component is not associated with a single rotation state, but originates from a transition process involving a change of rotation as well as of vibration quanta. h’evertheless, on account of the selection rules for such transitions, deduced from the correspondence argument, the resulting fine structures of the infra-red absorption bands are of the general type predicted by Bjerrum and soon verified by observation. The complete analysis of these bands, first made possible by the refined methods of quantum mechanics, allows a n unambiguous determination of the moments of inertia of the molecule in its different vibration states, and, in consequence, a detailed insight into the spatial configuration of the nuclei. I n the case of the band spectra of the optical region, we have to do with transitions involving an essential modification of the electron binding, responsible for the energy exchanges occurring in chemical reactions, and the analysis of these spectra, based on the combination principle and the correspondence argument, has given us valuable information regarding such reactions. The

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methods of quantum mechanics have been here especially helpful for the understanding of the homopolar bonds, as was first shown by Heitler and London. The chemical “ bond ” thus appears as essentially connected with such aspects of atomic stability as do not allow of a n unambiguous visualisstion by means of space-time pictures. Although no quantitative results have hitherto been obtained in this field, the underlying ideas are, no doubt, sufficiently advanced t o furnish, in connexion with the analysis of band spectra, a trustworthy basis for the discussion of the large amount of material regarding organic compounds, disentangled by chemists with such unfailing intuition. f t is very tempting to enlarge upon this point which would be in itself a suitable subject for a lecture on chemistry and quantum theory, but this would oblige me t o enter into details of a more technical character than the plan of the present lecture permits. Before proceeding to other problems, however, we shall briefly discuss the important conclusions which can be drawn from the interpretation of molecular spectra regarding the so-called quantum statistics in their relation t o the nuclei. A starting point here was found by Heisenberg and Hund in the peculiar alternations inintensity occurring within the rotation bands of the spectra of molecules consisting of two identical atoms. Corsesponding to the fact t h a t the rotation of such symmetrical molecules, on classical theory, will not give rise t o any radiation of the rotation frequency, but only t o radiation of double frequency, the rotation states will, on quantum mechanics, split into two non-combining sets, characterised respectively by even and uneven values of the rotation quantum number. As was shown by Dennison, this result receives a striking confirmation from Eucken’s measurements of the specific heat of hydrogen a t low temperatures which for a long time had resisted all attempts a t an interpretation on quantum theoretical statistics. Indeed, owing to the impossibility of transitions between the two types of rotation state of the hydrogen molecule, there will only be thermal equilibrium, under t h e conditions of these measurements, within each of the two sets of states, but not between them. Even at the very lowest temperatures, molecules will then be present in rotation states of both sets, which, in analogy with the classification of the stationary states of the helium atom, are called orthoand para-states. Only under special conditions facilitating the establishment of thermal equilibrium, Bonhoffer and Euckeri have recently succeeded in bringing all molecules into the lowest para-state, analogous t o the normal helium state. For the quantitative interpretation of these remarkable phenomena, it is necessary t o assume t h a t the protons obey the same exclusion

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principle as the electrons, in the sense t h a t all wave-functions of a hydrogen molecule are not only antisymmetrical in the space and spin co-ordinates of the electrons, but also in those of the protons, defined in a n exactly analogous way. This conclusion is in complete agreement with the intensity variations within the hydrogen rotation bands, the analysis of which has given for the moments of inertia of the molecule values identical with those derived from the theory of specific heat. The study of the helium band spectrum, however, has disclosed a new important feature. I n fact, it was found t h a t the wave-functions in this case were symmetrical in the space co-ordinates of the two nuclei, for which no spin had to be taken into account. We meet here with the same kind of statistics as t h a t first introduced by Bose to account for Planck’s law of black-body radiation on the basis of the idea of photons. Notwithstanding this formal similarity, the striking departure from classical ideas of statistics with which we have here to do presents, from the point of view of correspondence, an important difference in the cases of photons and of material particles like helium nuclei. I n the first case, this departure is connected with the symbolic character of the photon idea, already emphasised ; in fact, in the limit where the quantum of action can be neglected, and thus any trace of this idea disappears, the kind of statistics under consideration reduces to the classical treatment of electromagnetic radiation fields. I n the case of material particles on the other hand, which are well-defined concepts from the classical point of view, the new quantum statistics find no unambiguous application within the scope of ordinary statistical mechanics in which the existence of the action quantum is neglected and the particles are treated as individual dynamical entities. This situation appears also from the very circumstance that in quantum mechanics we have two radically different kinds of statistics, namely, besides the Bose-statistics, the so-called Fermi-statistics which rest 011 the exclusion principle. The essentially non-visualisable character of these statistics has been no hindrance, however, to their great fruitfulness in the most varied atomic problems. Thus, in the hands of Sommerfeld, the Fermi-statistics have been fundamental for the understanding of the electric conduction in metals and allied phenomena ; and, as was recently shown by Mott, the Bose-statistics are necessary to account for the scattering of ,x-rays in helium. I n this very cursory account it is impossible to give a proper impression of the beauty and consistency of the new quantum mechanics, in which nobody can take greater delight than those who have followed from the primitive stages the evolution of ideas

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which I have tried t o sketch in this lecture. It is true that this development has gradually carried us farther away from the ideals which inspired the ancient philosophers of the atomistic school and which have been so immensely fruitful for the development of chemical and physical science. This disillusion, however, has in return led us to a more comprehensive and, I venture t o say, more open-minded view regarding natural phenomena. Indeed, we recognise in the existence of the quantum of action a n inherent limitation, as regards the problem of atomic constitution, not only of all concepts of classical physics, but even of the ideas underlying our account of everyday experience. I n fact, the unambiguous application of such fundamental concepts as space and time is essentially limited on account of the finite interaction between the object and the measuring tools, which, as a consequence of the existence of the elementary quantum, is involved in any measurement. To appreciate this point, we must remember that this interaction cannot be taken fully into account in the description of the phenomena, since the very definition of the space-time frame implies the neglect of the reaction of the object on the measuring instruments. Thus, any attempt t o fix the space-time co-ordinates of the constituent particles of an atom would ultimately involve an essentially uncontrollable exchange of energy and momentum with the measuring rods and clocks which prevents an unambiguous correlation of the dynamical behaviour of the atomic particles before the observation with their later behaviour. Inversely, every application of conservation theorems, for instance to the energy balance in atomic reactions, involves an essential renunciation as regards the pursuance in space and time of the individual atomic particles. I n other words, the use of the idea of stationary states stands in a mutually exclusive relationship to the applicability of space-time pictures. This situation corresponds exactly to the formalism of quantum mechanics, according t o which the numerical values of two dynamical variables cannot in general be simultaneously determined, the limits for their unambiguous evaluation being given by the peculiar reciprocal relations known as the indeterminacy principle of Heisenberg. This principle defines the latitude in the application of classical concepts, necessary for the comprehension of the fundamental laws of atomic stability which are beyond the reach of these concepts. The essential indeterminacy in question must therefore not be taken t o imply a oneaided departure from the ideal of causality underlying any account of natural phenomena. The use of energy conservation in connexion with the idea of stationary states, for instance, means an upholding of causality particularly striking when we realise that the very

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idea of motion, on which the classical definition of kinetic energy rests, has become ambiguous in the field of atomic constitution. As I have stressed by the argumentation mentioned, space time co-ordination and dynamical conservation laws may be considered as two complementary aspects of ordinary causality which in this field exclude one another t o a certain extent, although neither of them has lost its intrinsic validity. I n this sense we recognise, as I mentioned a t the beginning of this lecture. in the very attitudes of physicists and chemists respectively two complementary viewpoints equally indispensable for the comprehension of the laws of nature. To appreciate the r61e of the concept of probability in atomic theory, it is moreover important t o remember t h a t the complete control over the course of events aimed a t in the classical description of natural phenomena involves the essential assumption of a perfect liberty in the choice of the initial conditions. I n such cases, however, as the occurrence of elementary transition processes where we have not even the possibility of defining the initial conditions in the classical sense we must be satisfied to have recourse t o probability considerations in the sense of the correspondence argument. Xotwithstanding the essentially new situation created by the discovery of the quantum of action, the characteristic feature with which we have here to do is not unfamiliar in atomic theory. A typical example is afforded by the statistical theory of heat, according to which the very concept of temperature stands in an exclusive relation to a detailed description of the behaviour of the atoms in the bodies concerned. It is just this point, implied in Maxwell’s law of velocity distribution and especially conspicuous in Gibbs’s treatment of statistical thermodynamics, which allows us to solve the apparent contradiction between the law of increase of entropy and the general reversibility of the individual mechanical processes which are involved in Boltzmann’s interpretation of entropy in terms of probability. I n fact, thermodynamical irreversibility, as exhibited in the levelling of temperatures, does not mean that it reversal of the course of events is impossible, but t h a t the prediction of such a reversal cannot be part of any description involving a knowledge of the temperatures of the various bodies. This situation presents it remarkable analogy with the peculiar irreversibility characteristic of the description of quantum mechanics. Indeed, the reversibility of the classical laws of motion is formally upheld in the quantum symbolism, but the indeterminacy in the use of classical concepts defining the state of a system a t a given time implies a n essential irreversibility in the physical interpretation of this symbolism. I n thermodynamics as well as in quantum

QUANTUM THEORY O F ATOMIC CONSTITUTION.

377

mechanics, the description contains a n essential limitation imposed upon our control of the events which is connected with the impossibility of speaking of well-defined phenomena in the ordinary mechanical sense. Of course, this limitation has a quite different origin in the two cases. I n fact, in statistical thermodynamics, we have in the first place not to do with a failure of the mechanical concepts in accounting for the details of the events, but with the incompatibility of such a detailed account with a definition of temperature. I n quantum mechanics, on the other hand, we are concerned with the essential incompatibility between the elementary laws of atomic stability and the use of the classical mechanical concepts on which all measurements must be interpreted. Indeed, as we have seen, t h e view-point of ‘’ complementarity in the description of atomic phenomena is forced upon us by t h e existence of the quantum of action in a similar way as the view-point of relativity in classical physics by the finite propagation of all electromagnetic interactions. I n this sense, quantum mechanics may be said t o represent the next step in the development of our tools of a n adequate description of the natural phenomena. The scope of the quantum mechanical symbolism is essentially confined, however, to problems where the intrinsic stability of the elementary electrical particles can be left out of consideration in a similar way as in t h e classical electron theory. I n this connexion, it must not be forgotten t h a t t h e existence of the electron even in classical theory imposes an essential limitation on the applicability of the mechanical and electromagnetic concepts. Indeed, the finite propagation of electromagnetic forces brings with it the existence of a fundamental length, the so-called “ electron diameter,” defining a lower limit for the extension of the region where the idealisation according t o which the electron is considered as a charged material point is justifiable. Not only would a concentration of the charge of the electron within a smaller space result in an essential modification of its mass, but we even meet here witha limitation of theunambiguous use of the idea of inertial mass. I n fact, we lose any simple bask for a sharp separation between ponderomotoric forces and radiative reactions when we consider processes in which the electron undergoes a relocity change of the same order as the velocity of light within a length of path equal t o the electron diameter. It is true t h a t such considerations lose their significance t o a large extent on account of the existence of the quantum of action which imposes an essential limit t o the analysis of motion. The fertility of quantum mechanics as applied t o t h e problem of atomic stability lies just in the fact t h a t the linear dimensions of the regions ascribed t o even the firmest electron-bindings outside the nucleus are still very large compared

378

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with the classical electron diameter. At the same time the theory of Dirac, already referred to, which represents a most important step towards the adaptation of the symbolism t o the point of view of relativistic invariance, has disclosed new aspects of the fundamental difficultiesinvolved in the reconciliation of the intrinsic stability of the electron with the existence of the quantum of action. I n fact, Dirac's formalism implies the possibility of transition processes from state\ corresponding t o normal properties of the electron t o so-called states of negative energy for which the sign of its mass-charge ratio is reversed, the energy changes iiivolved exceeding the critical value which, from Einstein's well-known relation, corresponds t o the inertial mass of the electron. Transitions of this type should, on the theory. occur so frequently t h a t even hydrogen atoms would be in-\tan+aneously destroyed with emission of radiation of very high frequency. Dirac himself has made an interesting attempt t o overcome these difficulties by a n extension of the formalism which permits exclnsion of the unwanted transition processes by assuming that all states of negative energy are ordinarily filled up in a similar way as the completed electron groups in atomic structures. Such considerat ion3, however, would seem t o trespass the limit of applicability of the correspondence argument, and the difficulties inherent in m y symbolism resting on the idealisation of the electron as a charged material point appear also most instructively in the recent attempt of Heisenberg and Pauli t o build up a theory of electromagnetic fields on the lines of quantum mechanics. Their formalism leads, in fact, t o consequences inconsistent with the finite mass of the electron and the small coupling between atoms and electromagnetic radiation fields, on which rests the interpretation of the empirical evidence regarding spectra, based on the idea of stationary states. Under these circumstances, we are strongly reminded t h a t the whole attack on atomic problems leaning on the correspondence argument is an essentially approximatice procedure, made possible only hy the smallness of the ratio between thc square of the elementary unit of electric charge and the product of the velocity of light and the quantum of action which allow us t o a large extent to avoid t h e difficulties of relativistic quantum mechanics in considering the behaviour of extra-nuclear electrons. Like the ratio between the masses of t h e electron and of the proton, this is a non-dimensional constant fundamental for our whole picture of atomic phenomena, the theoretical derivation of which has been the object of much interesting speculation. Although we must expect t h a t the determination of these constants will be a n integral part of a general consistent theory in which the existence of the elementary electric particles and the existence of the quantum of action are both naturally incor-

QUAXTUM THEORY O F ATOMIC CONSTITUTION.

379

porated, these problems would appear t o be out of reach of the present formulation of quantum theory in which the complete independence of these two fundamental aspects of atomicity is an essential assumption. This situation must above all be kept in mind when we turn t o the problem of the constitution of atomic nuclei. The empirical evidence regarding the charges and the masses of these nuclei, as well as the evidence concerning the spontaneous and the excited nuclear disintegrations, leads, as we have seen, to the assumption that all nuclei are built up of protons and electrons. Still, as soon as we inquire more closely into the constitution of even the simplest nuclei, the present formulation of quantum mechanics fails essentially. For instance, it is quite unable to explain why four protons and two electrons hold together t o form a stable helium nucleus. Evidently we are here entirely beyond the scope of any formalism based on the assumption of point electrons, as it also appears from the fact that the size of the helium nucleus, as deduced from the scattering of a-rays in helium, is of the same order of magnitude as the classical electron diameter. J u s t this circumstance suggests that the stability of the helium nucleus is inseparably connected with the limitation imposed on classical electrodynamics by t h e existence and the stability of the electron itself. This means, however, that no direct attack on this problem, based on the usual correspondence argument, is possible as far as the behaviour of the intra-nuclear electrons is concerned. As regards the behaviour of the protons, the situation is essentially different, since their comparatively large mass permits of an unambiguous use of the idea of space co-ordination even within nuclear dimensions. Of course, in absence of a general consistent theory accounting for the stability of the electron, we cannot make any direct estimate of the forces which hold the protons in the helium nucleus, but it is interesting to note that the energy liberated by the formation of the nucleus, as calculated from the so-called mass-defect by means of Einstein’s relation, is in approximate agreement with the binding energy of the protons t o be expected on quantum mechanics from the known nuclear dimensions. Indeed, this agreement indicates that t h e value of the ratio of t h e masses of the electron and the proton plays a fundamental part in the question of the stability of atomic nuclei. I n this respect, the problem of nuclear constitution exhibits a characteristic differencefrom that of the constitution of the extranuclear electron configuration, since the stability of this configuration is essentially independent of the mass-ratio. When we pass from

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the helium nucleus to heavier nuclei, the problem of nuclear constitution is, of course, still more complicated, although a certain simplification is afforded by the circumstance that the a-particles can be considered t o a large extent t o enter as separate entities into the constitution of these nuclei. This is not only suggested by the general facts of radioactivity, but appears also from the smallness of the additional mass defect, expressed by Aston’s wholenumber rule for the atomic weights of isotopes. The main source of knowledge regarding the constitution of atomic nuclei is the study of their disintegrations, but important information is also derived from ordinary spectral analysis. As was mentioned, the hyperfine structures of spectral lines allow us to draw conclusions concerning the magnetic moments and angular momenta of the atomic nuclei, and from the intensity variations in band spectra we deduce the statistics obeyed by the nuclei. A s might be expected, the interpretation of these results falls largely outside the scope of present quantum mechanics, and, in particular, the idea of spin is found not t o be applicable to intra-nuclear electrons, as was first emphasised by Kronig. This situation appears especially clearly from the evidence concerning nuclear statistics. It is true that the fact, already mentioned, that the helium nuclei obey the Bose statistics is just what was t o be expected from quantum mechanics for a system composed of an even number of particles which, like the electrons and protons, satisfy Pauli’s exclusion principle. But the next nucleus for which data concerning statistics are available, namely the nitrogen nucleus, obeys also the Bose statistics, although it is composed of an uneven number of particles, namely 14 protons and 7 electrons, and thus should obey the Fermi statistics. Indeed. the general experimental evidence concerning this point seems to follow the rule that nuclei containing an even number of protons obey the Bose statistics, while nuclei containing an uneven number of protons obey the Fermi statistics. On the one hand, this remarkable “passivity ” of the intra-nuclear electrons in the determination of the statistics is a very direct indication, indeed, of the essential limitation of the idea of separate dynamical entities when applied t o electrons. Strictly speaking, we are not even justified in saying that a nucleus contains a definite number of electrons, but only that its negatire electrification is equal t o a whole number of elementary units, and, in this sense, the expulsion of a p-ray from a nucleus may be regarded as the creation of an electron as a mechanical entity. On the other hand, the rule just mentioned regarding nuclear statistics may be considered, from this point of view, as a support for the essential validity of a quantum mechanical treatment of the behaviour of t h e a-particles and protons in the nuclei. Actually, such a treatment

QUANTUM THEORY OF ATOMIC CONSTITUTION.

381

has also been very fruitful in accounting for their part in spontaneous and controlled nuclear disintegrations. I n the ten years that have elapsed since Rutherford’s fundamental discoveries, a large amount of most valuable material on this subject has been accumulated, owing, above all, to the great exploration work in the new field carried on in the Cavendish Laboratory under his guidance. Now, from the theoretical standpoint, it is one of the most interesting results of the recent development of atomic theory that the use of probability considerations in the formulation of the fundamental disintegration law, which for its time was a quite isolated and very bold hypothesis, has been found to fall entirely in line with the general ideas of quantum mechanics. Already at the more primitive stage of the quantum theory, this point was touched upon by Einstein in connexion with his formulation of the probability laws of elementary radiation processes, and was further stressed by Rosseland in his fruitful work on inverse collisions. I t is the wavemechanical symbolism, however, which first offered the basis for a detailed interpretation of radio-active disintegrations, in complete conformity with Rutherford’s deduction of nuclear dimensions from the scattering of a-rays. As was pointed out by Condon and Gurney, and independently by Gamow, the wave-formalism leads, in connexion with a simple model of the nucleus, to an instructive explanationof the law of a-ray disintegration as well as of the peculiar relationship, known as the rule of Geiger and Nuttall, between the mean life-time of the parent element and the energy of the a-ray expelled. Gamow, especially, succeeded in extending the quantum mechanical treatment of nuclear problems to a general qualitative account of the relationship between u- and y-ray-spectra, in which the ideas of stationary states and elementary transition processes play the same part as in the case of ordinary atomic reactions and the emission of optical spectra. I n these considerations, the aparticles in the nuclei are treated similarly to the extra-nuclear electrons in the atoms, with the characteristic difference, however, that the u-particles obey the Bose statistics and are kept within the nucleus bytheir owninteraction, while theelectrons, obeying the Fermi statistics, are held in the atom by the attraction of the nucleus. This is, among other causes, responsible for the smallness of the rate of energy emission, as y-radiation, from excited nuclei which is even comparable with the rate of mechanical energy exchange between such nuclei and the surrounding electron clusters, the so-called internal conversion. I n fact, in contrast to an atom built up of separate positive and negative particles, a nucleus-like system composed only of a-particles will never possess an electric moment, and, in this respect the additional protons and negative electrific-

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ation of actual nuclei can hardly be expected t o make much difference. Apart from such simple applications of the correspondence argument, our ignorance of the forces acting on the a-particles and protons in the nuclei, which must be assumed to depend essentially on the negative electrification, prevents a t present theoretical predictions of a more quantitative character. A promising means of exploring these forces is afforded, however, by the study of controlled disintegrations and allied phenomena. As far as the behaviour of u-particles and protons is concerned, it may therefore be possible t o build up gradually, by means of quantum mechanic;;, a detailed theory of nuclear constitution, from which in turn we may get further information about the new aspects of atomic theory Dresented by the problem of negative nuclear electrification. As regards this last question, much theoretical interest has recently been aroused by the peculiar features exhibited by the ?-my ezpulsions. On the one hand, the parent elements have a definite rate of decay, expressed by a simple probability law, just as in the case of the u-ray disintegrations. On the other hand, the energy liberated in a single P-ray disintegration js found t o vary within a wide continuous range, whereas the energy emitted in an a-ray disintegration, when due account is taken of the accompanying electromagnetic radiation and the mechanical energy conversion, appears to be the same for all atoms of the same element. Unless the expulsion of p-rays from atomic nuclei, contrary to expectation, ir not a spontaneous process but caused by some external agency, the application of the principle of energy conservation to p-ray disintcgrations would accordingly imply that the atoms of any given radioelement would have different energy contents. dlthough the corresponding variations in mass would be far too small to be detected by the present experimental methods, such definite energy differences between the individual atoms would be very difficult to reconcile with other atomic properties. I n the first'place, we find no analogy to such variations in the domain of non-radioactive elements. I n fact, as far as the investigations of nuclear statistics go, the nuclei of any type, which have the same charge and, within the limits of experimental accuracy, the same mass, are found to obey definitd statistics in the quantum mechanical sense, meaning that such nuclei are not t o be regarded as approximately equal, but as essentially identical. This conclusion is the more important for our argument, because, in absence of any theory of the intra-nuclear electrons, the identity under consideration is in no way a consequence of quantum mechanics, like the identity of the extra-nuclear electronic configurations of all atoms of an element in a given stationary state, but represents a new fundamental feature of atomic stability. Secondly,

Q U A N T U M THEORY OF ATOMIC CONSTITCTION.

383

no evidence of a n energy variation of the kind in question can be found in the study of the stationary states of the radioactive nuclei involved in the emission of U - and y-rays from members of a radioactive familj? preceding or following a P-ray product. Finally, the definite rate of decay, which is a common feature of u- and p-ray disintegrations, points, even for a pray product, t o a n essential similarity of all the parent atoms, in spite of the variation of the energy liberated by the expulsion of the p-ray. I n absence of a general consistent theory embracing the relationship between the intrinsic stability of electrons and protons and the existence of the elementary quanta of electricity and action, i t is very difficult t o arrive at a definite conclusion in this matter. At the present stage of atomic theory, however, we may say that me have no argument, either empirical or theoretical, for upholding the energy principle in the case of p r a y disintegrations, and are even led t o complications and difficulties in trying to do so. Of course, a radical departure from this principle would imply strange consequences, in case such a process could be reversed. Indeed, if, in a collision process, an electron could attach itself t o a nucleus with loss of its mechanical individuality, and subsequently be recreated as a $-ray, we should find that the energy of this p-ray would generally differ from that of the original electron. Still, just as the account of those aspects of atomic constitution essential for the explanation of the ordinary physical and chemical properties of matter implies a renunciation of the classical ideal of causality, the features of atomic stability, still deeper-lying, responsible for the existence and the properties of atomic nuclei, may force us to renounce the very idea of energy balance. I shall not enter further into such speculations and their possible bearing on the much debated question of the source of stellar energy. I have touched upon them here mainly to emphasise that in atomic theory, notwithstanding all the recent progress, we must still be prepared for new surprises. I n judging the present situation of atomic theory, it is essential t o recognise t h a t the whole classical description of natural phenomena depends on the intrinsic stability of ordinary material bodies, and t h a t therefore we must not be too much surprised when in t h a t part of science where this stability itself is the object of investigation we shall meet with new aspects of natural philosophy. I n striving with the unsolved difficulties involved in this situation, we are above all encouraged by the example of men like Faraday who in their wandering on untrodden paths knew how t o find reliable guidance to the disclosure of Nature’s secrets in Nature herself. The unfamiliar character of the views t o which

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such endeavours lead will naturally often appear mysterious, but, as Helmholtz so forcefully emphasised, just Faraday’s general scientific method allowed him more than anybody else t o contribute t o the great aim “ t o purify science from the last remnants of metaphysics.” I n concluding this lecture, I may be allowed t o express the hope t h a t modern endeavours in atomic theory have not in this respect betrayed the great example which Faraday has set us, and t h a t the new aspects of natural philosophy, which tend t o harmonise the knowledge collected by chemists and physicists in their respective fields, far from involving any mysticism foreign t o the spirit of science will be found t o have contributed to the great common aim.

-

PRINTEUIN GREAT B R I T A I N BY R I C H A R D C L A Y & Soh’s,L I M I T E D . BUNGAY, SUFPVLK.

INTRODUCTION

The letters to and from Bohr, quoted in the Introductions to Parts 1-111, are reproduced here in the original language, and arranged in alphabetical order according to correspondents. Letters in Danish are followed by a translation. As in the previous volume, this also applies to the German letters in the Pauli correspondence. The editors have used their discretion tacitly to correct “trivial” mistakes, e.g., in spelling and punctuation. We have tried, however, to preserve “characteristic” mistakes. In the reproduction of the letters we have attempted to make the lay-out of letterheads etc. correspond as closely as possible to that of the original letters. The list preceding the letters contains references to the pages in the Introductions where the letters are quoted so that the reader can readily find the context in which a particular letter has been quoted.

PART

IV:SELECTED

CORRESPONDENCE

(MAINLY 1926- 1930)

CORRESPONDENCE INCLUDED Reproduced p. Translation p.

Quoted p

310 312 315

310 312 315

CHARLES G . DARWIN Bohr to Darwin, 15 November 1930 Bohr to Darwin, 18 November 1930 Bohr to Darwin, 21 December 1930

-

-

PAUL A.M. DIRAC Bohr to Dirac, 24 March 1928

44

44

PAUL EHRENFEST Ehrenfest to Goudsmit, Uhlenbeck, and Dieke, 3 November 1927 (extract)

415

37

37

418

21

21

ALBERT EINSTEIN Bohr to Einstein, 13 April 1927 RALPH H. FOWLER Bohr to Fowler, 14 April 1926 Bohr to Fowler. 26 October 1926

42 1 423

9 14

WERNER HEISENBERG Heisenberg to Dirac, 27 April 1927 Bohr to Heisenberg, [December 19281

17 424

24

17 24

425 428

427 429

197 196

-

HENDRIK A. KRAMERS Bohr to Kramers, 30 November 1929 Bohr to Kramers, 7 December 1929

*

A carbon copy of this letter is in the Niels Bohr Archive. A copy of the original letter was kindly provided by the Princeton University Library with the permission of the Hebrew University. Kramers Scientific Correspondence.

’

PART IV: SELECTED CORRESPONDENCE (MAINLY

1926-1930)

Reproduced p. Translation p . Quoted p CARL W. OSEEN Bohr to Oseen, 5 November 1928

430

189

189

432 435 436 437 437 438 44 1 444 448 449 450 45 1 329 455

32 41 42 43 43 440 443 446 194

32 41 42 43 43 53 193, 309 194, 309 194 326 326 321 329 329

WOLFGANG PAUL1 Pauli to Bohr, Pauli to Bohr, Bohr to Pauli, Pauli to Bohr, Bohr to Pauli, Pauli to Bohr, Bohr to Pauli, Pauli to Bohr, Bohr to Pauli, Bohr to Pauli, Pauli to Bohr, Bohr to Pauli, Bohr to Pauli, Pauli to Bohr,

17 October 1927 13 January 1928 15 January 1928 10 March 1928 13 March 1928 16 June 1928 1 July 1929 17 July 1929 31 July 1929 15 January 1947 28 January 1947 16 May 1947 20 May 1947 29 May 1947

-

MAX PLANCK Planck to Bohr, 14 July 1929

456

192

457

8

ERNEST RUTHERFORD Bohr to Rutherford, 27 January 1926

’

ERWIN SCHRODINGER Schrodinger to Bohr, 23 Bohr to Schrodinger, 2 Schrodinger to Bohr, 5 Bohr to Schrodinger, 23

October 1926 December 1926 May 1928 May 1928

459 462 463 464

-

46 48

12 14 46 48

Draft. In English. We are grateful to the Cambridge University Library for providing us with a copy of this letter.

PART IV: SELECTED CORRESPONDENCE (MAINLY

1926-1930)

Reproduced p. Translation p.

Quoted p.

OTTO STERN Stern to Bohr, 9 August 1947 (extract) Stern to Bohr, 31 October 1947 Preparations for letter, Bohr to Stern, 19 November 1947 a) Notes, 1/11 October 1947 b) Draft of letter, 1 November 1947 Bohr to Stern. 19 November 1947

’

461 468 469 469 470 472

Notes and draft in Aage Bohr’s handwriting. These notes have not yet been microfilmed. ’We are grateful to David H. Templeton for providing us with a copy of this letter.

PART IV: SELECTEDCORRESPONDENCE (MAINLY 1926-1930)

CHARLES G. DARWIN 15 November 1930 [Carbon copy with handwritten postscript] See p. [3101.

BOHR TO DARWIN,

BOHR TO DARWIN,

[Carbon copy] See p. [312]. BOHR TO DARWIN,

[Carbon copy] See p. [315].

18 November 1930

21 December 1930

PAUL A.M. DIRAC BOHR TO DIRAC,

[Carbon copy] See p. [44].

24 March 1928

PAUL EHRENFEST 3 November 1927 [Carbon copy with handwritten formulae and addition]

EHRENFEST TO GOUDSMIT, UHLENBECK AND DIEKE,

Leiden, 3 November 1927. Lieber Goudsmit und Uhlenbeck, lieber Dieke!

...

Bruessel-Solvay war fein! Lorentz, Plank, Einstein, Bohr, Heisenberg, Kramers, Pauli, Dirac, Fowler, Brillouin, Bragg, Compton, Langmuir, Schroedinger, De Broglie, Curie, Wilson, Richardson, Knudsen, Debye und ich. Alle ganz ueberragend BOHR. Erst ganz und gar nicht begriffen (Born war auch) dann Stueck fuer Stueck alle ueberwindend. Natuerlich wieder arg Bohrische Beschwoerungsterminologie. Unmoeglich durch andere Resummierbar (Armer Lorentz als Dolmetscher zwischen den einander absolut nicht begreifenden Englaendern und Franzosen. Bohr resummierend. Und Bohr mit Hoeflicher Verzweiflung reagierend) (Jede Nacht um 1 Uhr kam Bohr zu mir noch aufs Zimmer um bis Drei mir nur noch EIN EINZIGES WORT zu sagen.) Herrlich war es fuer mich den Zweigespraechen

PART IV: SELECTED CORRESPOKDENCE (MAIXLY

1926-1930)

zwischen Bohr und Einstein beizuwohnen. Schachspielartig. Einstein immer neue Beispiele. Gewissermassen Perpetuummobile zweiter Art um die UNGENAUIGKEITSRELATION zu durchbrechen. Bohr stets aus einer dunklen Wolke von Philosophischen Rauchgewoelkes die Werkzeuge heraussuchend um Beispiel nach Beispiel zu zerbrechen. Einstein wie die Teuferln in der Box: Jeden morgen wieder frisch herausspringend. Oh das war koestlich. Aber ich bin fast rueckhaltlos pro Bohr Contra Einstein. Er verhaelt sich nun exact gegen Bohr wie die vertheidiger der absoluten Gleichzeitigkeit sich gegen ihn verhielten. In einer der aller naechsten Ablieferung Naturwissenschaften findet ihr Bohr Aufsatz mit Hauptgedanken. Er hat unter Verbesserung der durchlaufenden Fehler von Heisenberg die Ungenauigkeitsrelation nach vorne geschoben aber ihr in wunderbar einfacher Weise eine ganz wunderbare Universalitaet gegeben. Etwa folgendermaassen: Denkt erst allein an LICHTPROBleme. Dann zunaechst rein aus WELLEN KINEMATIK, die folgende Unschaerfe (Zum Beispiel) A t - Av 1. Je kurzer zeitliche Dauer eines Wellensignales desto unschaerfer seine Frequenz definiert (analog Wellenzahl und reziproke Wellenlaenge8) Daraus weiter wegen Planckh Einstein Relationen E = hv, p = - (Moment) die “Reziproken UngenauigkeitsA relationen’ ’

-

Allgemein tritt also ZUNAECHST IM GEBIET DES LICHTES die Reziproke Ungenauigkeit der Raumzeitdaten gegen ueber den Dynamischen Daten hervor

x y z t contra p 4 r

E

(Im exponenten der Wellenfunction treten sie gerade in der Verbindung 277i h

-( X l P l + x2P2 + X 3 P 3 + X 4 P 4 )

auf .) Soweit also fuer LICHT. Aber nun zeigen solche Effecte wie besonders der Comptoneffect, dass bei Wechselwirkung von Licht und beweglicher Materie DIE ERHALTUNGSSAETZE fuer den Impuls-Energievector gelten. ALSO folgt fuer jede solche Wechselwirkung, dass sich dank der Erhaltungssaetze Cf. the Introduction to Part I . ref. 45.

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1926-1930)

(! ! ! ! ! ! ! ! ! ! !) die obigen reziproken Ungenauigkeitsrelationen vom Licht auf die Materie uebertragen (! ! ! ! ! ! ! BRAVO BOHR ! ! ! ! ! !). Darueber muesste man ganz verzweifelt sein (sieh j a den verzweiflungsversuch von Slater Kramers Bohr) wenn nicht gerade De Broglie-Schroedinger, rnit der Wellen und Born-Heisenberg-Dirac rnit der nicht permutatieven Matrizzenrechnung gerade auch von der Materieseite mit “Unbestimmtheiten entgegenkaemen” , Und zwar nicht etwa rnit einer Unbestimmtheit von anderer Breite als die Optik sonder Wonder boven Wonder wieder mit der Breite h. Also Bohr: Geradezu unverdient prachtvolle Harmonie! !! ! Und nun kann man rnit vollem Vertrauen rnit Huelfe der Erhaltungssaetze die Ungenauigkeitsrelation in ganz beliebige Winkel der Physik sich fortpflanzen lassen. Zum Beispiel vor allem von ganz kleinen Koerpern (Electronen) auf beliebig grosse. Zum Beispiel auf ein ganzes Mikroskop! Betrachte ja einfach den Zusammenstoss zwischen einem Electron und dem Mond. Die Erhaltungssaetze sorgen dafuer dass die Unbestimmtheit der Dynamischen Groessen vom Electron auf den Mond uebergeht. Man uebersieht so leicht dass auch fuer grosse Koerper die h Unschaerfe von x c o n t r a p gilt. Weil man ja x und GESCHWINDIGKEIT so scharf zugleich bestimmen kann. Aber die Ungenauigkeit der Geschwindigkeit muss ja rnit der grossen Masse multipliziert werden um die Unbestimmtheit des Moments zu liefern. Bohr hat noch sehr huebsche Dinge in Privatdiscussionen mit Einstein entwickelt. Z.B. dieses: Grosse massige feste Bezugssysteme rnit unverstoerbaren Uhren sind besonders geschickt x y z t festzulegen. Aber zugleich unfaehig Impuls oder Energieuebertragungen anzuzeigen. Das ist die Weise wie in der klassischen Mechanik die reziproke Ungenauigkeitsrelation sich (schwer bemerkbar aber voellig deutlich aeussert). Indem man einen sich kraftfreien bewegenden kleinen Koerper jede Stunde einmal recht schwach beleuchtet, kann man recht schoen jede Stunde seine Lage bestimmen und die dazwischen liegende Geschwindigkeit und Moment rnit enormer Genauigkeit berechnen. Dadurch SCHEINT die Ungenauigkeitsrelation verletzt. Das ist aber nur ein Missverstaendnis. Man hat hier nur das Moment fuer Zwischenzeit BERECHNET aber nicht auch gemessen. Ferner bemerkt man auch, dass diese Ortsmessungen um 0 1 2 3 Uhr NICHT genau sondern mit Comptonstossungenauigkeit das Moment VOR 0 Uhr und NACH 3 Uhr zu berechnen gestatten. Ueberhaupt sollte man sich gegenueber “Gedankliche Verfolgung des Partikels zwischen den Beobachtungsmomenten” ebenso ablehnend verhalten wie gegen “Verfolgung eines Lichtcorpuscels zwischen Emission und Absorption durch das Wellenfeld hindurch” (ich hoffe mich rnit dieser Formulierung nicht gegen Bohrs Auffassungen zu Versuendigen). In Naturwissenschaftenartikel werdet ihr sehen wie Bohr auf “Complementaere Beschreibung” aller Erfahrung herumreitet (einerseits die mathematisch

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926- I 930)

eindeutige MEHRDIMENSIONALE oder Matrizzige Rechenmaschine ueber den sorgfaeltig abgechlossen bleibenden Bauch eines isolierten Systems (definiert, eindeutig scharf aber jenseits aller Beobachtung und aller x y z t Beschreibung). Anderseits das furchtbar rohe (naemlich mindestens h starke Eingreifen in diese Idylle bei jeder Beobachtung) “Schaffung des Partikels in x y z t” aber mit Ungenauigkeitsrelation. Bohr sagt: Wir verfuegen vorlaeufig nur ueber diejenigen Worte und Begriffe die uns eine solche Complementaere Beschreibungsweise liefern. Aber schon sehen wir wenigstens dass die beruehmten INNEREN WIDERSPRUECHE der Quantentheorie nur dadurch entstehen, dass wir mit dieser noch nicht genuegend revidierten Sprache operieren (ich weiss sicher dass Bohr ueber diese letzte Formulierung von mir GANZ VERZWEIFELT sein wuerde). Nun liest es eben selber!

...

Euer Euch herzlich liebender P . Ehrenfest

... Translation, see p. [37]. ALBERT EINSTEIN BOHR TO EINSTEIN,

[Typewritten]

13 April 1927

UNIVERSITETETS INSTITUT FOR

BLEGDAMSVEJ DEN

15,

KQBENHAVN 0.

13. April 1927.

TEORETISK FYSIK

Lieber Einstein, Vor seiner Ferienreise zu den bayrischen Gebirgen hat Heisenberg mich gebeten, Ihnen ein Exemplar der von ihm erwarteten Korrektur einer neuen Abhandlung in der Zeitschrift fur Physik zu schicken, da er hoffte, dass sie Sie interessieren wurde. Diese Abhandlung, die ich hiermit schicke, bezeichnet wohl einen ausserst bedeutungsvollen Beitrag zu der Diskussion der allgemeinen Probleme der Quantentheorie. Da der Inhalt in enger Beziehung steht zu den Fragen, die ich die grosse Freude gehabt habe, einige Male mit Ihnen zu diskutieren, zuletzt wahrend der unvergesslichen Tagen in Leiden bei der Lorentzfeier,

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mochte ich gerne die Gelegenheit benutzen, einige Bemerkungen mitzuschicken, die in Beruhrung stehen mit dem Problem, das Sie neuerdings in den Sitzungsberichten der Berliner Akademie erortert haben9. Seit langem ist es ja erkannt, wie innig die Schwierigkeiten der Quantentheorie rnit den Begriffen oder vielmehr rnit den Worten verkniipft sind, die bei der gewohnlichen Naturbeschreibung benutzt werden, und die alle in den klassischen Theorien ihren Ursprung haben. Diese Begriffe geben uns ja nur die Wahl zwischen Charybdis und Scylla je nachdem wir unsere Aufmerksamkeit auf die kontinuierliche oder diskontinuierliche Seite der Beschreibung richten. Gleichzeitig fiihlen wir doch, dass die durch unsere eigenen Gewohnheiten bedingten Hoffnungen uns hier in Versuchung fiihren, da es ja bis jetzt immer moglich gewesen [ist] uns zwischen den Realitaten schwimmend zu halten, so lange wir bereit sind, jeden gewohnten Wunsch als Opfer zu bringen. Eben dieser Umstand, dass die Begrenzung unserer Begriffe so genau rnit der Begrenzung unseres Beobachtungsvermogens zusammenfallt, erlaubt, wie Heisenberg betont, Widerspriiche zu vermeiden. In Verbindung rnit der Lichtquantenfrage ist es dann wesentlich, die wohlbekannten Paradoxe mit der physikalischen Begrenzung des Begriffs eines monochromatischen ebenen Wellenzugs in Verbindung zu bringen. Rein geometrisch gehort ja zu der Beschreibung eines Wellenzuges eine gewisse Unbestimmtheit der Wellenlange, die eine Beschreibung der endlichen Ausdehnung in der Fortpflanzungsrichtung erlaubt, ebenso wie eine Ungenauigkeit in der Parallellitat der Strahlen eine Begrenzung des Querschnitts des Wellenzugs bedingt; alles nach den wohlbekannten Gesetzen der optischen Zeitbestimmung und der Abbildung durch optische Instrumente. Wird die Unsicherheit der Schwingungsfrequenz mit Av bezeichnet, so ist ja die Zeit, welche die Wellen brauchen, um eine bestimmte Stelle zu passieren, mindestens von 1

der Grossenordnung A t = -. Bezeichnet ferner AA die Unsicherheit der

Av

A2 Wellenlange, so ist die Grossenordnung der Minimallange des Zuges A x = - , AA

A

und die der Minimalbreite A y = - , wo &

E

ein Winkel ist, der die Divergenz der

Lichtstrahlen angibt. Diese Unsicherheit in der geometrischen Beschreibung der Wellen und folglich in der Moglichkeit der Beobachtung der Lichtquanten steht also in einer eigentiimlichen umgekehrten Relation zu der Genauigkeit, rnit h welcher die Energie E = h v und der Impuls I = - der Quanten definiert sind.

A

’Cf. the Introduction to Part I , refs. 16 and 17.

PART IV: SELECTED CORRESPONDENCE (MAINLY

so haben wir A E A t = h A v .

-he. A

A &

-=h,

1 Av

-= h

hAA und AI, AX=-A2

1926-1930)

A2 =hunddI,dy= .A2

alles im Einklang mit der allgemeinen Relation der gleichzeitigen

Unsicherheit konjugierter Variablen, die nach Heisenberg eine direkte Konsequenz der mathemathischen Gesetze der Quantenmechanik bildet. Durch die neue Formulierung ist die Moglichkeit gegeben, die Forderung der Erhaltung der Energie mit den Konsequenzen der Wellentheorie des Lichts im Einklang zu bringen, indem nach dem Charakter der Beschreibung die verschiedenen Seiten des Problems nie gleichzeitig zum Vorschein kommen. In dieser Verbindung glaube ich, dass auch das von Ihnen in der Berliner Akademie diskutierte Paradoxon der spektralen Zerlegung des von einem bewegten Atom ausgesandten und durch einen Spalt senkrecht zur Bewegungsrichtung beobachteten Lichts umgangen werden kann. Betrachten wir zuerst das Problem vom Gesichtspunkt der Wellentheorie, so finden wir, dass die Unbestimmtheit der Frequenz, die von der Begrenzung der Beobachtungszeit herruhrt, von der V

Grossenordnung A v = - ist, wo v die Geschwindigkeit des Atoms und a die a

Breite des Spalts bedeuten. Damit steht im Einklang, dass die Lichtbeugung durch den Spalt dazu Anlass gibt, dass Licht, das vom bewegten Atom in einen

A

gewissen endlichen Richtungsbereich - ausgesandt wird, in der zu der Bewegung a

senkrechten Richtung zur Beobachtung gelangt; man findet ja fur den durch den v A v Dopplereffekt bedingten Frequenzbereich wieder A v = v - * - - -. Betrachten c a a wir andererseits die Energiebilanz, so finden wir, dass die Moglichkeit, auf Grund des verbreiterten Frequenzbereiches unter Umstanden ein etwas grosseres oder kleineres Lichtquant durch Photoeffekt nachzuweisen, mit dem Umstand zusammenhangt, dass Bewegungsenergie vom Atom fortgenommen oder zugefuhrt werden kann durch Strahlungsruckstoss in einer Richtung, die von der senkrechten Beobachtungsrichtung abweicht. Dass nicht nur eine statistische, sondern eine individuelle Energiebilanz beobachtet werden kann, hangt damit zusammen, dass, wie Sie in Ihrer Fussnote andeuten, eine etwaige “Lichtquantenbeschreibung” nie explizite den geometrischen Verhaltnissen des “Strahlungsganges” gerecht werden kann. In uberaus geistreicher Weise zeigt Heisenberg wie seine Unsicherheitsrelation verwertet werden kann nicht nur in dem tatsachlichen Ausbau der Quantentheorie sondern auch fur die Beurteilung von deren anschaulichen Inhalt. Insofern als diese Relation eine direkte Konsequenz des quantenmechanischen Formalismus ist, bildet das ganze ein sehr geschlossenes System, wenigstens wenn man sich auf mechanische Erscheinungen beschrankt. Bei einem so padagogisch

-

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gefarbten Begriff wie die Anschaulichkeit kommt es mir jedoch lehrreich vor, sich immer daran zu erinnern, wie unentbehrlich bei dem jetzigen Stand der Wissenschaft die Begriffe der kontinuierlichen Feldtheorie sind. Wenn wir nur von Partikeln und Quantensprungen sprechen, ist eine einfache Einfuhrung in die Theorie, die auf einem Hinweis auf die Begrenzung der Beobachtungsmoglichkeiten basiert ist, schwer zu finden, denn die erwahnte Unsicherheit ist ja nicht allein an das Vorhandensein von Diskontinuitaten gebunden, sondern eben an die Unmoglichkeit ihrer genauen Beschreibung, denjenigen Eigenschaften der materiellen Teilchen und des Lichts zu Folge, die in der Wellentheorie zum Ausdruck kommen. Die Reprasentation eines Elektrons durch eine Gruppe von de Brogliewellen ist ja ganz analog zu der eines Lichtquants durch eine Gruppe von elektromagnetischen Wellen. So gelten alle obigen Relationen auch in diesem Fall. Nur folgt unmittelbar aus der der Unsicherheit im Elektronenimpuls entsprechenden Unsicherheit in der Gruppengeschwindigkeit, dass die Gruppe mit der Zeit sich auch in der Fortpflanzungsrichtung verbreitert, alles genau so wie es Heisenberg auf Grund der Quantenmechanik in Anschluss an Diracs Matrixtransformationstheorie ausfuhrt. Ich will Sie aber nicht langer qualen. Uber die ganze wundervolle Entwicklung lasst sich ja unendlich reden. Wie schon ware es, wenn ich wieder ein Ma1 mundlich uber alle diesen Sachen mit Ihnen sprechen konnte. Wie ich verstanden habe, ist es die Absicht von Heisenberg, Sie auf seiner Ruckreise in Berlin zu treffen zu versuchen. Schon lange habe ich die Absicht gehabt, meine Gedanken uber die allgemeinen Fragen in einem kleinen Aufsatz zu klaren zu versuchen, aber die Entwicklung geht so sturmisch, dass alles wieder alltaglich wird. Doch hoffe ich bald einen solchen Aufsatz fertigzustellen. Mit den freundlichsten Grussen Ihr N. Bohr Translation, see p. [21]. RALPH H. FOWLER BOHR TO FOWLER,

[Carbon copy]

14 April 1926

[K~benhavn,]14th of April [19]26. Dear Fowler, Many thanks for your kind letter. It was a great pleasure to my wife and myself to hear that you and Eileen are thinking of coming to Denmark this summer.

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Your visit shall be most welcome at any time. In June, July and August, however, my wife and the children will be staying in the cottage in Tisvilde. You know yourself the very modest arrangement, but if you and Eileen will come and stay with us there we shall be delighted. It would of course be great fun for us to have you there some time for ourselves. I am thinking, however, that you perhaps would find your stay in Denmark more profitable if you could come at a time where the work in the institute was going on at a more normal rate. In June most people will have left, but I suppose that Heisenberg will still be in Copenhagen and possibly Kramers also. If you could come for instance in September you would certainly be more sure to find people here, and at that time it should also be a great pleasure for my wife and myself to have you and Eileen staying with us in our home in Copenhagen. But please come just at the time which suits your plans best. I was also very glad indeed for what you wrote about Dirac. He shall certainly be welcome to work here next term. We are all full of admiration for his work. Dirac’s coming will also be a great pleasure to Heisenberg, who will arrive here to the first of May in order to overtake Kramers’ post. Already last autumn when I spoke with him about his taking up of this post he suggested a visit of Dirac as most desirable. However, I did not venture to make a direct proposal myself to this extension. I am looking forward very much to learn more closely about the recent work of Dirac on the Compton scattering. It looks indeed that we are on a high road to advance now. At present we are here having a visit of Pauli, who has just in these days succeeded in proving that the beautiful method of determining energy values for stationary states proposed by Schrodinger in the last issue of the Annalen der Physik will always lead to results identical with those derived by the methods of Heisenberg, Born and Dirac. It looks even that Schrodinger’s method may offer a simplification in the calculations especially as regards the determination of the transition probabilities. Also Klein is here at present and has brought with him some very original and interesting ideas as regards the possibility of obtaining a greater unity in the principles of the quantum theory by using a system of representation with five dimensions. In contrast to this kind of work Thomas is in these days occupied with some old-fashioned orbital calculations which, thanks to the spinning electron, may not be quite so useless as one might be inclined to think some time ago. Thomas’ stay here has certainly been most helpful for the work in the institute, and we shall be sorry to miss him when he leaves at the end of May. With best wishes from us all and kindest regards to you and Eileen and the Rutherfords from my wife and myself, Yours, [Niels Bolir]

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BOHR TO FOWLER,

[Carbon copy]

1926-1930)

26 October 1926

[K~benhavn,]October 26th [19]26. Dear Fowler, Thanks for your kind letter. It was a great pleasure for us all to have you here again and I enjoyed very much our discussions. I am sorry not to have written earlier and hope that you have not missed your paper too much in these weeks, but ever since you left we have had a very busy time with the visits of Schrodinger and Mme Curie, and first in the last days I have been able to read your paper carefully and discuss it with Dirac, Heisenberg and Klein. We have all taken great interest in it and think that it will be very welcomed by everybody. As regards your asking for criticism the suggestion we can offer concerns only minor points and deals not with the real content of your paper but with the discussion going on as regards the present state of the quantum theory. Thus Heisenberg points out that regarding the problem of the statistics of a radiation field it is hardly justified to lay too much stress on the distinction between the equilibrium distribution and the fluctuations, since a rigorous treatment on the quantum mechanics, as shown by Born, Heisenberg and Jordan, gives automatically the fluctuations which Einstein deduced by a thermodynamical consideration from Planck’s law. Another point concerns the analogy and difference between a radiation field and a monatomic gas. Here you will have seen yourself already that Dirac in his last paper has also treated the gas from the Schrodinger point of view. It would appear, however, that Dirac’s reluctance in preferring a definite boundary condition is unnecessary. In fact, from adiabatic consideration it would appear that the only rational condition is that used in your paper. This question has been closely studied by Hund in connection with the problem of combination of atoms into molecules. He has here obtained most beautiful results and shown how all difficulties hitherto involved in adiabatic arguments disappear completely from the point of view of the wave mechanics. This may also be of importance for the problem touched upon in your last paragraph of a rational connection between the characteristics of the wave equation and the “probability” of the presence of an atom in a given part of space. We had great pleasure of the visit of Schrodinger. The discussions centered themselves gradually on the problem of the physical reality of the postulates of the atomic theory. We all agreed that a continuity theory in the form indicated in his last paper at a number of points leads to expectations fundamentally different from those of the usual discontinuity theory. Schrodinger himself continued in his hope that the idea of stationary states and transitions was altogether avoidable, but I think that we succeeded at least in convincing him that for the

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fulfilment of this hope he must be prepared to pay a cost, as regards reformation of fundamental concepts, formidable in comparison with that hitherto contemplated by the supporters of the idea of a continuity theory of atomic phenomena. I understood that Schrodinger had been working under the impression that the essential characteristics of the matrix mechanics was the final recognition of the impossibility of ascribing a physical reality to a single stationary state, but I think that this is a confounding of the means and aims of Heisenberg’s theory. Just in the wave mechanics we possess now the means of picturing a single stationary state which suits all purposes consistent with the postulates of the quantum theory. In fact, this is the very reason for the advantage which the wave-mechanics in certain respects exhibits when compared with the matrix method which in other respects is so wonderfully suited to bring out the true correspondence between the quantum theory and the classical ideas. After the discussions with Schrodinger it is very much on my mind to complete a paper dealing with the general principles of the quantum theory such as I spoke of already during your visit. First in these days, however, I have got a chance to settle down to the work. All in the laboratory wishes to be kindly remembered to you. With kindest regards and best wishes to Eileen and yourself and the Rutherfords from my wife and yours, [Niels Bohr] WERNER HEISENBERG HEISENBERG, [December 19281 [Draft in Margrethe Bohr’s handwriting]

BOHR TO

[Hornbak, December 19281 K a r e Heisenberg! Allerede for lange siden burde jeg have svaret paa Dit rare Brev; men jeg vil da alligevel ikke lade Aaret gaa ud uden at takke Dig for den store Glade, Du gjorde 0s alle med Dit Beserg. Sjalden har jeg f0lt mig i inderligere Harmoni med noget Menneske, og jeg glader mig endnu, naar jeg tanker tilbage paa vore Spadsereture og Diskussioner og ikke mindst paa den Aften hvor vi sammen herrte Herffdings smukke Foredrag om Sokrates. Sammen med min Tante og Hans og Christian har min Kone og jeg tilbragt Dagene mellem Jul og Nytaar herude i Hornbzek. Det har varet en dejlig Hviletid og jeg har haft rig Lejlighed ti1 at tzenke over, hvad Aaret har bragt og drermmer om, hvad det nye vil bringe. Jeg sidder og tanker paa, hvor meget vi

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allesammen har l a r t siden dengang vi under Dit farste Besag heroppe sammen vandrede gennem Hornbak paa Vej ti1 Tisvilde og med Afbrydelser med Stenkastning og Springning over Grafter fantaserede over Fysikkens og Livets Problemer. Hvordan er det gaaet Dig selv med Arbejdet i Efteraaret. Fru Maar fortalte forleden, at Du og Pauli atter mente at ajne lysere Udsigter. Ogsaa Klein har arbejdet meget med Relativitetsproblemet og har nogle Tanker paa hvilke han haaber at bygge en mere tilfredsstillende Formulering. Jeg er spandt paa at herre, hvad han har faaet ud deraf, naar han kommer tilbage fra sin Juleferie. Jeg selv har beskaftiget mig en Del med Spargsmaalet om Elektro[n- ?]magnetismens Uiagttagelighed og Pauliprincippet; men det er endnu meget uklart. Trods Paulis Advarsler er jeg ogsaa stadig forberedt paa en yderligere Begransning af Energibegrebets Anvendelighed. I den sidste Tid har Gamow beskaftiget sig indgaaende med de kontinuerte P-Straale-Spektra; men a1 Sergen efter andre Udveje har hidtil bestyrket min Overbevisning om, at Vanskelighederne ligger meget dybt. Hvordan er det gaaet Jer med Supraledningsevnen? Blochs smukke Arbejde, som Du saa venligt sendte mig og som jeg havde megen Glade af, belarte mig jo om, at den af mig antydede Udvej ikke var farbar. Iavrigt har jeg anvendt min meste Tid i Efteraaret med at filosofere over Kvanteteoriens Grundlag og haaber nu i Juleferien at udarbejde en Fremstilling af mine Tanker, der selv om de ikke bringer noget egentlig nyt, maaske dog kan bidrage ti1 Begrebernes Afklaring. I Forbindelse hermed har jeg t a n k t over mange almindelige Problemer ogsaa udenfor Fysikken, som jeg haaber, at vi inden altfor lange skal faa Lejlighed ti1 at tale naermere sammen om. For denne Gang vil jeg slutte med mine hjerteligste 0nsker om et gladeligt Nytaar for Dig selv og Dine Foraldre og mange venlige Hilsner fra Hjem ti1 Hjem, Din hengivne N. Bohr.

Translation, see p. [24] HENDRIK A. KRAMERS BOHR TO KRAMERS,

[Carbon copy]

30 November 1929

[Kabenhavn,] 30. November [ 19129. K a r e Kramers, Forlange siden skulde jeg have svaret paa Dit rare Brev, som gladede rnig meget; jeg har imidlertid haft saa travlt med min Afhandling ti1 Universitetets

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Aarsskrift og andre Forpligtelser, at jeg aldrig kom dertil. Som Du vil se af det Eksemplar af Aarsskriftet, som jeg samtidig sender, fik jeg ikke Tid ti1 at udarbejde en ny, sammenhangende Fremstilling af Atomteoriens Problemer, som jeg egentlig havde t a n k t , men har benyttet en dansk Oversattelse af mine sidste almindelige Artikler. Under Arbejdet dermed har jeg ofte med Taknemmelighed mindedes a1 Din H j a l p i de mange Aar og ikke mindst vort falles Slid med den ferrste af Artiklerne, der j o nasten blev en Slags Afslutning paa vort direkte Samarbejde. I Indledningen har jeg forsragt at give Udtryk for den Stemning, hvoraf jeg i 0jeblikket beherskes, og som trods a1 Forskel i Tid og Vilkaar ikke er saa rneget forskellig fra den Stemning, der betog 0s begge i vort Samarbejdes allerferrste Aar. Jeg har t a n k t meget over de alrnindelige Sperrgsmaal, siden vi sidst saas, og haaber at vi snart igen faar Lejlighed ti1 at tale rigtig sammen om alle de Sperrgsmaal, der ligger 0s begge paa Hjerte. Du skrev, at Du maaske kunde komme ti1 Danmark i Juleferien. Hvad Tid tankte Du at komme, og hvor lang er Din Ferie? Sagen er, at Klein og vist ogsaa jeg er borte fra Kerbenhavn i de frarste Uger efter Jul. Klein rejser med sin Familie ti1 Aarhus, og jeg har t a n k t paa at rejse ti1 Norge, da jeg tranger meget ti1 en rigtig Ferie. Nu ved jeg slet ikke, hvad Dine og Din Families Planer er, men hvis Du allerede kunde komme ti1 Kerbenhavn omkring Midten af December Maaned, vilde det v a r e storartet. Casimir rejser ti1 Holland ti1 den Tid, og hvis det passer Dig, kan Du bo paa Instituttet i de Varelser, som han for Tiden bebor. Vi har haft stor Fornerjelse af Casimir. Netop i disse Dage er han i F a r d med at afslutte et Arbejde over det kvanteteoretiske Kreiselproblem, som jeg tror vil more Dig. Det er en Fortsattelse af et Arbejde af Klein, der om nogle Dage udkommer i Zeitschrift fur Physik. Efter Kleins Ide kan Problemet behandles umiddelbart ud fra Kvantemekanikkens Korrespondensgrundlag, og hans og Casimirs Undersragelser har givet 0s alle rnegen Anledning ti1 at t a n k e over Kvantemekanikkens Vasen og at beskaftige 0s med Dine mangesidige og dybtgaaende Undersragelser over Kreiselproblemet. Vi har lige i disse Dage haft et forfardeligt rart Beserg af Ehrenfest med mange ivrige Diskussioner. Iravrigt har jeg lige herrt fra Dirac om nogle meget interessante og dristige Betragtninger, hvormed han haaber at undgaa Vanskelighederne i den relativistiske Kvantemekanik. Han mener at kunne redde Energisatningen, men jeg er endnu ikke helt sikker paa, at dette lader sig grare. Om dette og om andet haaber jeg imidlertid, at vi snart kan snakke sammen. Jeg vil jo v a r e rneget glad for snarest muligt at herre om Dine Planer og Rejsemuligheder. Med mange venlige Hilsener fra 0s alle ti1 Dig og Din hele Familie, Din hengivne [Niels Bohr]

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Translation [Copenhagen,] November 30, 1929 Dear Kramers, Long ago I should have answered your nice letter that gave me great pleasure. However, I have been so busy with my article for the University Year Book and other obligations that I never got around to it. As you will see from the copy of the Year Book, which I am forwarding simultaneously, I have not been able to find time to work out a new, coherent exposition of the problems of atomic theory, as I had actually intended, but I have used a Danish translation of my most recent general articles. During this work I have often recalled with gratitude all your help through the many years and not least our toil together with the first of the articles, which nearly came to mark a sort of termination of our direct collaboration. In the introduction I have tried to give expression to the mood that presently holds me under its sway, and which, notwithstanding all difference in time and conditions, is not so very different from the mood that stirred us both during the very first years of our collaboration. You wrote that perhaps you could come to Denmark during the Christmas holiday. When were you thinking of coming, and how long is your holiday? The point is that Klein and probably I myself are away from Copenhagen during the first weeks after Christmas. Klein is going with his family to Aarhus and I have thought of going to Norway as I badly need a real holiday. Now I do not know at all what your and your family’s plans are, but if you could already be in Copenhagen around the middle of December it would be splendid. Casimir is going to Holland at that time, so, if it suits you, you may live at the Institute in the rooms which he occupies at present. Casimir has given us great pleasure. Any day now he is about to finish a paper on the quantum theoretical problem of the top, which I think you will enjoy. It is a continuation of a paper by Klein, which is going to appear in a few days in Zeitschrift fur Physik. According to Klein’s idea, the problem can be treated directly on the correspondence basis of quantum mechanics, and his and Casimir’s investigations have given all of us occasion to think about the nature of quantum mechanics and to occupy ourselves with your many-sided and fundamental investigations on the problem of the top. In these days we have had an extremely nice visit from Ehrenfest with many lively discussions. By the way, I have just heard from Dirac about some very interesting and bold considerations, through which he hopes to avoid the difficulties in relativistic quantum mechanics. He believes that he can save energy conservation, but I am not yet quite sure that this is possible. I hope, however, that we may soon talk together about this and other things. I should be very hap-

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py to hear from you as soon as possible about your plans and travel possibilities. With many kind regards from us all to you and all your family, Yours sincerely, [Niels Bohr] BOHR TO KRAMERS,

[Typewritten]

7 December 1929

UNIVERSITETETS INSTITUT FOR

15, K0BENHAVN 7 . December 1929.

BLEGDAMSVEJ

DEN

0.

TEORETISK FYSIK

Kzere Kramers. Tak for Dit rare og indholdsrige Brev. Vi er alle kede af, at Du ikke kan komme fOr Jul, men jeg haaber bestemt, at vi paa en eller anden Maade mart skal ses. Siden Du sidst var i K~benhavn,har jeg taenkt meget over de almindelige Sp~rgsmaal,som Du b e r ~ r e ri Dit Brev, og jeg skal om faa Dage sende Dig en Korrektur af det Foredrag, jeg holdt ved Naturforskerm~deti Sommer, og hvor jeg er gaaet lidt narmere ind paa Aarsagsproblemet. For en Ugestid siden holdt jeg i ~ v r i g tet Foredrag derom i en kabenhavnsk Forening, der kalder sig Selskabet for Filosofi og Psykologi, og jeg larte ikke saa lidt af den paaf~lgende Diskussion. Navnlig ved jeg nu bedre, hvilke Punkter der falder ikke-Fysikere for Brystet, og jeg tror ogsaa, at jeg netop derfor ved den Lejlighed fandt bedre Ord ti1 at ware paa Indvendinger end tidligere. Siden jeg skrev sidst, har vi her haft mange Diskussioner om Diracs nye Tanker, og jeg sender indlagt en Kopi af hans Brev, samt en Kopi af et Svar, jeg skrev ti1 ham for nogle Dage siden. Jeg sender Brevet, fordi Du s p ~ r g e rom narmere Oplysninger, men jeg ved ikke, hvorvidt han onsker det udbredt i s t ~ r r e Kredse, da Sagen jo kun befinder sig paa Begyndelsesstadiet. Vi er jo alle forberedt paa, at han atter denne Gang kan have gjort et betydningsfuldt, nyt Fund; men som Du vil se af mit Svar, finder jeg det denne Gang vanskeligt at falge ham, i det mindste f0r man i nzermere Enkeltheder forstaar, hvordan han kan komme ud over de ojensynlige Vanskeligheder, som en uendelig Elektrontzethed vil berede. Du vil ogsaa se, at jeg ikke har sluppet min gamle Overbevisning, at det er Lidenheden af Forholdet e2/hc, der skal bringe 0 s ud af Ufaret. Jeg er ogsaa mere og mere overbevist om, at Vejen vil fare 0s ud over Energibegrebets Granse. Jeg paatzenker at fremstille mine Synspunkter i en liile Note og skal v a r e glad for at hme, om Du har nogen naermere Beregning af Sandsynligheden af de Diracske Overgangsprocesser for Brintatomet. Som Du ser af mit Brev, har Klein og jeg indset, at den tidligere Overslagsregning, som

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jeg tror stammer fra Heisenberg, giver altfor store Vmdier. Jeg tror jo ikke paa Overgangenes Realitet; men jeg ansker kun at gare mig det helt klart, hvorledes vi kan se bort fra dem uden at komme i Strid med Korrespondenssynspunktet. Klein og Casimir var som jeg selv yderst interesseret i, hvad Du skrev om Dine nyeste Arbejder. I Forbindelse med deres eget Arbejde, der iavrigt er af ren metodisk Karakter, havde de ogsaa selv t z n k t lidt paa nogle af disse Spmgsmaal uden dog at vaere kommet ti1 nye Resultater. Jeg kan ikke forstaa, at Kleins lille Note endnu ikke er kommet. Casimir venter Korrektur paa sin Artikel i disse Dage og skal straks sende Dig et Eksemplar. Der er ingen af os, der ved noget bestemt at ware paa Dit Spargsmaal om Grunden ti1 den interessante Omvending af D-Dubletterne i de natriumlignende Spektre. Med mange venlige Hilsener fra 0s alle, Din hengivne Niels Bohr

Translation Copenhagen, December 7 , 1929 Dear Kramers, Thank you for your nice and detailed letter. We are all sorry that you cannot come before Christmas, but I definitely hope that we shall meet soon one way or another. Since you were last in Copenhagen, I have pondered the general questions that you touch upon in your letter, and in a few days I am going to send you the proofs of the lecture that I gave at the gathering of scientists this summer, where I discuss the causality problem a little more closely. By the way, I gave a lecture about this to a Copenhagen association, which calls itself the Society for Philosophy and Psychology, and I learned a great deal from the ensuing discussion. In particular, I now know better which points non-physicists resent, and I also believe that for this very reason I found on this occasion better words than previously to answer the objections. Since I last wrote we have had many discussions here about Dirac’s new ideas, and I include a copy of his letter as well as a copy of an answer that I sent him a couple of days ago. I am sending the letter because you ask for more detailed information, but I d o not know whether he wants it distributed in wider circles, since the matter is still only in a preliminary stage. We are all prepared for the event that he once again may have made an important new discovery. But, as you will see from my reply, I find it difficult to follow him this time, at least until one understands in more detail how he can get around the obvious difficulties to which an infinite electron density will give rise. You will also see that I have not given up my old conviction that it is the smallness of the ratio e2/hc which

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will get us out of the mess. I am also more and more convinced that the road will lead us beyond the limits of the energy concept. I plan to present my points of view in a small note, and I shall be happy to learn whether you have any more detailed evaluation of the probabilities for the Dirac transition processes in the hydrogen atom. As you can see from my letter, Klein and I have come to realize that the earlier estimate, which I believe is due to Heisenberg, yields values which are much too large. I do not of course believe in the reality of the transitions, but I only wish to make it quite clear to myself, how we can disregard them without getting into conflict with the correspondence viewpoint. Klein and Casimir, as I myself, were very interested in what you wrote about your latest work. In connection with their own work, which by the way is of a purely methodological character, they themselves had also thought a little about these questions, but without arriving at any new results. I cannot understand why Klein’s small note has not yet appeared. Casimir is expecting the proofs of his article any day now and will immediately send you a copy. None of us can give a definite answer to your question about the reason for the interesting inversion of the D-doublets in the sodium-like spectra. With many kind regards from us all, Yours sincerely, Niels Bohr CARL W. OSEEN BOHR TO OSEEN, 5

[Carbon copy]

November 1928

[Kabenhavn,] 5. November [19]28. K a r e Ven, Du kan tro, jeg blev glad for Dit venlige Brev og for at hare, at Du interesserer Dig for de samme Spargsmaal, som optager 0s hernede for Tiden. Jeg haaber, Du har faaet Sartrykkene, som jeg sendte for nogle Dage siden. Jeg skamrner mig meget over ikke at have skrevet far og takket for dit Brev, men vi har netop i disse Dage varet meget optaget med Diskussion af de Paradokser, som er kommet ti1 Syne ved Bestmbelserne for at samarbejde Relativitetsteorien og Kvanteteorien, og jeg vilde meget gerne have kunnet fortalle Dig lidt derorn. Det maa imidlertid alligevel vente, da vi endnu ikke har kunnet klare Tankerne. I min Artikel i Naturwissenschaften drejer det sig jo om en almindelig Indstilling, der har ligget mig paa Sinde i alle de Aar, jeg har beskaftiget mig med Kvanteteorien, men som farst de sidste Aars store Udvikling, der har muliggjort en modsigelsesfri Fremstilling af Erfaringerne, har givet Midler ti1 at udtrykke.

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Som vi jo allerede har talt om for Aar tilbage, er Vanskelighederne i a1 Filosofi den Omstandighed, at vor Bevidstheds Virkeform forudsatter et Krav ti1 Indholdets Objektivitet, medens dog Tanken om Subjektet, om vort eget Jeg, er en Del af vor Bevidstheds Indhold. Det er netop saadanne Vanskeligheder, som vi har faaet saa klart et Eksempel paa i den af Kvantepostulatets Vasen fordrede Karakter af Naturbeskrivelsen. Langtfra at sarge over at vi i Atomfysikken ikke kan faa vore szedvanlige Onsker med Hensyn ti1 Naturbeskrivelsen opfyldt, tror jeg, at vi berr g k d e 0s over den nye Belaring med Hensyn ti1 de menneskelige Anskuelsesformers BegrEnsning, som Virkningskvantets Opdagelse betyder. Allersimplest indser man maaske Uforeneligheden af Rum-Tidsbilleder med Kravet om Impuls- og Energibevarelse, naar man betanker, at Grundlaget for Rum-Tidsbeskrivelsen er Benyttelsen af faste Legemer og uforstyrrelige Uhre, der efter deres Vasen netop betyder en Resignation med Hensyn ti1 Impuls- og Energiomsatningen. Skal vi kunne sige, at en Partikel ti1 given Tid har vzeret paa et givet Sted, maa vi vide, at en eller anden narmere angivet Blaxder har vzeret aabnet og lukket paa angivne Tider. Det kan nu let eftervises, at den Usikkerhed i Beskrivelsen, der hidrerrer fra Materie- eller Lysberlgernes Spredning ved Blanderen og Forstyrrelsen af Berlgesystemets spektrale Sammensatning ved Aabningen og Lukningen netop svarer ti1 vort Ukendskab ti1 henholdsvis den Bevagelsesmzengde, der ommttes mellem Individet og de faste Legemer, hvortil Blanderen er fastet, og den Energi som omsattes med den Maskine eller Uhr, som beserrger Aabningen og Lukningen. For Rum-Tidsangivelsen betaler vi derfor aabenbart med Bruddet i Impuls-Energibeskrivelsen. Det er jo ikke min Mening her at gennemgaa alle de gamle Sager. Jeg haaber, at Du undskylder min Begejstring, der ikke mindst holdes i Live ved den Utilfredshed, der fra forskellig Side stadig kommer ti1 Orde overfor Tingenes Tilstand i Atomfysiken. Selv Einstein har hidtil stillet sig noget kerligt overfor Kvanteteoriens nyeste Udvikling. I Forbindelse med en Diskussion af den rigtignok overdrevne Form af Statistik, som blev foreslaaet herfra for nogle Aar tilbage, ytrede Einstein ti1 Pauli noget saadant som, at han var forberedt paa mange Overraskelser, men den Tanke “dass der liebe Gott wurfelt” kunde han dog ikke rigtig lide. Naturligvis ferlte jeg mig starkt truffen deraf; men da jeg nu sidste Vinter traf Einstein i Bryssel, forsergte jeg som Gensvar at sige, at allerede de gamle jerdiske Profeter jo saa godt vidste, hvor vanskeligt det var at beskrive Guds V m e n ud fra vore menneskelige Begreber. Kun for saa vidt vi forlanger at henferre alt ti1 Rum-Tidsbilleder, kan der v a r e Tale om at sammenligne Naturlovene med Terningspil. Det er jo netop Erkendelsen af den af selve disse Love betingede Begramming i vore Anskuelsesformer, der tillader 0s at anvende Aarsagsbegrebet ti1 dettes yderste Granse. Ja, man kan maaske sammenfatte de sidste Aars vidunderlige Fremskridt i Kvanteteoriens Form og Ind-

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hold deri, at det har vist sig muligt at anvende alle klassiske Begreber paa modsigelsesfri Maade. Tilfredsheden med Kvanteteoriens Formulering galder jo dog, som allerede sagt i Begyndelsen af Brevet, kun saa vidt vi ser bort fra Relativitetskravet. Trods det vidunderlige Fremskridt med Hensyn ti1 Bestrabelserne for at tage tilb~rligt Hensyn ti1 dette, som vi skylder Dirac, aabenbarer der sig her hver Dag nye Vanskeligheder. Et szrlig lzrerigt Eksempel er, som vist Waller har fortalt, fornylig fremdraget af Klein, der er valdig godt inde i de symbolske Metoder, der for Tiden danner Grundlaget for Arbejdet paa disse Omraader. Det synes, som om de Vanskeligheder, vi her har med at gerre, ligger meget dybt og forlanger en endnu s t ~ r r eResignation med Hensyn ti1 Anskuelighed, end den vi allerede har maattet vanne 0s til. Dog herom ved jeg som sagt endnu ikke noget afg~rende at fortalle. Jeg lznges meget efter at se Dig igen og tale rigtig med Dig, og jeg har tankt lidt paa, dersom jeg kan faa Tid dertil, og dersom jeg har noget nyt at fortdle, engang i Vinter at komme paa et lille B e s ~ gti1 Sverige. Med mange venlige Hilsener ti1 Din Hustru og Dig selv og alle fzlles Venner i Upsala fra min Hustru og Din hengivne Ven [Niels Bohr]

Translation, see p. [189]. WOLFGANG PAUL1 PAULI TO BOHR,

[Handwritten]

17 October 1927

Institut fur Theoretische Physik an der Universitat Hamburg

Hamburg 36, den 17. Okt. 1927 Jungiusstrane 9

Viele Grune von Stern! Sehr verehrter lieber Herr Professor! Nunmehr sende ich Ihnen die Korrekturen, Ihrem Wunsche geman “mit allen kritischen Bemerkungen, die mir einfallen” , Vorher mochte ich aber bemerken, daB ich im allgemeinen sehr einverstanden bin sowohl mit der ganzen Tendenz der Note wie auch mit den meisten Einzelheiten. Besonders ist mir klar geworden, dan die statistische Deutung der theoretischen Resultate immer an der

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Stelle hereinkommt, wo man ein abgeschlossenes System in zwei Teile teilt, die man dann als beobachteten Gegenstand bezw. M e h s t r u m e n t interpretiert, und dann fragt, was man uber den einen Teil ohne Kenntnis des anderen aussagen kann. Wenn man die Resultate der Matrixformulierung der Quantentheorie so interpretiert, scheint mir durchaus die Moglichkeit vorhanden, sie unmittelbar mit der Erfahrung in Beziehung zu setzen. Deshalb scheint rnir Ihre Angabe (auf Seite 6 ) “streng genommen ist ihr innerer Zusammenhang auf Kosten jeder unmittelbaren Deutung der Rechenresultate gewonnen” nicht zutreffend oder wenigstens zu Miherstandnissen AnlaD gebend. Nun will ich der Reihe nach die paar Stellen angeben, die rnir noch verbesserungsfahig scheinen. Rein sprachliche Verbesserungsvorschlage habe ich an den Rand der Korrekturbogen geschrieben. Auf S.4 oben ist die einzige Stelle, die ich unklar finde. Dies liegt aber wohl zum Teil daran, dan sie auch sprachlich nicht ganz korrekt ist. (Wenn ich sie aber eben wieder lese, verstehe ich sie schon etwas besser). Weiter unten auf S.4, wo von der “unstetigen Verkleinerung der raumlichen Begrenzung der Felder” die Rede ist, konnte man vielleicht der Deutlichkeit halber noch etwas hinzufugen. Dies ist ja gerade ein Punkt, der bei Heisenberg nicht ganz befriedigend war; es schien dort die “Reduktion der Pakete” ein bischen mystisch. Nun ist ja aber zu betonen, dal3 solche Reduktionen zunachst nicht notig sind, wenn man alle Messungsmittel rnit zum System zahlt. Urn aber Beobachtungsresultate uberhaupt theoretisch beschreiben zu konnen, mu13 man fragen, was man uber einen Teil des ganzen Systems allein aussagen kann. Und dann sieht man der vollstandigen Losung von selbst an, dal3 die Fortlassung des Beobachtungsmittels in vielen Fallen (nicht immer naturlich) formal durch derartige unstetige Reduktionen ersetzt werden kann. Von Seite 6 (Matrizenformulierung) habe ich bereits oben gesprochen. Seite 7 (oben): Was soll in diesem Zusammenhang eigentlich “wenn moglich” bedeuten? Seite 9 (oben). Ich bin naturlich ganz einverstanden damit, da13 bei diskreten Eigenwerten betreffend die Unsicherheitsrelation eine nahere Untersuchung notig ist. Es scheint mir aber, da13 die auf x und p x bezugliche Relation

A X Ap,- h auch hier richtig bleibt, wenn man (unabhangig vom Energiewert) diejenige allgemeine Losung ins Auge fafit, die zu einer gewissen Zeit to auf ein gewisses Gebiet A x bechrankt ist und sie dann spektral zerlegt. Von der ersten Relation A t A E - h scheint mir allerdings, darj sie bei diskreten Energiewerten keinen Sinn mehr hat. Vielleicht soll man also nur von der Relation At A E - h (nicht aber von

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der Relation A x Ap,- h ) behaupten, daD “ihre Formulierung wesentlich auf der Annahme einer kontinuierlichen Folge von Eigenwerten beruht .” Seite 9 (unten). Dort steht der Satz: “Die Verwendung der Rechenresultate beruht darauf, dal3 man sich bei Vergleich rnit der Erfahrung immer auf die Betrachtung von nur drei von den raumartigen Koordinaten beschranken kann.” Dies halte ich fur nicht richtig u. ich mochte dagegen opponieren, weil mir dieser Punkt besonders am Herzen liegt. Wenn man ein System rnit mehreren Teilchen hat, so kann nach meiner Meinung die Eigenfunktion ty(ql,...,qiV) im mehrdimensionalen Konfigurationsraum durch statistische Verwertung von Beobachtungsresultaten prinzipiell empirisch ermittelt werden; und dies gilt auch von der Funktion selbst, nicht nur von Integralmittelwerten dieser Funktion uber alle Koordinaten bis auf drei. Das Quadrat des Betrages dieser Funktion bedeutet die Wahrscheinlichkeit dafur, daD gleichzeitig eines der Teilchen die Koordinaten qi’), q y ) , q:l), ein anderes die Koordinaten qi2),qr), q$2)... etc. hat. Man kann nun die Ermittelung dieser Funktion im Prinzip auf die Beobachtung freier Teilchen reduzieren, wenn man StoDversuche macht, wobei das zu untersuchende System gleichzeitig von mehreren Teilchen gestorjen wird. Man kann dann z.B. fragen, wie grolj bei parallelem Einfall aller stoflenden Teilchen die Wahrscheinlichkeit dafur ist, daD gleichzeitig eines der Teilchen mit gewisser Energie in gewisser Richtung, ein anderes wieder rnit anderer Energie in anderer Richtung U.S.W.nach dem StoD fortfliegt (was nicht unabhangige Ereignisse sind). Wohl hat man es hier wieder letzten Endes rnit freien Partikeln zu tun, aber rnit mehreren u. es scheint mir ganzlich irrefuhrend, zu sagen, daD man sich hier im Ganzen auf drei (raum- oder impulsartige) Koordinaten beschranken kann. (Dies hatte ich mir auch genau uberlegt als ich jene FuDnote (in meiner Arbeit uber Gasent [artung] u. Paramagnetismus 9a) uber die ty-Funktionen im mehrdimensionalen Raum schrieb.) Ich habe hier naturlich hauptsachlich die (wenigen) Punkte hervorgehoben, rnit denen ich nicht einverstanden war, wahrend ich fast gar nicht uber die Stellen sprach, die mir besonders gefallen haben. Dadurch konnte leicht ein falscher Eindruck entstehen. Ich bin uber Ihre Arbeit im GroDen u. Ganzen sehr froh und glaube, dal3 die oben besprochenen Stellen leicht noch verbessert werden konnen! - Besonders froh bin ich uber den SchluDsatz, wo Sie sagen, dafl die stationaren Zustande ebensosehr oder wenig real sind wie die Partikeln selber. Denn ich hoffe sogar, dal3 die Atomistik der elektrischen Ladung sich spater einmal analog auffassen lassen wird, wie die Existenz stationarer Zustande, so daD die

9a

W . Pauli, Uber Gasentartung und Paramagnetismus, Z . P h y s . 41 (1927) 601-623

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Zahl der Elementarladungen in einem gewissen Gebiet als Quantenzahl erscheinen wird. - Aber dies ist noch in weiter Ferne! Dagegen bin ich mit dem relativistichen Mehrkorperproblem in den letzten Tagen vie1 weiter gekommen u. glaube bestimmt, dal3 seiner Losung keine grundsatzlichen Schwierigkeiten mehr entgegenstehen! Wenn Sie schon Freitag in Hamburg waren, ware es schon! Dann konnte ich Ihnen auch noch helfen bei der Korrektur u. wir konnten sie eventuell von hier aus abschicken. Dies sol1 aber nur ein unverbindlicher Vorschlag sein. Jedenfalls hoffe ich bald von Ihnen zu horen! Mit den herzlichsten GruDen an Sie u. auch an Klein Ihr W. Pauli

Translation, see p. [32].

PAULI TO BOHR,

[Handwritten]

13 January 1928

Institut fur Theoretische Physik an der Universitat Hamburg

Hamburg 36, den 13. I. 1928 JungiusstraDe 9

Lieber Bohr! Vielen Dank fur Ihren lieben Brief. Wir freuen uns alle sehr auf Ihr Kommen, naturlich sind ich und die anderen Physiker (Lenz, Stern, Koch) den ganzen Monat Januar (sogar bis Ende Februar) hier in Hamburg. Die Papierscheren und Leimtopfe des Institutes werden aufs beste fur Sie vorbereitet werden. Ich bin eigentlich ganz froh, dal3 Sie das Manuskript geandert haben. Nach einiger Zeit gefiel mir namlich das alte nicht mehr besonders gut, namentlich die “Komplementaritat von kausaler u. raumzeitlicher Beschreibung” schien mir noch mehr der Erlauterung bedurftig, und die statistische Deutung der theoretischen Rechenresultate schien mir zu plotzlich eingefuhrt. Jedenfalls freue ich mich schon, das neue Manuskript zu sehen, und habe die beste Absicht, mich so kritisch wie nur moglich dazu einzustellen (das Resultat davon wird sein, dal3 sie Ihre Satze weiter verlangern werden). Uber alles andere werden wir uns j a mundlich unterhalten, vielleicht sind Sie so gut und schreiben uns dann noch die genaue Stunde Ihrer Ankunft, damit wir Sie eventuell abholen konnen. Viele GruDe an Ihren Bruder’O (die Mathematiker lo

For a biographical note on Harald Bohr, cf. Vol. 1, p .

XVIII.

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u. ich werden vie1 Freude davon haben, ihn hier in Hamburg begrunen zu konnen) u. seien Sie selbst sehr herzlich begrunt von Ihrem getreuen W. Pauli N.S. Ich sehe, ich hatte vielleicht immer Du statt Sie schreiben sollen; habe, bitte, Geduld, mit der Zeit werde ich mich schon daran gewohnen.

Translation, see p. [41]. BOHR TO PAULI,

[Handwritten]

15 January 1928

Hotel Pratschli, Arosa. 15-1-1928. K a r e Pauli Du gjorde rnig en stor Glade med Dit Brev. Jeg ~ n s k e rnetop Kritik, og jeg haaber at Du vil synes at Afhandlingen er forbedret. Eller for at udtrykke rnig mere beskedent vil jeg sige, at jeg tror at jeg forstaar forskellige vasentlige Punkter bedre, end da [vi] sidst saas. Jeg kommer ti1 Hamburg fra Gottingen paa S ~ n d a gd. 22de Januar K1. 545 Eftermiddag. Vil Du v a r e saa rar at bestille et roligt Varelse ti1 mig i det samme Hotel, hvor jeg sidst boede*. Hvis det passer Dig, kunde vi maaske allerede den f ~ r s t eAften gennemgaa Afhandlingen i den ny Form. Saa kunde vi de naste Dage tale om Enkeltheder, jeg har t a n k t en del over Sagerne, medens jeg har staaet paa Ski her i Arosa, og kommer derfor ikke helt uforberedt; men navnlig glader jeg mig ti1 at l a r e om alt nyt. Jeg er meget taknemmelig for Tilbudet om Papirsakse og Klisterb~tter,og hvis det kan naas, vil jeg gerne sende Korrekturen af Sted fra Hamburg, da det hele dog engang skal have en Ende. Harald kommer ti1 Hamburg den 24deJanuar om Eftermiddagen; det var morsomt, om vi den Aften kunde slaa 0s sammen en lille Kreds af Matematikere og Fysikere og - selvom Tidspunktet er forsinket - fejre det nye Aars Indtraden og d r ~ f t eHaab og Erfaringer. Med mange Hilsner fra 0 s begge ti1 alle falles Venner i Hamburg. Din hengivne Niels Bohr

* bei Darnrntor, wo ich dann aussteigen werde. (Jeg ser at jeg ubevidst allerede over rnig paa Rettelserne i den tyske Korrektur. Min Adresse i Gottingen vil v m e Prof. Courant, Wilhelrn Weberstrasse.)

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Translation, see p. [42]. BOHR,10 March 1928 [Handwritten]

PAULI TO

Institut fur Theoretische Physik an der Universitat Hamburg

Hamburg 36, den 10. Marts 1928 Jungiusstrane 9

Lieber Bohr! Gestern Abend bin ich wieder nach Hamburg zuruckgekommen und Stern hat mir ausgerichtet, daD Du hier durchgekommen bist. Ich mochte nun gerne von Dir horen, wie er nun mit Deiner Arbeit steht, prazise formuliert: ob die letzte Korrektur nun wirklich abgeschickt ist und Du nichts mehr damit zu tun hast. Denn einerseits glaube ich, daI3 falls Du noch nicht ganz fertig bist mit der Arbeit, wir alle beide nicht vie1 von meinem Besuch in Kopenhagen hatten, andrerseits mochte ich sehr gerne kommen, um mit Dir (und auch mit Klein) uber verschiedene physikalische und unphysikalische Dinge gemutlich zu plaudern. Konntest Du also so gut sein, mir moglichst bald kurz zu schreiben, ob alles in Ordnung ist? - Wenn ja, wurde ich etwa am 14., 15. oder 16. nach Kopenhagen kommen (den genauen Tag depeschiere ich dann noch). Viele GruI3e und hoffentlich auf baldiges frohes Wiedersehen! Dein W. Pauli

Translation, see p. [431.

BOHR TO PAULI,

[Typewritten]

13 March 1928

UNIVERSITETETS INSTITUT

FOR

15, KBBENHAVN 0. 13. Marts 1928.

BLEGDAMSVEJ

DEN

TEORETISK FYSIK

K m e Pauli, Som aftalt sendte jeg Korrekturen af Sted fra Holland og venter nu blot en af Dagene at faa et sidste Aftryk for at se, om alle Rettelser er rigtigt indferrte. Selv om jeg saaledes ikke kan give et entydigt Svar paa Dit Sp~rgsmaal,haaber jeg alligevel, at Du er tilfreds, og at vi kan glzede 0 s ti1 at se Dig her meget snart.

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Hvis Du i S k r z k efter Fortidens Erfaringer onsker yderligere Garantier, kan jeg j o sende Dig et Telegram, saa snart jeg har faaet den allersidste Korrektur sendt tilbage; men jeg synes, at Du burde have saa stor Tillid ti1 Fremtiden, at Du omgaaende skrev eller telegraferede ti1 mig og fortalte, naar Du kunde komme; men hvad Du end gor, og naar Du end kommer, ved Du, at jeg skal v a r e lykkelig for at have Dig her og foruden over Fysikken sammen glade 0s over andre Ting. Jeg havde en rar Rejse i Holland og stor Glade af Bes~getbaade hos Kramers og Ehrenfest. Siden min Hjemkomst har jeg haft travlt med at forberede en Opposition ti1 Jacobsens Disputats, der skal finde Sted paa Torsdag. Jeg har endnu ikke hort, hvordan det er gaaet Klein i Cambridge; jeg tznker, at han er her tilbage i Lerbet af en Uges Tid. Mange venlige Hilsener fra 0s alle, Din Niels Bohr. P.S. Jeg kan iovrigt fortaelle, at vi nu har hele seks Drenge; men du skal ikke v z r e bange for, at de skal forstyrre Roen for Dig i Kerbenhavn.

Translation, see p. [43].

PAULI TO BOHR,

[Handwritten]

16 June 1928

Physikalisches Institut der Eidg. Technischen Hochschule Zurich

7 , 16. Juni [19]28. Gloriastrasse 35 ZURICH

Lieber Bohr! Entschuldige, dal3 ich Deinen Brief vom 14/V erst so spat beantworte. Eben durch unsere Gesprache uber die Frage der Einseitigkeit der Zeitrichtung bei meinem letzten Besuch in Kopenhagen veranlant, habe ich mir in letzter Zeit einiges uberlegt uber die Frage, unter welchen Voraussetzungen und in welcher Allgemeinheit vom Standpunkt der neueren Quantenmechanik aus ein “HTheorem” vom Anwachsen der Entropie abgeleitet werden kann. Ich hatte die Antwort auf Deinen Brief immer wieder hinausgeschoben, weil ich erst daruber zu einiger Klarheit kommen wollte. Jetzt glaube ich, folgende Antwort auf die “H-Theorem”-Frage geben zu konnen: 1) Man kann das Anwachsen der Entropie allgemein zeigen, das heil3t nicht nur fur spezielle Systeme (wie das ideale Gas), sondern unabhangig von deren spezieller Beschaffenheit. 2) Eine besondere

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Annahme uber “Ungeordnetheit” ist dabei aber doch nicht entbehrlich (entgegen meiner ursprunglichen Hoffnung). Diese dem Stoflzahlansatz der klassischen Mechanik ensprechende Annahme, laflt sich wellenmechanisch immer so fassen, da13 die Phasen gewisser Gruppen von Eigenschwingungen unabhangig voneinander sind. Ich will in den nachsten Tagen versuchen, die Sache zusammenzuschreiben und mochte Dir dann eine Kopie senden. In der Quantenelektrodynamik bin ich gar nicht mehr vorwarts gekommen (Heisenberg ubrigens auch nicht). Die Schwierigkeiten, von denen ich bei meinem Besuch in Kopenhagen erzahlte, scheinen doch von sehr tiefliegender Art zu sein und ich glaube jetzt, dafl sie erst durch eine prinzipiell neue Idee umgangen werden konnen. (War Dirac in Kopenhagen? Was meint er zur jetzigen Situation?) - Ich glaube, ich mul3 die prinzipiellen Fragen vorlaufig ruhen lassen und mich inzwischen mit anderen Problemen beschaftigen. Hat Klein etwas Neues herausgebracht? - Heisenberg hat mir von einer sehr schonen, neuen Theorie des Ferromagnetismus geschrieben, die er jetzt herausgefunden hat. Kronig ist dieses Semester hier und laflt Dich auch schon griiflen; er hat eine recht gute Arbeit iiber Bandenspektren hier geschrieben (die insbesondere eine Theorie der sogenannten “Storungen” der Bandenterme enthalt). Im Herbst wird er doch wohl wieder zu Kramers zuruckgehen. Uberhaupt gefallt es mir hier in Zurich sehr gut. Herr Scherrer ist nicht nur menschlich sehr nett, sondern auch ein sehr guter Physiker. Und von Weyl habe ich jetzt schon so vie1 gelehrte Gruppentheorie gelernt, dafl ich die Arbeiten von Wigner u. Heitler wirklich verstehen kann. Die Nachfolge Schrodingers ist noch immer nicht entschieden, aber ich habe Hoffnung, dafl Wentzel herberufen und auch herkommen wird; das ware sehr nett fur mich. Wenn ich jetzt Deinen Brief nochmals durchlese, denke ich auch an Deinen Artikel. Also ich habe schon lange nicht so gelacht wie bei der Lekture des komischen Kommentars, den die Redaktion der Nature Deinem Artikel vorangestellt hat! Nachdem in kurzen Satzen eine uberflussige historische Ubersicht uber die Entwicklung der Quantentheorie in den letzten Jahren ohne die Erwahnung des Namens von de Broglie gegeben ist, folgt eine undeutliche Umschreibung folgender Stimmung: “Wir englischen Physiker waren schrecklich froh, wenn die im folgenden Artikel vertretenen Ansichten in Zukunft sich als unzutreffend erweisen wurden. Da aber Herr Bohr ein netter Mensch ist, ware eine solche Freude nicht liebenswurdig, und da er ein beruhmter Physiker ist und ofter Recht als Unrecht hat, bleibt uns nur eine geringe Chance fur die Erfullung unserer Hoffnungen” . Jedenfalls habe ich dieses aus dem Nature-Kommentar herausgelesen, und ich dachte mir: “Sancta simplicitas!” - Es freut mich immer von Dir zu horen! Herzliche Grufle an Dich sowie auch an Klein, den nicht mehr funfdimensionalen von Deinem W. Pauli

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1930)

Translation Zurich, June 16, 1928 Dear Bohr, Excuse me for being so late in answering your letter of May 14. Prompted by our conversation about the one-way character of the time direction during my last visit to Copenhagen, I have recently considered the question: from what presuppositions and in which generality is it possible from the standpoint of the new quantum mechanics to derive an “H-theorem” for the increase of entropy? I have kept postponing the answer to your letter because I first wanted to arrive at some understanding of this. Now I believe that I can give the following answer to the “H-theorem” question: 1. One can demonstrate the increase of entropy in general, that is, not only for special systems (like the ideal gas), but also independently of their particular nature. 2 . In this connection an assumption about “disorder” is however not dispensable (contrary to my original hope). This assumption, which corresponds to the “Stosszahlansatz” in classical mechanics, may within wave mechanics always be expressed by stating that the phases of certain groups of proper vibrations are independent of each other. In the next few days I shall try to write it up and I may then send you a copy. In quantum electrodynamics I have made no more progress at all (neither, by the way, has Heisenberg). The difficulties that I spoke about during my visit to Copenhagen seem still to be of a very deep nature, and I now believe that they can only be circumvented by a fundamentally new idea. (Was Dirac in Copenhagen? What does he think about the present situation?) I think I must let the fundamental questions rest for the time being and in the meantime tackle other problems. Has Klein produced anything new? Heisenberg has written to me about a very beautiful new theory of ferromagnetism, which he has now developed. Kronig is here this term and sends you his best greetings. He has written quite a good paper here about band spectra (which in particular contains a theory of the so-called “perturbations” of the band terms). Still, in the autumn he will probably return to Kramers. All in all I like it very much here in Zurich. Mr Scherrer is not only a very nice man but also a very good physicist. And from Weyl I have learned so much erudite group theory that I can really understand the works of Wigner and Heitler. No decision has yet been taken concerning Schrodinger’s successor, but I cherish the hope that Wentzel may be nominated and also actually come here. It would be very nice for me. As I now read your letter again, I also think about your paper. Honestly, I have not laughed so much for a long time as when reading the ludicrous comment with which the editors of Nature have prefaced your article. After a superfluous

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historical survey, in short sentences, of the development of quantum theory in recent years, in which the name of de Broglie is not mentioned, there follows an obscure paraphrase of the following mood: “We British physicists would be awfully pleased if in the future the points of view advocated in the following paper should turn out not to be true. Since, however, Mr Bohr is a nice man, such a pleasure would not be kind. Since moreover he is a famous physicist and more often right than wrong, there remains only a slight chance that our hopes will be fulfilled.” In any case, this is how Z read the Nature commentary, and I thought to myself, “Sancta simplicitas!” I always enjoy hearing from you! Cordial greetings to you as well as to Klein, the no longer five-dimensional, from your W. Pauli

BOHR TO PAULI,

[Carbon copy]

1 July 1929

[Kerbenhavn,] 1. Juli [19]29. K a r e Pauli, Jeg beherver ikke at sige, at jeg skammer mig over min lange Tavshed. Ikke saa meget over at jeg endnu ikke har faaet sagt, hvor glade vi var for Dit Beserg; det kunde Du ikke undgaa selv at faa et starkt Indtryk af; men Skabnen har virkelig varet meget unaadig med Hensyn ti1 Afslutningen af de forskellige smaa Noter, som jeg havde lovet at sende. Sagen er imidlertid, at jeg i de sidste Maaneder har arbejdet temmelig flittigt med en Undersagelse over de statistiske Problemer i Kvanteteorien. Jeg synes virkelig, at jeg nu kan fremstille Iagttagelsessperrgsmaalet vasentlig klarere. Man kan saaledes ferlge Reciprociteten af Individualitetsbegrebet og Superpositionsprincippet ud i vidtgaaende Konsekvenser. Almindeligt kan man vise, at enhver Brug af det farste Begreb begranser Anvendelsen af det sidste Princip som en umiddelbar Ferlge af det Fasetab, som enhver Iagttagelse medferrer; omvendt indskranker enhver konsekvent Anvendelse af Superpositionsprincippet Mulighederne for en anskuelig Tydning baseret paa Individualitetsprincippet, saaledes som det fremfor alt kommer ti1 Udtryk ved den kvanteteoretiske Behandling af Systemer med flere ens Partikler. Alt dette indeholder jo intet egentlig nyt. Dog tror jeg som sagt, at jeg kan fremstille det paa en Maade, hvoraf ogsaa andre, maaske endda ogsaa Du? kan l a r e lidt. Jeg havde t a n k t at skrive derom i Planckheftet, men det blev altfor langt og tog saa megen Tid, at jeg i sidste 0jeblik af min Artikel maatte udelade a1 Fysik og holde mig ti1 den rene Filosofi og det endda kun i Antydninger. Jeg haaber imidlertid, at Fysikerne vil se med Overbarenhed derpaa, og

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926-1930)

Niels Bohr sailing with Niels Bjerrum, Ole Chievitz and Holger Hendriksen.

at Planck selv i det mindste vil f0le, hvor varmt min lille Artikel var ment. For at Du kan se, at det ikke var helt Usandhed med L ~ f t e tom de andre Noter, sender jeg indlagt to Brudstykker. Det ene er Begyndelsen ti1 en Note om den magnetiske Elektron, som jeg maatte lagge ti1 Side for Planckheftet, den anden er et lille Stykke om 8-Straalespektrene, som lange har ligget mig paa Sinde, og som jeg i de sidste Dage har faaet renskrevet, uden at jeg dog har kunnet bekvemme mig ti1 at sende det af Sted, da det giver saa lidt positivt og kun er saa skitsemassigt udfmt. Jeg skal blive glad ved at h ~ r eDin Mening om det altsammen, ganske ligegyldig af hvor strenge eller hvor milde Udtryk Du fder Dig foranlediget ti1 at anvende. Jeg selv tranger meget ti1 Ferie og ti1 i Qjeblikket helt at glemme Fysikken. Jeg sejler i Aften paa en 14 Dages Tur ti1 Qstersoen med Bjerrum og Chievitz. Hvordan staar det ti1 med Dine egne Sommerplaner? Du ved, hvor glade vi skal blive, hvis Du lagger Vejen om ad Danmark. I Qjeblikket er Du jo midt i Fysikerkonferensen i Zurich; der bliver sikkert meget at h e , og jeg vilde gerne v a r e kommet ti1 Stede, hvis jeg ikke trangte saa stzerkt ti1 Ferie. Vil Du hilse alle mange Gange fra mig, ikke mindst Geiger, som jeg var umaadelig glad over at v a r e sammen med og bo hos for et Par Uger siden i Kiel. Jeg skrev ti1 ham forleden, men er bange for, at det ikke naaede ham f0r hans Rejse ti1 Zurich. Nu vil jeg som sagt glemme Fysikken helt, og fplrst naar jeg

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kommer tilbage p r ~ v eat fuldende en systematisk Afhandling over: “Statistik og Reciprocitet i Kvantemekanikken” (jeg haaber ikke, at Du er altfor vred for den nye Navneforandring; den er imidlertid, saavidt jeg kan se, baade sagligt og pzedagogisk velbegrundet). Maaske p r ~ v e r jeg saa ogsaa at udferre de skitsemassige Brudstykker lidt nzermere, men det kan vi jo altid f O r den Tid skandes om. Med mange venlige Hilsener fra alle i K~be nhavn. Din [Niels Bohr]

Translation [Copenhagen,] July 1, [ 19129 Dear Pauli, I need not say how ashamed I feel because of my long silence. Not so much because I have not yet told you how happy we were about your visit; you could hardly have avoided getting a strong impression of that yourself. But fate has really been very ungracious as regards my finishing the various little notes that I had promised to send. However, the fact is that during recent months I have worked rather diligently on an investigation of the statistical problems in quantum theory. I really feel that I can now present the question of observation substantially more clearly. Thus one can pursue the reciprocity of the concept of individuality and the superposition principle to far-reaching consequences. One can show in general that any use of the former concept limits the application of the latter principle as an immediate consequence of the loss of phase resulting from every observation. Conversely, any consistent application of the superposition principle limits the possibilities of a visualizable interpretation based on the principle of individuality, as it finds expression above all in the quantum theoretical treatment of systems with several identical particles. All this contains of course nothing really new. Yet, as already said, I feel that I can present it in a manner from which others also - perhaps even you? - may learn a little. I had intended to write about it in the Planck issue, but it got much too long and took so much time that at the last moment I had to leave all physics out of my article and stick to pure philosophy, and even that only by way of allusions. I hope however that the physicists will look upon it with indulgence and that Planck himself at least will appreciate how sincerely my little article was meant. In order that you may see that it was not altogether untruthful to promise the other notes, I enclose two fragments. One is the beginning of a note on the magnetic electron, which I had LO put aside because of the Planck issue. The other is a little piece about the P-ray spectra, which I have had in mind for a long time, and which

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has been typed in the last few days, but I have not yet made up my mind to send it off, since it yields so few positive results and has been written so sketchily. I shall be happy to hear your opinion about all of this, no matter how severe or how mild the expressions which you feel it appropriate to use. I am myself very much in need of a holiday and of entirely forgetting physics for the moment. Tonight I am sailing on a two-week tour to the Baltic together with Bjerrum and Chievitz. What are your plans for the summer? You know how happy we should be, if your path would lead you to Denmark. At the moment you will already be in the midst of the physics conference in Zurich. Certainly, there will be much to learn, and I would have liked to be present, if I had not so badly needed a holiday. Please, give my greetings to everybody, not least to Geiger. I enjoyed immensely being together and staying with him in Kiel a couple of weeks ago. I wrote to him the other day, but I am afraid that it did not reach him before his trip to Zurich. As already mentioned, I shall now try to forget physics completely, and only when I come back try to complete a systematic treatise on “Statistics and Reciprocity in Quantum Mechanics”. (I do hope that you are not too angry with the new change of name. As far as I can see, however, it is well justified, factually as well as pedagogically.) Perhaps I am also going to try to work out the sketchy fragments in a little more detail, but about that we will be able to quarrel in due course. With many kind regards from everybody in Copenhagen. Yours, [Niels Bohr]

PAUL1 TO BOHR,

[Handwritten]

17 July 1929

Physikalisches Institut der Eidg. Technischen Hochschule Zurich

ZUFUCH 7,

17. Juli 29 Gloriastrasse 35

(Bitte schreib an diese Adresse; ich reise zwar nachste Woche ab, aber es wird nachgeschickt !) Lieber Bohr, Vielen Dank fur Deinen ausfuhrlichen Brief, der mir vie1 Freude gemacht hat. Hoffentlich bist Du inzwischen von Deiner Segeltour gut erholt zuruckgekommen. Die kleine Note uber das magnetische Elektron hat mir so gut gefallen, dal3 ich

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es auflerordentlich bedauert habe, darj sie nicht fertiggestellt und zum Druck abgeschickt ist. Wenn ich Dir einen Rat geben darf, so ist es der, vor allem anderen diese Note abzuschicken und dies moglichst bald zu tun! - Anders ist es mit der Note uber die P-Strahlen. Ich mu13 sagen, da13 sie mich wenig befriedigt hat. Sie fangt schon so unerfreulich an mit der Besprechung der unsinnigen Bemerkungen von G.P. Thomson und die Englander werden daraus nur den falschen SchluB ziehen, da13 Du diese Bemerkungen fur wichtig haltst. Dann kommt die unangenehme Einfuhrung des Elektronendurchmessers d ; ich meine nicht, da13 das geradezu unerlaubt ist, aber es ist immer eine gewagte Sache. Man miinte dann auch berucksichtigen, dal3 fur nahezu mit Lichtgeschwindigkeit eL bewegte Elektronen infolge der Lorentzkontraktion d vie1 kleiner ist als moc2’

mo ) wenigstens in der Langsdimension. In Zurich ( j m hat uns Frl. Meitner einen schonen Vortrag uber die experimentelle Seite der namlich - m = n$

-

Frage gehalten und sie hat mich beinahe davon uberzeugt, da13 man das kontinuierliche P-Strahlspektrum nicht durch sekundare Prozesse (y-Strahlemission etc.) erklaren kann. Wir wissen also wirklich nicht, was da 10s ist! Du weirjt es auch nicht, kannst nur Grunde dafur angeben, warum wir nichts verstehen. Du schreibst ja selbst, der Zweck der Note sei der, zu betonen, “eine wie geringe Grundlage wir besitzen fur eine theoretische Behandlung des Problems des PStrahl-Zerfalles”. Hier mu13 ich aber die Frage aufwerfen, ob ein so negativer Zweck es uberhaupt rechtfertigen kann, eine Note zu publizieren! Lasse die Note also jedenfalls noch recht lange liegen. Und lasse die Sterne in Frieden strahlen! Betreffend die Moglichkeit, uber “Statistik und Reziprozitat in der Quantenmechanik” etwas Neues sagen zu konnen bin ich zwar etwas skeptisch, aber ich zweifle nicht, da8 alles was Du daruber sagen wirst, schon und interessant sein wird*. Ich glaube auch, da13 Du glucklicher und zufriedener sein wirst, nachdem Du die neue Abhandlung geschrieben haben wirst. - Von Deinem Artikel im Planck-heft war ich gerade deshalb sehr befriedigt, weil alle Physik ausgelassen war; das war einmal etwas Neues, Originelles und Anregendes! (Ob es richtig ist, daruber zu urteilen, mu13 auch ich den Psychologen und Fachphilosophen uberlassen, ich fuhle mich da ebenso - oder vielleicht sogar noch mehr - als Laie wie Du.) - Gegen die Namensveranderung Reziprozitat statt Komplementaritat habe ich gar nichts, nur hatte ich eine nahere Darlegung der Griinde, die Dich dazu gefuhrt haben, in dem Artikel gewunscht. - - Ich fahre wahrscheinlich Anfang August mit Hecke nach Sud-Schweden. Die Versuchung, Dich dann etwa Mitte August in Deinem Haus ohne Wasserleitung zu besuchen * Du siehst, ich sage nicht “geistreich und interessant”.

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(ich wurde viele Kerzen mitbringen!) ist grol3. Ich will Dir aber keine Muhe machen, konnte j a ein paar Tage in Tisvilde im Hotel wohnen! Ich vermute ubrigens, daJ Klein in einem anderen Haus ohne Wasserleitung und ohne Licht (ausgenommen den Fall, dal3 das Haus der Frau Maar gehort) in der Nahe von Dir wohnen wird; und es wurde mich auch sehr freuen, ihn zu sehen. Ich konnte mit einem kurzeren Aufenthalt in Danemark noch den anderen Zweck verbinden, selbst eine Arbeit zusammenzuschreiben, die Oppenheimer und ich gemeinsam in Zurich gemacht haben und die eine Weiterfuhrung der Arbeit von Heisenberg und mir uber Quantenelektrodynamik enthalt. Sehr befriedigt bin ich von der ganzen Theorie von Heisenberg und mir nicht (obwohl ich schon glaube, dal3 sie “gewisse Ziige” mit einer kunftigen richtigen Theorie gemeinsam haben wird). Insbesondere macht die Eigenenergie der Elektronen vie1 grol3ere Schwierigkeiten als Heisenberg anfangs gedacht hat. Auch sind die neuen Resultate, zu denen unsere Theorie fuhrt, uberhaupt sehr durftig und die Gefahr liegt nahe, darj die ganze Angelegenheit allmahlich den Kontakt mit der Physik verliert und in reine Mathematik ausartet. (Ich werde sehr gerne horen, was Du daruber meinst; dann kannst Du kritisieren und die Reziprozitat wird vollstandig hergestellt sein!) - Trotz allem habe ich aber das starke Bedurfnis, gewisse Gedankengange noch weiter und zu Ende zu denken! Was sind Deine Reiseplane fur nachstes Jahr? Wie hast Du Dich beziiglich der Amerikareise entschieden? Viele herzliche Grul3e von Deinem W. Pauli Empfehlungen an Deine Frau! Translation Zurich, July 17, 1929 Dear Bohr, Many thanks for your detailed letter which gave me much pleasure. I hope that in the meantime you have returned well rested from your sailing trip. I liked the little note on the magnetic electron so much that I deeply regretted that it has not been completed and sent off for printing. If I may give you a piece of advice, then it would be to send off this note before anything else, and to do it as soon as possible! It is quite different with the note about the P-rays. I must say that this gave me very little satisfaction. It already starts so depressingly with a reference to the nonsensical remarks by G.P. Thomson, and from this the people in England will only draw the erroneous conclusion that you regard these remarks as important. Then comes the unpleasant introduction of the electron

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diameter, d. I do not mean to say that this is actually illegitimate, but it is always a risky matter. One should then also take into account that for electrons moving almost with the velocity of light, d becomes, because of the Lorentz contraction, e2 namely ,at least in the longimuch smaller than 2 , ml,c tudinal direction. In Zurich Miss Meitner ga;e us a beautiful lecture about the experimental aspect of the question, and she almost convinced me that the continuous P-ray spectrum cannot be explained by secondary processes (pray emission etc.). So we really don’t know what is the matter here. You don’t know either, and can only state reasons why we understand nothing. After all, you write yourself that the purpose of the note is to emphasize “how little basis we possess at present for a theoretical treatment of the problem of P-ray disintegrations”. But here I must raise the question whether such a negative purpose can at all serve as a justification for publishing a note! In any case let this note rest f o r a good long time and let the stars shine in peace! True enough, I am somewhat sceptical as regards the possibility of saying anything new on “Statistics and Reciprocity in Quantum Mechanics”, but I do not doubt that everything you will say about it will be beautiful and interesting*. I also believe that you will feel more happy and satisfied when you have written the new paper. I was so very satisfied with your article in the Planck issue, precisely because all physics was omitted. For once this was something new, original and exciting! (Whether it is correct, I also must leave to the judgement of the psychologists and the professional philosophers. I feel myself here just as much a layman as you do - or perhaps even more.) I have no objections at all as regards the change of name to reciprocity instead of complementarity, only I should have liked a more detailed explanation in the article of the reasons that have led you to this change. At the beginning of August I shall probably go with Hecke to southern Sweden. The temptation is great to visit you then around the middle of August in your house without water pipes (I would bring plenty of candles along). However I will not cause you any trouble; I could easily stay a couple of days in a hotel in Tisvilde! By the way, I assume that Klein will be staying nearby in another house without water pipes and without light (except for the case that the house belongs to Mrs Maar), and it would also be a great pleasure for me to see him. I could combine a short stay in Denmark with another purpose, namely to write up an investigation which Oppenheimer and I carried out together in Zurich and which contains a continuation of the work by Heisenberg and myself on quantum electrodynamics. I am not really satisfied with the whole theory of * You see that I do not

say “subtle and interesting”.

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Heisenberg and myself (although I do believe that it will turn out to have “certain features” in common with a future correct theory). In particular, the selfenergy of the electron causes much greater difficulties than Heisenberg had thought at the beginning. Furthermore, the new results to which our theory leads are altogether very modest and there is a danger that the whole matter gradually loses contact with physics and degenerates into pure mathematics. (I should very much like to hear what you think about it. Then you may criticize, and the reciprocity would be completely restored!) In spite of everything, I feel a strong urge to follow up certain lines of thought to their end! What are your travelling plans for next year? What have you decided as regards the journey to the United States? Many cordial greetings from your W. Pauli Greetings to your wife.

31 July 1929 [Handwritten draft]

BOHR TO PAULI,

3 1-7-1929. Lynghuset, Tibirkelunde. pr. Tisvildeleje.

K a r e Pauli! Tak for Dit rare lange Brev, som jeg, lige hjemkommen fra Sejltur og Fodtur i Jylland, finder her i Tisvilde. Hvor vilde det v z r e morsomt om Du i Forbindelse med Din Ferierejse ti1 Sverige kunde b e s ~ g e0s her nogle Dage. Jeg langes efter at snakke med Dig om mange Spmgsmaal. Hvad den lille Artikkel ti1 Planckheftet angaar, er jeg bange for at Navneforandringen var en Dumhed; Sagen har saa mange Sider rent pzdagogisk set, og jeg spekulerer lidt paa at sende et Brev ti1 Naturwissenschaften for at forklare at min Kzrlighed ti1 Kunstord ikke saa meget skyldes en Trang ti1 Mystik som Bestrzbelsen for at undgaa denne ved selve Sprogets Hjselp. Du synes vist heller ikke at dette er meget klart, men det kan vi jo tale nzrmere om, naar Du kommer. Det passer storartet om Du vil skrive paa et Arbejde her; Du b e h ~ v e rikke at tage “Lys” med, vi har nu elektrisk Belysning herude. Jeg haaber at Du bliver lznge; Du ved jo ogsaa at Du altid kan bo paa Institutet. Mange venlige Hilsner fra Margrethe og hele Familien Din N. Bohr

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930)

Translation, see p. [194].

BOHR

‘ro PAULI, 15 January 1947

[Carbon copy]

[Kobenhavn,] January 15, [ 19147, Dear Pauli, I need not say how much we all enjoyed the visit of you and Franca and how happy I was again to be able to talk with you about all the scientific and human problems which are so much on the mind of us both. Since you left we have had the sorrow to lose our dear relatives who you know were so ill and also my old friend Chievitz” with whom all my life I have been so closely connected. Under these circumstances I have first in these days been able to finish the abstract of my talk at the Cambridge meeting last summer. In the manuscript which I enclose I have tried, while retaining the essence of the arguments, to express myself in a more general and cautious way. I shall be very glad to learn what you think about it and for any criticism. As Stern may have told you, Aage has written to him about the collision problems we discussed at the Kolloquium, but I was sorry not to have opportunity to learn more from Stern about his thoughts regarding the thermodynamics of the observation problem. The idea that any observation must necessarily involve an increase in entropy has been much discussed and I remember that already in the discussions with Stern and you in Hamburg, when you helped me with the “proofs” of the old paper on complementarity, I stressed the principal irreversibility of the concept of observation. More specifically, any observation must make use of some registering device, whether through a photographic plate or directly by the retina of the eye, which involves processes of amplification by which free energy is spent. I know that also Teller is interested in these problems which were discussed in Los Alamos. I shall be glad if either you or Stern would write to me what has come out of your discussions in Zurich. With kindest regards to you and Franca, and also to Stern, from us all, Yours ever, [Niels Bohr]

I ’ Ole Chievitz (1883-1946), distinguished surgeon and professor of medicine; a prominent figure in the Danish resistance movement during the German occupation. Bohr’s eulogy at his funeral was published in the memorial volume: Ole Chievitz, Nordisk Boghandel, Copenhagen 1956.

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28 January 1947 [Typewritten with handwritten insertion]

PAULI TO BOHR,

Physikalisches Institut der Eidg. Technischen Hochschule Zurich

7 , Jan. 28, 1947. Gloriastrarje 35

ZURICH

Dear Bohr, Many thanks for your letter and for the abstract of your talk at the Cambridge meeting. I read this abstract with the greatest interest and it seems to me that in its present form your views are much clearer expressed. So I hope, that it will appear soon just as it is now. Obviously only the future can show whether your view on the open questions is correct. Perhaps I had still more stressed the necessity of new ideas (in contrast to a procedure on the old lines with better mathematics) than you did already. On the other hand I am looking as critical as you on this idea of a so-called “universal length”. If this length - let us call it lo - is understood to be of geometrical nature, such theories or models will always lead to strange consequences for large momenta of the order h / l o in a field of purely classical experiments where the quantum of action should not play any role. Recently we discussed here in Zurich a mathematically “ingenious” proposal of Snyder, which, however, seems to be a failure for reasons of physics of the type just mentioned. The discussions which I had here with Stern (he left Zurich a few days ago) concerned the quantitative side of the connection of the concepts of entropy and of observation, a connection which, as we all agree, is of a very fundamental character. The problem arises whether there is a well defined minimum of the increase in entropy, independent of the particular experimental arrangement in use, if a certain quantity (“observable”) is measured. Our discussions seemed to indicate that this is actually the case, although we did not reach yet any final conclusion. The increase in entropy can easily [be] computed if one starts with a “mixture” as initial states and changes it into a “pure case” by constatation of the value of a certain quantity. If, however, before and after the measurement the observed system is in pure cases the situation seems to be more difficult to judge. (Example: a single particle in a closed box. Before the measurement it is supposed to be in a certain eigenstate with a sharply fixed value of the energy. Then one observes the place of the particle in space, perhaps by constatation that it is in a certain partial volume. What one can [can one] say about the amount of the increase in entropy through this measurement, independent of a particular experimental arrangement?) We discussed different experimental arrangements, but we are not sure how general our preliminary results are.

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Stern and I are both trying to continue the considerations of the problems, possibly by correspondence. Needless to say how anxious I am to hear what you know yourself on this quantitative side of the increase in entropy by observations and how grateful I would be if you could write to me your views on this problem. I feel that you might know the answer already or at least that you may find it out quicker than we. Franca and I enjoyed our meeting in Copenhagen and the staying in your house tremendously. I am extremely sorry that I have to add my participation in your sorrow for the loss of your relatives and of your friend, of whom I knew myself Mrs. Adler’’ and Dr. Chievitz quite well. I shall always remember them as friendly and fine personalities. With kindest regards from both of us to you, to your wife and the whole family Yours ever W. Pauli

BOHR TO PAULI,

[Typewritten]

16 May 1947

UNIVERSITETETS INSTITUT FOR

15, KBBENHAVN May 16, 1947.

BLEGDAMSVEJ DEN

0.

TEORETISK FYSIK

Dear Pauli, I was very glad indeed for your letter and not least it was a great comfort to learn that you still feel the same deep responsibility for my education as in the old days in Copenhagen. Needless to say that I have certainly no better conscience now, but in spite of all I am afraid that I have no less optimism than in those days when we also had some occasion to make the best out of the natural impatience of editors. The actual situation for good or bad is that for some weeks I have been almost drowned by “unforeseen circumstances” of partly practical and partly scientific origin; but let me at once say that I shall not for the moment annoy you with the meson troubles for which the responsibility, after I wrote to you, is entirely in the hands of Weisskopf and his learned friends from whom I have not yet

’’

Hanna Adler ( 1 8 5 9 - 1 9 4 7 ) , a sister of Bohr’s mother; founder and principal of a progressive private school. She was very close to Bohr who wrote the preface to a memorial volume, Hanna A d k r og hendes skole (Hanna Adler and Her School), G.E.C. Gads Forlag, Copenhagen 1 9 5 9 .

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heard, although to my consternation I have in the New York Times by a journalist reviewing a Colloquium in Harvard been cited as “a rescuer of this ill-fated particle staggering from a deadly blow”. Nevertheless, I have thought a good deal of the Einstein article13 and looked up some of the old literature and, not least, read a new book about Einstein’s life by Philipp Frank, which neither physically nor philosophically may be too good, but certainly gives a most interesting picture of Einstein’s personality and the unbalanced situation in Germany between the wars. I feel more and more that it may be a sensible task to give a proper account of the epistemological discussions through the years and, especially, that it may be a welcome opportunity for us all once more to learn Einstein’s reactions. I have just made a start with the writing down and, if I can get it finished, I shall surely be most eager to have your advice and criticism. In the meantime I look forward very much to see your own article. Of course, I was much interested to learn about the plans for a special issue of “Dialectica” on the epistemological problems and, if I can manage, it should certainly be a pleasure to me to contribute an article14. To my mind, the situation is far more clear than generally assumed, and such tools as three-valued logics I consider rather as complications, since a consistent representation of all axiomatic and dialectic aspects of the situation can be given in simple daily life language. I should, however, be glad to wait with a definite answer until I have come further with the article on which I am working and which perhaps, if it will not be ready in time for the Einstein volume, could be used to some such other purpose. As regards the problem of the entropy increase connected with observations I have in the last days been thinking anew over the situation and have also revived my reminiscence of earlier discussions on this point. You will remember our talks with Stern in Hamburg at the time you so kindly assisted me in gradually working my first article on complementarity from proof back to manuscript. The question was then of Boltzmann’s ideas regarding the direction of time and my point was that the very concept of observation entails an irreversibility in principle. In the following years, this problem came into the foreground in connection with the question of the consistency of the interpretation of the quantum mechanical formalism and in continuation of our discussions in Warsaw I had during the war l 3 N. Bohr, Discussions with Einstein on Epistemological Problems in Physics in Albert Einstein, Philosopher-Scientist (ed. P.A. Schilpp), The Library of Living Philosophers, Vol. VII, Evanston, Illinois 1949. Cf. Vol. 7 . l 4 N. Bohr, On the Notions of Causality and Complementarity, Dialectica 2 ( 1 9 4 8 ) 3 1 2 - 3 1 9 . Cf. Vol. 7 .

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some talks in America with v. Neumann who still felt some uneasiness about the apparent arbitrariness in the distinction between the objects and the measuring instruments, and with Teller who just like Stern, as I gather from you, was endeavouring to look for the elucidation of the paradoxes in a more quantitative connection between thermodynamics and the observational problem. After reconsidering the question, I feel myself that the whole question is purely epistemological and therefore of qualitative rather than quantitative character. O n the one hand, it is evident that any practical observational arrangements, making use of photographic plates, cloud chambers or direct sensual impressions, involve a mechanism of amplification in the working of which free energy is spent in amounts out of all proportion with the energy exchanges characterizing the individual atomic processes under investigation. On the other hand, it is equally clear that, for the interpretation of [the] quantum mechanical formalism and the elucidation of the paradoxes involved, the problem is how to account consistently for the phenomena defined by means of measuring agencies and recording devices which serve to fix the external conditions and register the experimental results and which, for this purpose, are to be treated as ideal classical instruments. Of course, it is true that the constitution and operation of the instruments is ultimately subject to the laws of atomic mechanics and that a consideration of this point may perhaps eventually prove a guide for the overcoming of the still unsolved difficulties in quantum theory, but I am sure we agree that this point has as little to do with the questions for which Einstein feels such uneasiness as the final clarification of the still unsolved problems of cosmology has to do with futile criticisms of the foundation of relativity theory. The irreversibility in any observational problem has its root in a certain degree of complication of the interaction of the object with the measuring agencies and, trying to make the situation more clear to me, I have considered experimental arrangements where the critical element of irreversibility may be arbitrarily far removed from the final macroscopic recording. For instance, we may, for the localization of a particle, instead of catching it directly on a photographic plate, allow it to enter through a small hole in a box from which the probability of escaping is vanishingly small and where, therefore, the presence of the particle can be ascertained in some suitable way at a later time. The degree of irreversibility here depends on the complicated character of the state of motion of the particle in the box, and the problem presents a certain analogy with the entropy increase accompanying irreversible expansion of a gas from a smaller volume u to a larger volume I/ which pro gas molecule is given by k log V/u. From such considerations it follows that there will be a close correspondence between the degree of complexity required and the degree of irreversibility prac-

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tically demanded and that, under optimal circumstances, the unavoidable entropy increase may be brought down to the order of k, representing the limit for the unambiguous use of the very concept of entropy. Here, I have in mind such considerations about the complementary relationships between thermodynamical and mechanical concepts as I tried to indicate in my old Faraday lecture. Just as such considerations offer a consistent attitude to the well-known paradoxes of irreversibility in thermal phenomena, so it appears to me that, notwithstanding the obvious qualitative relationship between such phenomena and the irreversibility of observations, we may more adequately regard thermodynamical considerations and the essence of the observational problem as different complementary aspects of the description. I do not know how far such remarks meet the problems in your mind and I am afraid that you may feel that I have got complementarity on the brain, but I know we both consider it a task for physicists to develop a way of talking which is suited not to hinder, but to stimulate the search for ordering experience. Of course, there must be a division of labour corresponding to the different temperaments which, after all, are themselves complementary in the deepest sense. I shall be very interested to hear your reactions and I shall endeavour to incorporate a more orderly discussion of the general arguments in the account which I hope to send you soon. I need not add that I would enjoy tremendously some time to get opportunity to discuss all such points thoroughly with you and I should, therefore, be glad to hear a little about your plans and about the chances for our meeting in the near future. If not before, I hope that you will be able to join us for the little conference which we have planned for the later part of September and to which we expect visits of a few of the old friends of the Institute. With kindest regards from us all, also to Franca, Yours ever, Niels Bohr

BOHR TO PAULI,

[Typewritten] See p. [329].

20 May 1947

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PAULI TO BOHR,

[Typewritten]

29 May 1947

Physikalisches Institut der Eidg. Technischen Hochschule Zurich

7, May 29, 1947. Gloriastrasse 35 ZURICH

Dear Bohr, I thank you very much for your three very interesting and stimulating letters of May 16, 20 and 24th. Regarding the Einstein article I am awaiting your further news and instructions. I am glad to hear that you started now on a much shorter article and I also wish to add some remarks “complementary” to my last letter about the impatience of the editor. The weak spot in the editor’s position is the circumstance that the time of the appearance of the Einstein volume is not connected with any anniversary of Einstein. For this reason the editor will never be able to prove the necessity to bring the volume out at a certain date and many authors will let him wait. Weyl, for instance, who has just arrived for a visit in Zurich, told me that he has not yet written his promised article. So I don’t see any reason why the editor should not wait a bit for your contribution, too. On the other hand politeness toward Einstein himself (who has written already his own biographical contribution) is limiting the waiting time for the publication of the volume. Therefore I am sharing at the moment your optimism, although I am prepared for more of your bulletins about “circumstances”. I am also very glad for your friendly attitude toward the plan of an issue of “Dialectica” on complementarity. As a result of your letter, I definitely agreed to overtake the redaction of this particular issue, which, I hope, will appear in one year. I also agree with your view on three valued logic, but it seems to me good to have a discussion about it in which the physicists should say, why they think that it is superfluous. Could perhaps your original plan of a longer article, which now turned out to be too long for the occasion of the Einstein volume, be used for this issue of the “Dialectica”? The most difficult topic of your letters is the problem of the connection between the entropy increase with every observation and the paradoxes of the observational problem in quantum mechanics. I am very grateful that you discussed so many details of these thorny questions in your letter and I am inclined to accept your point of view that the two mentioned sides of the problem should be considered as different complementary aspects of the description. I have, however, some hope that it may be possible to say about the unavoidable entropy increase, under optimal circumstances, in case of an observation something more quantitative than the statement that it may be brought down to

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the order of k (which is certainly true). I have not yet a definite opinion about this point and I hope to come back to it in another letter in a few days. Meanwhile I have sent to Stern a copy of the passages of your letters which are dealing with this observational problem. I was very much interested in the plan of a little conference in Copenhagen in the later part of September, in which I would very much like to participate. I have an invitation to give some lectures in Sweden in September and it should be easy to coordinate this plan with the conference in Copenhagen. Please let me know in time the details about it. With kindest regards to you and your family, also from Franca Yours ever W. Pauli MAX PLANCK PLANCK TO BOHR,

[Handwritten]

14 July 1929

Geh. Reg. Rat Prof. Dr. M. PLANCK Berlin-Grunewald Wangenheimstr . 21 14.7.29. Lieber verehrter Herr Kollege! Zu den grossten Freuden, die mir die Feier meines Doktorjubilaums gebracht hat, gehort die Ueberreichung des bei dieser Gelegenheit herausgegebenen Heftes der “Naturwissenschaften”, und der Wert dieser Gabe fur mich beruht zum grossen Teil darauf, dass auch Sie mir die Ehre erwiesen haben, einen Artikel beizusteuern, besonders wenn ich bedenke, was sonst alles an Arbeit aur Ihre Zeit lastet. Umso lebhafter ist mein Dank fur diesen neuen Beweis Ihrer freundschaftlichen Gesinnung, die ich, wie Sie wissen, von ganzem Herzen erwidere. Der Inhalt Ihres Artikels ist, wie alles, was Sie schreiben, so tief durchdacht, dass ich jetzt nicht meinerseits versuchen will, an einzelnes anzuknupfen; das konnte nicht ohne eine langere Auseinandersetzung geschehen. Vielleicht bietet sich spater einmal Gelegenheit dazu. Es gibt da noch ein reiches Feld von Gedankengangen. Als kleines Zeichen meines Dankes bitte ich Sie den gleichzeitig an Sie abgehenden Abdruck eines Vortrags von mir freundlich anzunehmen. Vielleicht wer-

PART IV: SELECTED CORRESPONDENCE (MAINLY 1 9 2 6 - 1 9 3 0 )

den Sie mit einigem darin nicht einverstanden sein. Aber das ist wohl bei der Neuheit dieser Gebiete nicht so unnaturlich. In jedem Falle wird es niitzlich sein, wenn ein jeder seine Meinung moglichst deutlich ausspricht. Nehmen Sie mit Ihrer werten Frau Gemahlin einen warmen herzlichen Gruss von Ihrem stets treu ergebenen Collegen M. Planck ERNEST RUTHERFORD BOHR TO RUTHERFORD,

[Typewritten]

27 January 1926

15, K0BENHAVN January 27th 1926.

UNIVERSITETETS INSTITUT

BLEGDAMSVEJ

FOR

DEN

0.

TEORETISK FYSIK

Dear Sir Ernest, My wife and I send you and Lady Rutherford our best wishes for the New Year. We hope that you have had a good journey, and that you are both well, and also that you found everything in Cambridge alright. Here we have had a very busy time all the autumn with the enlargement of the institute, but now I hope that we shall have more quiet working conditions for some time. For the rest we are going to have a great change, as Kramers who has been in Copenhagen for the last 10 years has accepted a professorship in theoretical physics in Utrecht. It will be a great loss for us and not least for me personally. Under the circumstances, however, we are very happy to have succeeded as his successor in the lectureship to acquire Heisenberg, a young German of whose gifts and achievements Fowler has certainly told you. In fact, due to the last work of Heisenberg prospects have with a stroke been realized, which although only vague[ly] grasped have for a long time been the centre of our wishes. At the same time as we now see the possibility of developing a quantitative theory of atomic structure, another, and as I believe very important contribution to the atomic theory has been made by two young Dutchmen Goudsmit and Uhlenbeck, who have got the idea that the origin of the fine structure of spectral lines is to be traced in a spin of the electron round its own axis. Indeed, this idea allows a remarkable way to overcome the apparently grave difficulties in the spectral theory which have puzzled us so much in later years. In fact, these difficulties gradually created that scepticism as to the unambiguous use of mechanical pictures even for qualitative conclusions to which I have tried to give expression in

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the lecture I gave in Cambridge last spring. It looks now that at any rate in the qualitative way characteristic for the correspondence principle we shall be able by means of such pictures supplemented with the new idea to account comprehensively for the building up of atoms and for their properties; of course in so far as they are independent of the structure of the nucleus. At any rate I am at present just as optimistic as regards this problem as I was when I gave my first lectures in Cambridge 5 years ago. As Fowler will have told you, we have lately in Copenhagen been interested in the problem of capture of electrons by a-particles. A pupil of Fowler, Mr. Thomas, who is working here has made some calculations on this problem which suggest the desirability of measurements of capture in hydrogen for higher velocity than those studied in your original work. If you are not contemplating such measurements in the Cavendish Laboratory, J a c o b ~ e n ' ~whose , work on the life time of recoil particles you may remember, might like to try such an experiment. We are quite aware of the difficulties, but have spoken a little of the ways to overcome them. Otherwise the experimental work in the institute is as usual mainly concerned with various spectral problems. Hevesy, however, makes an exception being occupied with an attempt of separating the isotopes of potassium on a large scale. He has got an ingenious arrangement, where he can evaporate and condense his potassium a large number of times within the same reservoir. The apparatus, however, is rather dangerous and has broken down several times already. Also Hevesy will leave Copenhagen in a near future, as he has accepted a post as professor of physical chemistry in Freiburg. He will probably leave this summer and we shall miss him very much indeed. He will at that time have been visiting Denmark for 6 years and has worked with unbelievable industry and as you know with a good deal of success. With kindest regards from us all and not least from my wife and yours, N . Bohr P.S. I shall write to Fowler in a few weeks when I have had opportunity to discuss some problems more closely with Goudsmit, whom we expect to see in Copenhagen in a few days.

'' For a biographical note on J.C. Jacobsen, cf. Vol. 5 , p . [95].

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ERWIN SCHRODINGER SCHRODINGER TO BOHR,

[Typewritten]

23 October 1926

Prof. Schrodinger Zurich 6, Huttenstrasse 9

Zurich, am 23. Oktober 1926.

Lieber und hochverehrter Herr Professor Bohr! Es ist keine kleine Dankesschuld, die ich Ihnen und Ihrer hochverehrten Gemahlin gegenuber fuhle, sowie gegen alle die, welche wahrend meines Aufenthaltes in Kopenhagen so lieb und gutig, so fursorgend und hilfsbereit gegen mich waren. Aus mehr als einem Grunde wird mir diese Woche in unvergesslicher Erinnerung bleiben - ich brauche das kaum naher auszufuhren. Allein der Eindruck der einzigartig schonen Stadt mit ihrer lieblichen und grossartigen Umgebung wurde ein anhaltender und bleibender gewesen sein. Allein das liebe, sonnige, gastfreundliche Heim rnit seinen liebenswurdigen Bewohnern, das mich Unbekannten wie einen alten Freund aufnahm und rnit Traulichkeit umgab, war ein Erlebnis, dessen das Herz nicht wieder vergisst. Aber nun ist diese Stadt, dieses Haus, diese Familie - es sind die des grossen Niels Bohr, er selbst ist es, dem ich alle Freundlichkeit danke, ich darf stundenlang mit ihm uber die Dinge sprechen, die mir so sehr am Herzen liegen, und hore von ihm selbst, welche Stellung er gegenwartig zu den zahlreichen Versuchen einnimmt, ein Stuck weiter zu bauen auf dem breiten tragfahigen Fundament, das er der modernen Physik gegeben hat. Das ist fur einen Physiker, der es auch rnit dem Herzen ist, ein wirklich unvergangliches Erlebnis! Es ist moglich, dass die Starrheit, rnit welcher ich in unseren Wechselgesprachen schliesslich doch immer an meinen “Wunschen” fur eine kunftige Physik festhielt, Ihnen zuletzt den Eindruck hinterlassen haben, als hatten die allgemeinen und speziellen Einwande, welche Sie gegen meine Auffassung erhoben, keine bedeutende Wirkung auf mich ausgeubt. Das ist aber ganz sicher nicht der Fall. In einem gewissen Sinne kann ich sagen: die psychologische Wirkung dieser Einwande - besonders der zahlreichen speziellen Falle, in denen meine Auffassung offenbar zunachst mit der Erfahrung kaum in Einklang zu bringen ist - ist bei rnir wahrscheinlich noch grosser als bei Ihnen selbst. Und zwar deshalb, weil Sie, wie mir scheint, doch einen gewissen vorlaufigen Ruhepunkt in der Auffassung finden, dass die ganzen scheinbar anschaulichen Bilder in Wahrheit nur symbolisch zu nehmen seien, vor allem dass - wie Dirac in seiner allerletzten Publikation ausfuhrt und auch Born vertritt - die “c”. die

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Amplituden oder Koeffizienten der einzelnen Eigenschwingungen, nur statistische Aussagen iiber das Verhalten einer grossen Anzahl gleichbeschaffener Systeme vermitteln und nicht das Verhalten eines Einzelsystems beschreiben. Ich vermag mich aber bei dieser vorlaufigen Losung durchaus nicht zu beruhigen. Sie scheint mir ebensowenig allgemein anwendbar, wie die meine. Wenn eine Lichtwelle auf eine grosse Zahl von Atomen auftrifft (etwa auf ein Gas), dann muss eben doch jedes einzelne Atom eine schwache Sekundarwelle abgeben, sonst versteht man die Schwachung und Dispersion der Lichtwelle nicht. Anderseits: wenn die Lichtwelle gerade die Resonanzfrequenz besitzt, dann miissen in der Tat blos einzelne wenige Atome eine bedeutende Veranderung erleiden (“in den oberen Zustand gehoben werden”). Hier liegt scheinbar ein Widerspruch vor und Sie sagen: hier reichen eben unsere bisherigen Worte und Begriffe nicht aus. Ich kann mich bei dieser Konstatierung nicht befriedigen und ich kann daraus nicht fur mich das Recht ableiten, mit widerspruchsvollen Aussagen weiter zu operieren. Man kann die Aussagen abschwachen, indem man z.B. sagt, die Atomgesamtheit verhalt sich “in gewisser Beziehung so, als o b ...” und “in gewisser Beziehung so, als ob ...”, aber das ist doch so zu sagen nur ein juristischer Behelf, der sich nicht in klares Denken umsetzen lasst. Ich halte es nicht fur ausgeschlossen, Bilder zu konstruieren, welche obiges Verhalten wirklich liefern. Die Strahlungsdampfung wurde bisher in keiner der neuen Theorien wirklich beriicksichtigt. Sie bildet aber eine in Wahrheit notwendige Erganzung jeder Theorie, auch der urspriinglichen, mit Elektronenbahnen operierenden, wie Sie oft betont haben. Fur viele Zwecke kann man von ihr absehen und tut das de facto stets, indem man Ergebnisse, die eigentlich nur aus der Strahlungsdampfung direct erschliessbar sind (Breite der Linie, Abklingungsdauer) entweder indirect auf Umwegen ableitet, oder nur qualitativ durch korrespondenzmassiges Zuriickgreifen auf die Klassik. - Vielleicht ist nun die Strahlungsdampfung, die Riickwirkung der selbst entsandten Welle auf das System, doch in ganz anderer Weise zu beriicksichtigen, als ich ursprunglich dachte, namlich nicht durch Hinzufiigen eines (nichtlinearen) Gliedes in der Wellengleichung, welches ohnehin ein empfindlicher Schonheitsfehler ware, sondern in ganz anderer Weise, etwa - nur ein Beispiel - durch Koppelung mit einem anderen System, dem “Aether”, das ein kontinuierliches Eigenwertspectrum von Null bis Unendlich hat. Ich habe noch gar keine bestimmten Vorstellungen in dieser Richtung und ich mochte Ihre Zeit nicht damit beanspruchen, dass ich Ihnen etwas vorphantasiere. Was mir vor Augen schwebt, ist nur die eine These: man darf, auch wenn hundert Versuche fehlschlugen, die Hoffnung nicht aufgeben, ich sage nicht durch klassische Bilder, aber durch logisch widerspruchsfreie Vorstellungen von

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der wirklichen Beschaffenheit des raumlich-zeitlichen Geschehens zum Ziele zu kommen. Es ist ausserordentlich wahrscheinlich, dass das moglich ist. Ausserordentlich wertvoll war mir Diracs letzte Arbeit, weil sie seinen interessanten Ideenkreis wenigstens teilweise in die mir verstandliche Sprache ubersetzt. Vieles finde ich freilich auch in dieser Darstellung noch sehr dunkel, z.B., um gleich am Anfang zu beginnen: nach der Feststellung auf S. 662

a

unten “bedeutet” ein p n stets -, Betrachten Sie nun auf der nachsten Seite aqn die auf Glg. (4) folgende Beziehung: d d d s = [x, F ] , in welcher x beispielsweise eines der p und die Grosse s beispielsweise die Zeit bedeuten kann. Welches ist nun die Bedeutung von

a

Was sol1 es heissen, den Operator - total nach der Zeit zu differenzieren? a4

Dirac hat eine ganz originelle und eigenartige Methode des Denkens, die gerade deshalb noch die wertvollsten Ergebnisse liefern wird, die uns anderen verschlossen sind. Aber er hat wohl keine Vorstellung davon, wie schwer die Lecture seiner Arbeiten einem gewohnlichen Menschen fallt, Lassen Sie mich nun, verehrter Herr Professor Bohr, Ihnen und Ihrer verehrten Gemahlin meinen tiefgefuhlten Dank wiederholen fur alle Cute und Freundlichkeit, die mir in Ihrem Hause geworden ist, fur alle Muhe, die ich Ihnen bereitet habe und fur die viele kostbare Zeit, die Sie mir gewidmet haben. Ich werde Ihnen das kaum je vergelten konnen; mein Trost ist, dass Sie funf Sohne haben, so dass ich vielleicht doch auf die Gelegenheit hoffen darf, spater einmal einem von diesen Freundschaft zu erweisen. Mit den ergebensten Empfehlungen und Grussen bleibe ich stets Ihr in warmster Verehrung ergebener E. Schrodinger

PART IV: SELECTED CORRESPONDENCE (MAINLY

BOHR TO SCHRODINGER,

[Typewritten]

1926-1930)

2 December 1926

UNIVERSITETETS INSTITUT

BLEGDAMSVEJ

FOR

DEN

15,

KBBENHAVN 0.

2. Dezember 1926.

TEORETISK FYSIK

Lieber Professor Schrodinger, Ich danke Ihnen sehr fur Ihre freundlichen Briefe. Es ist mir kaum moglich zu sagen, wie viel Freude und Belehrung wir alle hier von Ihrem Besuch gehabt haben, und wie hoch ich es schatze, Ihre personliche Bekanntschaft gemacht zu haben. Ich hoffe sehr, dass wir in den kommenden Jahren ofters Gelegenheit haben werden, uns wieder zu treffen und Gedanken auszutauschen. Was Ihren letzten Brief anbelangt, ist es uns allen hier eine grosse Freude, dass Dr. Fues im Fruhjahr in Kopenhagen arbeiten will, nicht nur wegen seiner schonen Arbeiten, die ich sehr schatze, sondern auch weil er eben aus Zurich kommt und so mit Ihren Gedanken und Ansichten besonders vertraut sein wird. Seit Ihrem Besuch haben wir hier die verschiedenen umstrittenen Fragen ofters und eingehend diskutiert. Eben in diesen Tagen ist Klein im Begriff eine Arbeit abzuschliessen uber die Moglichkeit, die Wellenmechanik fur die Auffassung der Atomvorgange zu verwerten, die mit Diskontinuitaten operiert 16. Fur Sie wird diese Arbeit inhaltlich wohl kaum viel neues bringen, aber ich denke, dass es Sie freuen wird zu sehen, wie gut die Wellenmechanik geeignet ist, die Korrespondenz zwischen der klassischen Elektrodynamik und der Quantentheorie hervortreten zu lassen. In der Tat lasst sich auf Grund der Wellenmechanik eine Korrespondenztheorie aufbauen, die in sich ebenso geschlossen ist wie die Matrixmechanik, die ihrerseits als eine auf der Korpuskelmechanik sich stutzende Korrespondenztheorie aufgefasst werden kann. Es ist dabei interessant zu sehen, wie der Begriff der Welle oder der Korpuskel als der mehr geeignete Begriff sich darbietet, je nach dem Platz in der Beschreibung wo die Annahme der Diskontinuitaten explizite auftritt. Nach meiner Auffassung ist dies leicht verstandlich, da die Definition jedes Begriffs oder viel mehr jedes Worts die Kontinuitat der Erscheinungen wesentlich voraussetzt und daher mehrdeutig wird, sobald diese Voraussetzung nicht aufrechterhalten wird. Dies ist aber wohl nur der von Ihnen verhasste Greuel der Unterirdischen, und ich brauche kaum zu betonen, mit wie grossem Interesse ich Ihre Bestrebungen folge, Ihre lichteren Hoffnungen zu ver-

l 6 0 . Klein, Elektrodynamik und Wellenmechanik vom Standpunkt des Korrespondenzprinzips, Z. Phys. 41 (1927) 407-442.

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wirklichen. Wenn Sie nicht in gewohnlicher Zeit und Raum die Gespenster ganz tot schlagen konnen, Iasst sich vielleicht in einer fiinfdimensionalen Welt in der Zukunft einen Vertrag schliessen. Von hier laisst sich sonst berichten, dass Heisenberg und Dirac schone weitere Beitrage zum Aufiau der Matrizenmechanik geliefert haben, und dass Hund auf Grund der Wellenmechanik einen, wie es scheint, sehr bedeutungsvollen Vorstoss zu einer allgemeinen Theorie des Molekiilbaus gegeben hat, Ich hoffe, dass Sie eine interessante und vergnugliche Reise in Amerika haben werden und sende zusammen mit meiner Frau die herzlichsten Griissen auch an Ihre Frau Gemahlin, die wir hoffen ein anderes Ma1 Sie nach dem Norden begleiten wird. Ihr sehr ergebener, N. Bohr

P.S. Ich habe gleichzeitig an Dr. Fues geschrieben. BOHR,5 May 1928 [Typewritten with handwritten footnote]

SCHRODINGERTO

PROFESSOR DR. ERWIN SCHRODINGER

BERLIN-GRUNEWALD, AM 5 . Mai 1928. CUNOSTRASSE

44

Sehr verehrter Professor Bohr! Verzeihen Sie die lange Hinauszogerung meiner Antwort, ich wollte die Angelegenheit doch durch briefliche Anfrage an die Kanzlei sicherstellen und habe erst soeben die Antwort des Rektors erhalten. Der Herr Rektor schreibt, dass er selbstverstandlich auf Grund Ihrer Empfehlung nun keinerlei Bedenken tragt, Herrn M ~ l l e zu r den Kursen zuzulassen. Herr Maller braucht sich also nur einfach anzumelden. Die Vorlage von Studienausweisen wird ja glaube ich ohnehin nicht verlangt. Vielen Dank fur den Sonderdruck Ihrer Arbeit, die ich iibrigens schon im Korrekturbogen, den mir Herr Planck freundlichst lieh, gelesen hatte. Ich habe neulich im Kolloquium iiber die Grundideen referiert und ware sehr neugierig, ob ich wohl einigermassen getroffen habe, was Sie meinen. Eine sehr merkwurdige Beziehung besteht, scheint mir, zwischen der Heisenbergschen Ungenauigkeitsrelation und der Behauptung diskreter Quantenzustande. Letztere ist eigentlich auf Grund der ersteren prinzipiell nicht erfahrungsmassig priifbar. Am besten erkennt man das in den Wirkungs- und Winkelvariablen. Da ist

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Lasst man nun fur d w den Bereich 1 zu, d.h. verzichtet man auf die Kenntnis der Winkelvariablen uberhaupt (weil doch in w alles periodisch ist mit der Periode l), so wird A J = h , d.h. gerade so gross wie der Unterschied des J i n benachbarten Quantenzustanden. Man kann das auch an einzelnen einfachen Fallen zeigen. Z.B. bei der Quantelung des idealen Gases. Lasst man als Spielraum fur den Ort des Molekuls das ganze Gasvolum zu, so wird die Unsicherheit des Impulses grossenordnungsmassig gleich der Impulsdifferenz benachbarter Quantenzustande. * Eine andere Bemerkung. Wenn Sie ein System, z.B. einen Massenpunkt, durch Angabe seiner p und q beschreiben wollen, so finden Sie, dass die Beschreibung nur mit einem begrenzten Grad von Genauigkeit moglich ist. Das scheint mir sehr interessant als Begrenzung fur die Anwendbarkeit der alten Erfahrungsbegriffe. Aber es scheint mir gebieterisch die Einfiihrung neuer Begriffe zu fordern, in denen diese Begrenzung nicht mehr besteht. Denn was prinzipiell unbeobachtbar ist, das sollte in unserem Begriffsschema uberhaupt nicht enthalten, sollte durch dasselbe nicht abbildbar sein. In dem adaquaten Begriffsschema darf es nicht mehr so aussehen, als sei unsere Erfahrungsmoglichkeit durch ungunstige Urnstande eingeschrankt. - Es wird aber gewiss sehr schwierig sein, dieses neue Begriffsschema aufzufinden, da, wie Sie so eindrucksvoll hervorheben, die erforderliche Neugestaltung die tiefsten Schichten unseres Erkennens, Raum, Zeit und Kausalitat betrifft. Mit den herzlichsten Wunschen fur Ihr und der Ihrigen Wohlergehen bin ich in aufrichtiger Ergebenheit Ihr E . Schrodinger

Translation, see p. [46]

BOHR TO SCHRODINGER,

[Carbon copy]

23 May 1928

[Kerbenhavn,] 23. Mai [19]28. Lieber Schrodinger, Ich danke vielmals fur Ihren freundlichen und inhaltsreichen Brief und bitte Sie zu entschuldigen, dass ich Ihnen erst jetzt antworte. Fur Ihre hilfreiche Bemuhung in Verbindung mit der Anfrage des Herrn Merller waren wir naturlich sehr *1 1 1

Q pdq = mu.21 = nh, mu = nh/2l, benachbarte Quantenzustande underscheiden sich also

irn Irnpuls urn h/21. Anderseit gibt d p dq = h rnit d q = I gerade dp = h / / = 2 . h / 2 / .

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dankbar. Herr Maller freut sich sehr, die Vortrage in Berlin diesen Sommer horen zu konnen. Es war mir auch eine besondere Freude, von Ihren Bemerkungen zu entnehmen, dass Sie sich nicht ganz ablehnend stellen zu Betrachtungen von der Tendenz, welche in meinem Artikel in Naturwissenschaften Ausdruck gegeben ist. Doch bin ich rnit Ihrer Betonung der Notwendigkeit der Entwicklung von “neuen” Begriffen kaum ganz einverstanden. Nicht nur haben wir, soweit ich sehe, bis jetzt keinerlei Anhaltspunkte fur eine solche Neugestaltung, sondern die “alten” Erfahrungsbegriffe scheinen mir untrennbar mit der Grundlage des menschlichen Anschauungsvermogens verknupft. Wohl hat der scheinbare Gegensatz des Superpositionsprinzips und des Individualitatspostulats die komplementare Natur der Raum-Zeit-Koordinaten und der Erhaltungssatze entschleiert. Ich glaube aber, dass es sich um eine in philosophischer Hinsicht konsequente und daher befriedigende Ausbildung der Grundlage unserer Naturbeschreibung handelt. Meiner Ansicht nach ist auch keine Rede von einer mehr oder weniger willkurlichen Begrenzung der Anwendbarkeit der klassischen Begriffe, sondern um [von] der Erkenntnis eines unvermeidbaren Zugs van Komplementaritat, der in einer Analyse des Beobachtungsbegriffs zum Vorschein kommt, und der in vielen Beziehungen an die Erkenntnis der allgemeinen Relativitat erinnert. Naturlich besitzen wir in der Quantentheorie noch nicht eine technische Ausrustung, die mit der in der Relativitatstheorie zu vergleichen ist. Ich glaube aber, dass auch in dieser Hinsicht die Quantentheorie sich einer gewissen vorlaufigen Abschliessung nahert. Ja, ich glaube, dass man schon sagen kann, dass jedem Gebrauch der klassischen Begriffe, der eine eindeutige Definition zulasst, auch eine physikalische Deutung zugeschrieben werden kann. In dieser Verbindung mochte ich zu Ihren Ausfuhrungen uber das Verhaltnis der Unsicherheitsrelationen zu dem Quantenpostulat folgendes bemerken: Wie ich in meinem Artikel zu zeigen versucht habe, sind diese Relationen als eine unvermeidbare Begrenzung der Definitionsmoglichkeiten des Raum-Zeit-Vektors und des Impuls-Energievektors der einzelnen Individuen anzusehen, indem sie einer Eigenschaft von Wellengruppen Rechnung tragen, die eine unmittelbare Folgerung des Superpositionsprinzips ausdruckt. In dem Fall der Wechselwirkung mehrerer Individuen und uberhaupt wenn von einer Quantisierung die Rede sein kann, mussen die Unsicherheitsrelationen immer mit Vorsicht benutzt werden. In dem von Ihnen erwahnten Fall einer Wirkungs- und Winkelvariablen besteht eben die Moglichkeit von Losungen der Wellengleichung, in welchen erstere wohldefiniert ist, und die angewandt werden konnen, ohne dass wir nach der gleichzeitigen Begrenzung der Winkelvariablen zu fragen brauchen. Ihre Bemerkung, dass eine Winkelvariable nie eine grossere Unsicherheit aufweisen kann, als der Periodizitatsmodul angibt, lasst sich, soweit ich es verstehe, in dieser Hin-

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sicht kaum heranziehen. Bei der Deutung von Experimenten mit Hilfe des Begriffs der stationaren Zustande haben wir es ja immer rnit solchen Eigenschaften eines Atomsystems zu tun, die von Phasenverbindungen uber eine grosse Anzahl nacheinanderfolgender Perioden bedingt sind. Eben in diesem Umstand ist ja die Definition und Anwendbarkeit der Eigenlosungen der Wellengleichung begrundet . In dem Fall wo gewohnliche Ortskoordinaten benutzt werden, ist in einer Eigenlosung die konjugierte Variable nicht eindeutig bestimmt, sondern weist einen endlichen Wertbereich auf, in solcher Weise dass das Produkt A p A q von der Grossenordnung nh ist, wo n die Knotenzahl bedeutet. Eine nahe Analogie zur Unsicherheitsrelation bei freien Partikeln hat man auch hier in dem Verhalten von Wellengruppen. So ist z.B. der experimentell nachweisbare Wertbereich der Variablen um so kleiner gegenuber dem Wertbereich der einzelnen Eigenlosungen, je grosser die Quantenzahl ist. Obwohl dieser Umstand einen naturlichen Ubergang von Mikro- zur Makromechanik darbietet, besteht wie im Artikel angefuhrt immer eine absolute Ausschliessung der Anwendung des Begriffs der stationaren Zustande und der Verfolgerung des Verhaltens der einzelnen Partikel im Atom. Diese Ausschliessung liefert meiner Ansicht nach ein besonders schlagendes Beispiel der allgemeinen komplementaren Natur der Beschreibung. Wie ich in meinem Artikel zu zeigen versucht habe, lasst sich dem Begriff der stationaren Zustande sowie der diskreten Energiewerte innerhalb ihres Anuendungsgebiets ein ganz bestimmter Sinn beilegen. Die Feststellung, dass ein Atom sich in einem bestimmten Zustand befindet, ist dabei immer mit dem Verzicht der Kenntnis der Phase der zugehorigen Eigenlosung verbunden. Eben in dieser Unbeobachtbarkeit der Phase haben wir, wie angefuhrt, wieder mit einem einfachen Beispiel der Konsequenzen des Superpositionsprinzips zu tun. In dem Artikel habe ich mich bestrebt, das Versagen von klassischen Bildern bei der quantentheoretischen Behandlung des Wechselwirkungsproblems moglichst stark hervorzuheben und zu betonen, dass unsere ganze Anschauungsweise auf die Abstraktionen der freien Individuen begrundet ist; ein Punkt, wo meiner Meinung nach das Verhaltnis zwischen klassischer Theorie und Quantentheorie besonders klar zu Tage kommt. Ich mochte noch hinzufugen, dass eben bei dem in Ihrem Brief beruhrten Fall der Quantelung eines Gases dieses Versagen ja so schlagend in den Paradoxien der neuen Statistik hervortritt. Ihre Anwendung der Unsicherheitsrelation auf diesen Fall verstehe ich jedoch nicht recht, weil ja hier die zur Koordinate konjugierte Impulsgrosse nicht einen eindeutigen Wert hat. In letzter Zeit habe ich uber einige weitere Fragen allgemeiner Art nachgedacht und hoffe bald in einer kleinen Note zeigen zu konnen, wie gewisse Paradoxien der quantentheoretischen Behandlung der Strahlungserscheinungen beleuchtet werden konnen durch die Bemerkung, dass die Festlegung einer Zeitrichtung aufs engste mit dem Beobachtungsbegriff zusammenhangt .

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Ich furchte, dass ich Sie mit diesen vielen Worten schon sehr gelangweilt habe. Sie mussen es aber auf die Rechnung meiner Begeisterung schreiben. Nach den jahrelangen Kampfen im Dunkeln fuhle ich vielleicht besonders stark die Erfiillung von alten Hoffnungen, welche die neuen Entdeckungen von Ihnen und anderen uns gebracht haben. Ich denke oft mit grosser Freude an unsere lebhaften Diskussionen in Kopenhagen und Brussel und hoffe, dass wir bald Gelegenheit bekommen werden, sie wieder aufzunehmen. Mit vielen freundlichen Grussen, auch an Planck und Einstein, mit denen Sie vielleicht den Inhalt dieses Briefes diskutieren. Ihr sehr ergebener, [Niels Bohr]

Translation, see p. [48]. OTTO STERN STERN TO BOHR,

[Handwritten]

9 August 1947 159 CRAGMONT AVE. 8, CAL., den 9. Aug. 1947

BERKELEY

Lieber Bohr, Pauli hat mir vor einiger Zeit die Kopie einiger Bemerkungen von Dir iiber die Frage der Entropievermehrung bei einer Messung geschickt. Weshalb glaubst Du, dass fur die Anwendung des Entropiebegriffs die Grenze von der Grossenordnung k ist? Die Entropieanderung k In V,/V2kann doch beliebig klein sein. Klassisch scheint mir die Idealisierung des Messprocesses als reversibel verniinftig, so weit entfernt von diesem Ideal auch die wirklichen Messungen sein mogen. Die Entropieabnahme des gemessenen Systems ist dann gleich der Entropiezunahme der Umgebung, des Messapparates, gleich k In V,/V2 ( Vl, V2Volumina im Phasenraum). Die Bedingung V, und V2 s h“ wurde dann den von Dir immer so stark betonten Satz bedeuten, dass letzten Endes immer mit “klassischen” Apparaten gemessen wird. Falls der Messapparat nicht als microcanonisches sondern canonisches System betrachtet wird, ist Vdurch Se In e d e (e Dichte im Phasenraum) zu ersetzen (bei grossem n praktisch indentisch), und Dein Satz wurde bedeuten, dass man die Summe iiber die Quantenzustande durch das Integral ersetzen kann. Quantentheoretisch: Wenn man an einem “reinen Fall” misst derart, dass das

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Resultat jeder Messung wieder ein reiner Fall ist, so ist die Entropie des gemessenen Systems vorher und nachher gleich Null, aber die Entropie der Umgebung hat zugenommen. Als “ideales” Minimum dieser Entropiezunahme kann - Pauli und ich hielten das in Zurich fur wahrscheinlich - die Entropie nk Exi In x, von n Exemplaren der Mischung angesehen, die dasselbe Resultat fur n Messungen ergeben wurde wie n Messungen am reinen Fall. Pauli glaubt das nicht mehr und berichtet dasselbe von Dir. Ich mochte auf jeden Fall glauben, dass man ein “ideales” Minimum der Entropiezunahme der Umgebung bei Messungen an einem reinen Fall definieren kann.

...

In der Hoffnung, dass es Dir und den Deinen gut geht, bin ich mit vielen herzlichen Grussen Dein Otto Stern STERN TO BOHR,

[Handwritten]

31 October 1947 Physikgebaude E.T.H., Gloriastr. 35 Zurich, d. 31. Oct. 1947.

Lieber Bohr, vielen Dank fur Deine Noten, die Pauli mir uberbrachte. Er hat mir auch noch von Euren letzten Gesprachen uber das Problem berichtet. Also: Der springende Punkt bei der Szilard’schen Maschine ist der, dass man (der Mechanismus) “wissen” muss, in welcher Richtung der Stempel bewegt werden soll. Wenn man den Stempel einfach durch das Molecul treiben lasst, so ist das ein irreversibler Process, wie das Ausstromen eines Gases ins Vacuum. Um die Entropieabnahme des Warmereservoirs zu erzielen, muss man auf den Stempel einen Druck ausuben, der entgegen[ge]setzt gleich dem Druck des Moleculs ist, resp. unendlich wenig kleiner. Nur dann wird dem Reservoir die Warmemenge k T l n V l / V 2entzogen. Der Mechanismus muss also messen, in welcher Halfte des Gefasses das Molecul ist, um den Stempel in der richtigen Richtung zu verschieben. Es ist wirklich nich Starrsinn von mir, wenn ich immer wieder diesen Punkt betone, er ist halt entscheidend fur den Zusammenhang von Messung und Entropie. Das Szilard’sche Resultat - dass die Messung, welche von 2 Moglichkeiten realisiert ist, im Mittel mindestens k In 2 Entropievermehrung in der Umgebung bedingt - kann verallgemeinert werden. Aber Du willst ja das Resultat nicht zugeben.

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Ich mochte nur noch ma1 sagen, weshalb ich glaube, dass diese Verallgemeinerung Bedeutung hat fur die Quantentheorie. Man kann nicht von der Entropiedifferenz reiner Faille (Entropie Null) reden. Aber wenn man durch Messung viele Exemplare desselben reinen Falles in andre reine Faille uberfuhrt (prinzipiell irreversibel), so kann man doch die dadurch bedingte minimale Entropiezunahme im klassischen Messapparat angeben. Das sollte weiterfuhren, vermute ich wenigstens. Ich will am 17. Nov. nach London fliegen, am 20. Nov. mit der “Queen Mary” nach U.S.A. Hoffentlich sehe ich Euch dort nachstes Fruhjahr. Ich mochte Dir und Deiner Frau nochmals recht herzlich fur die schonen Tage in Kopenhagen danken und bin mit den besten Grussen Dein Otto Stern.

PREPARATIONS FOR LETTER FROM BOHR TO STERN,

[Aage Bohr’s handwriting]

19 November 1947

a) Notes, 1/11 October 1947 The problems of observation in thermodynamics and in quantum mechanics are related to quite different physical problems. In thermodynamics, as far as we have to do with applications of statistical classical mechanics, observations involve in principle no finite interaction and, in “observing” positions and velocities of molecules, we just have to do with features of a description which is essentially different from considerations of infinite ensembles with continuously varying position and momenta co-ordinates. In the extreme case where all the positions and momenta are known, we have only rough analogies to thermodynamical phenomena defined by the notion of temperature and entropy, and all reversible properties of mechanical systems are, of course, maintained in principle [fundamental] contrast to the demands of entropy increase. Only in the so-called canonical ensemble, we have something approaching thermodynamics although in the case of varying temperatures, recourse to more general ensembles must be taken. The difference between the use of specified mechanical states and of infinite ensembles is, from the point of view of mechanics, of course only one of complication but the essential point at issue is the restriction imposed upon thermodynamical concepts in the former case. Only in the latter case any unambiguous use of such concepts is possible although, of course, we have, in case of finite numbers of molecules, to do with a generalization of thermodynamics characterized by the appearance of fluctuations.

I

oct [471

PART

II Oct 47

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I . Within the region of classical physics there is no obs. problem in that sense that we can neglect the interaction between object and meas. instr. in the process of observation. 2 . As long as we are speaking of thermod. problems framed in classical mechanical concepts, the problems of observation and knowledge of the state of the system are the same. 3. The unambiguous use of thermod. concepts like temp. and entropy is not compatible with an exact knowledge of the state of the system. In fact, a mechanical analogon to such concepts can only be contained in a comparison with an infinite ensemble of classical systems with continuously distrib. represent. of their states in phase space. In order that such a compar. shall hold not only in the limit of very large degrees of freedom but also for systems consisting of a few or even a single molecule, it is further necessary that the distribution representing a given temperature is canonical in the Gibbs' sense.

An. Szilard-Stern exper. The entropy exchanges implied in the obs. of the particle in one or the other half of the box, has however nothing to do with the irreversibility of observation. The problem is that if you want to describe the experiment by thermod., you must of course take into account also the recording instruments in the total thermod. system. If you want to describe it mechanically this is irrelevant but then there is of course no problem of a contradiction to the second law of thermod. which finds no room within mechanics. b) Draft of letter, 1 November 1947 l 7 Lieber Stern. Vielen Dank fur Deinen Brief. Ebenso wie bei Dir, ist es mein innerlicher Wunsch, dass wir, die im allgemeinen einander so wohl verstehen, nicht in dieser Sache uns ganz vorbeireden. Was die Szilard'sche Maschine anbelangt, ist es nur meine Bestrebung wirklich am einfachsten zu verstehen, warum sie nicht zu einem Widerspruch mit dem zweiten Hauptsatz wirken kann. Man kann sie naturlich so handhaben, dass man den Platz des Molekuls in einer der zwei Halften des Zylinders jedesmal konstatiert, bevor man den Stempel bewegt, aber meine Bemerkung war hier, dass es " The formulation of this draft is here and there rather careless. In order to make it more intelligible to the reader we have corrected obvious grammatical mistakes and suggested an amendment of a single incomplete sentence.

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nicht auf die direkte Wechselwirkung zwischen Molekul und Beobachtungsmittel ankommt, die ja nach der klassischen Mechanik so klein gemacht wird, als man wunscht. Aber [es kommt auf die] Registrierung und ihre Korrelation zur Stempelbewegung [an], die mit einer Entropievermehrung von anderen Korpern verbunden sein wird, weil es sich damit um eine Wahl zwischen verschiedenen Moglichkeiten handelt, und das betreffende Problem daher ganz analog ist mit dem Entropieunterschied eines Molekuls, von dem man nur weiss, dass es in einem Teil eines Volumens anwesend ist. Das ganze Problem kommt mir ubrigens etwas mehr verwickelt vor als bei erster Ansicht, weil es ja notwendig ist, die Temperaturbewegung des Stempels selber in Betracht zu ziehen. In dieser Verbindung verstehe ich nicht die Bemerkung in Deinem Brief, dass, wenn man ohne vorausgehende Beobachtungen das Molekul auf den Stempel direkt wirken lasst, man mit einer einfachen irreversiblen Expansion zu tun hat. Dies ware j a nur der Fall, wenn keine kinetische Energie dem Stempel wegen des Druckes zugefuhrt wird, aber wenn man annimmt, dass der Stempel vie1 schwerer als das Molekul ist, wird er nur langsam unter dem Einfluss der einseitigen Stosse des herumfliegenden Molekuls in Bewegung gesetzt, und die kinetische Energie seiner Schwerpunktsbewegung wird zuletzt genau der Arbeit entsprechen, die bei einer langsamen Herausziehung des Stempels gewonnen werden kann. Die oben beruhrte Verwicklung besteht darin, dass diese Arbeit in beiden Fallen nur den Betrag kTlog V,/V2erreicht und daher von derselben Grossenordnung ist als der Mittelwert der kinetischen Energie jedes Freiheitsgrades bei der Temperatur dieses Teils der Maschine. Dieser Punkt ist wohl aber ein mehr formaler, denn es wiirde sich prinzipiell um eine systematisch akkumulierende Abweichung eines reinen Temperturgleichgewichts handeln. Der Hauptpunkt meiner Bemerkungen in Kopenhagen war indessen, dass der Grund [dafur, dass] die Maschine auch ohne Beobachtung nicht wirken kann, ist einfach, dass die Zuruckfuhrung des Stempels zu seinem ursprunglichen Platz in der einen oder der anderen zweier moglichen Stellungen von einer Entropievermehrung der verbundenen Korper begleitet sein muss, weil es sich ja hier um genau dasselbe Problem handelt wie bei der Schatzung der Entropie von einem System bestehend aus einem Molekul, von dem man nur weiss, dass es in einem bestimmten Teil eines Volumens anwesend ist. Ich bin daher immer noch uberzeugt, dass die ganze Argumentation keine weiteren Konklusionen erlaubt als die fur das Verstandnis der thermodynamischen Gesetze wesentliche Komplementaritat zwischen mechanischen und thermodynamischen Begriffen. Wir sind naturlich ganz einig, dass diese Komplementaritat eben beruht auf die kleineren Forderungen zur Kenntnis der kinematischen und dynamischen Variablen, die [die] thermodynamischen Begriffe erlauben, und in dem Sinne sind wir alle einig, dass man es so ausdrucken kann, dass jede Vermehrung der Kenntnis dieser Variablen

PART IV: SELECTED CORRESPONDEKCE ( M A I N L Y 1 9 2 6 - 1 9 3 0 )

mit einer Entropieverminderung verbunden ist, aber die Hinsicht meiner Bemerkungen war eben die Bestrebung uns moglichst klar zu machen, dass solche Aussagen kein fur das Verstandnis der allgemeinen Thermodynamik fremdes Element darstellen. I n der Quantenteorie steht das Problem meines Erachtens (nach) in ganz analoger Weise, und ich bin insbesondere iiberzeugt, dass keine neue Erleuchtung der prinzipiellen Lage durch die Heranziehung thermodynamischer Argumente erreicht werden kann. Bei den prinzipiellen Fragen der Quantenteorie handelt es sich j a gar nicht u m eine Unvollstandigkeit der Beschreibung, aber [sondern] um den prinzipiellen Verzicht auf angewohntes Verlangen von Anschaulichkeit, den diese Beschreibung innehalt. Eben in dieser Hinsicht ist es hier noch mehr wesentlich in der Beschreibung zwischen den Objekten und den Messinstrumenten zu unterscheiden, und die bei der Beobachtung in Frage kommende Irreversibilitat kommt gar nicht direkt in den fur die Beschreibung der Objekte angepassten Formalismus ein, sondern nur primar in die Registrierungsinstrumente fur welche prinzipiell von thermodynamischen Problemen abgesehen werden kann. Dies war eben der Sinn in meinen vielleicht sehr dunklen alten Briefen a n Pauli. Ich sollte sehr froh sein, von Dir zu horen, was Du uber das Voranstehende denkst, und ich hoffe besonders, dass D u mir es klar machen kannst, in welchen Punkten unsere Einstellungen divergieren. Ich selber bin iiberzeugt, dass es [sich] u m Unterschiede der Wortwahl handelt, die ihre Wurzeln haben in den verschiedenen Seiten, von welchen wir uns dem Problem nahern. Ich bitte voraus urn alle Nachsicht mit meiner scheinbaren Unverbesserlichkeit. Mit freundlichen Grussen und den besten Wunschen von uns allen Dein stets

BOHR TO STERN,

[Typewritten]

19 November 1947

G1. Carlsberg, Valby Copenhagen, November 19, 1947.

Dear Stern, Thank you for your letter of October 31. I had hoped t o answer you before you left Zurich but, unfortunately, I have not been quite well in the last weeks. I need not say that I a m deeply interested in the problem and that by all my heart I wish that we - who used t o understand each other so well - shall not fail to agree also about this matter. Of course, I agree that the problem of observation is inseparately connected with the question of entropy and, in particular, that any increase in the accuracy

PART IV: SELECTED CORRESPONDENCE (MAINLY 1 9 2 6 - 1 9 3 0 )

of the knowledge of position and momentum coordinates of a mechanical system will entail a decrease in the entropy to be ascribed to its state. My point is, however, that all such relations are inherently included in the statistical mechanical interpretation of the laws of thermodynamics and, especially, in the mutually exclusive relationship between the consistent application of thermodynamical concepts, on the one hand, and of an exhaustive mechanical description, on the other hand. Also as regards the reasons for the impossibility of the working of a machine like that contemplated by Szilard, I think that we have always been in entire agreement, and my remarks aimed only at the simplest logical analysis of the situation. In such a problem, the way of expression can so easily give rise to misunderstandings and, to my mind, special caution is needed as regards the extent to which thermodynamical arguments have to be applied to all parts of the machine, the correlation of which just implies the compensation of the various apparent gains and losses of entropy. The more subtle question at issue is surely the relations of the observational problems in thermodynamics and quantum mechanics. The principal point would here seem to be that, as far as we can neglect the quantum, acquirement of knowledge about position or momenta should not involve a finite interaction between the objects and the measuring instruments and that we, as indicated above, have only to consider the implication of general thermodynamical principles. It is true that, in quantum mechanics, we have to do with a situation which, in epistemological respects, reminds of the apparent paradoxes in thermodynamics, but to my conviction we have reached so clear an understanding of both fields that, apart from illustrative analogies, no deeper insight in either field can be obtained by their comparison. Such a view does not, of course, imply that we can keep the two fields separate from each other in all problems, but also here, as for example in the distinction between pure and mixed states, we have so far as I can see only to do with a proper transcription of classical mechanical problems into the language of quantum theory. I hope that I have expressed myself more clearly and shall be very eager to hear your reaction to the attitude I have tried to describe. It is possible that I may come to the U.S.A. in the spring, in which case I shall certainly make an effort to get in touch with you so that we can talk of the prospects in physics as well as of the many other matters which are so deeply on the mind of us both. With the kindest regards and best wishes from us all, Yours ever, Niels Bohr

INTRODUCTION

The following documents are mainly from the years 1927-1931, which is the central period covered by the present volume. However, we have included two early documents on thermodynamics and statistical mechanics, since they are alluded to in the Introduction to Part 111. The folders listed below, with one exception, form part of the collection of Bohr Manuscripts in the Niels Bohr Archive. Unless otherwise indicated they are microfilmed under the designation “Bohr MSS” ; the corresponding microfilm number is given for each folder (abbreviation, mf.). The titles of the folders have been assigned by the cataloguers, as have all dates in square brackets. Unbracketed dates are taken from the manuscripts. Numbers in the margin facing an item indicate the pages on which the item is reproduced; they are followed by the letter E if only excerpts are given. Items for which English translations are given are indicated by the letter T and facsimile by the letter F.

I N V E N T O R Y O F R E L E V A N T M A N U S C R I P T S Ih’ T H E N I E L S B O H R A R C H I V E

[32(11-13?11E,T.F

1 Mekaniske Grundlag for Termodynamikken 16 October - 18 December 1912 Handwritten [N. Bohr and Margrethe Bohr], 76 pp., Danish, mf. 4. Lecture notes, entitled “Om det mekaniske Grundlag for Thermodynamikken” (On the Mechanical Foundations of Thermodynamics).

2 The Kinetic Theory of Matter I 15 October - 8 December 1915 Handwritten [N.Bohr], 22 pp., English, mf. 5 . Lecture notes, entitled “Lectures on the Kinetic Theory of Marrer”.

3 Como Lecture I [1927] Carbon copy, 7 p p . , English, mf. 11 Probably typescript of shorthand report of the Como Lecture, given on 16 September 1927. I ~ ~ I H X Y EI , T , F

4 Como Lecture 11 1927 Handwritten [O. Klein, N . Bohr, Margrethe Bohr and an unidentified handwriting], typewritten, and carbon copy, 195 pp., English, Danish, and German, mf. 11. Drafts and notes as preparation for an answer to Campbell (Nature 119 (1927) 779) and for the Como Lecture. Main titles: “Atomteori og Bolgemekanik” (Atomic Theory and Wave Mechanics), “Philosophical Foundations of the Quantum Theory”, “Fundamental Problems of the Quantum Theory”, “Uber die \Yellentheorie des Lichts und der Materie” (On the Wave Theory of Lighr and Matter), and “Zur Frage des begrifflichen Aufbaus der Quantentheorie” (On the Question of the Conceptual Structure of the Quantum Theory). Many pages are dated, carrying dates between 2 July and 13 September 1927 (except for 2 pages, apparently b> mistake dated 1926).

5 The Quantum Postulate and the Recent Development of Atomic Theory 12-13 October 1927 Carbon copy, 12 pp., English, mf. A H Q P no. 36. Manuscript found with a letter from Bohr to C . G . Darwin of 16 October 1927. Darwin Papers, deposited with the American Philosophical Society, Philadelphia, Pennsylvania.

[ ~ o ~ I - ~ IEo ~ I

6 Notes from Solvay Meeting 24-29 October 1927 Handwritten [H.A. Kramers, J.E. Verschaffelt and an unidentified

INVENTORY OF RELEVANT MANUSCRIPTS IN T H E NIELS BOHR ARCHIVE

handwriting], carbon copy, 40 pp., English, French, German, and Dutch, not microfilmed. Notes (partly elaborated) from the 5th Solvay Conference.

7 Wirkungsquanrum und Naturbeschreibung [ 19291 Typewritten, carbon copy, and handwritten [N.Bohr, H.B.G. Casimir and 0. Klein], 25 pp., German, Danish, and English, mf. 12. Notes for Bohr’s paper in the Planck issue of Naturv.issenschaften

8 Kausalitat und Objektivitat [ 19291 Typewritten and handwritten [N.Bohr and 0. Klein], 47 pp., German and Danish, mf. 12. Drafts and notes for unpublished papers. Main titles: “Kausalitat und Objektivitat” (Causality and Objectikity) and “Quantentheorie und Anschaulichkeit” (Quantum Theory and Visualizability). One page is dated 1 1 September 1929.

~

~

~

9 Magnetic ~ l Electron ~ 1929 ~ ~ ~ ~ Typewritten, carbon copy, and handwritten [N. Bohr and H.B.G. Casimir], 14 pp., English and Danish, mf. 12. lncomplete manuscripts of unpublished paper entitled “The Magnetic Electron”.

10 Statistik og Reciprocitet

[1929]

Typewritten and carbon copy with handwritten corrections [N Bohr], 13 pp., Danish, mf. 12. Parts of drafts of unpublished paper. On the envelope Casimir has uritren: “Statistik og Reciprocitet” (Statistics and Reciprocity).

11 Atomteoriens Principper 1929 Handwritten [H.B.G. Casimir], 5 pp., Danish, mf. 12. Outline and notes for the lecture to the Scandinavian Scientists in August 1929 (Principles of the Atomic Theory).

12 Kvanteteorien og de klassiske fysiske Teorier 1930 Typewritten and handwritten [C. Moller], 56 pp., Danish, mf. 12. Notes for 3 lectures on “The Quantum Theory and the Classical Physical Theories” gi\en to the Danish Physical Society on 10 February, 24 February, and 24 Varch 1930.

I N V E N T O R Y OF RELEVANT M A N U S C R I P T S I N T H E NIELS BOHR A R C H I V E

13 Rum- Tidsbeskrivelse og Bevarelsessztninger 1930- 1931 Typewritten, carbon copy, and handwritten [Margrethe Bohr, N. Bohr, 0. Klein and Betty Schultz], 74 pp., Danish, German, and English, mf. 12. Notes, drafts, and calculations dated between 31 March 1930 and 12 March 1931. Main titles: “Iagttagelsesproblemet” (The Problem of Observation), “Zur Begrenzung der Zeit-Raumbeschreibung in der Atomtheorie” (On the Limitation of the Time-Space Description in the Atomic Theory), and “Space-Time Coordination and Conservation Principles”.

14 Cambridge Lectures 1930 Typewritten with handwritten corrections and formulae [unidentified handwriting], 54 pp., English, mf. 12. Notes for 3 lectures on “The Principles of Atomic Theory”, given at the Cavendish Laboratory on 28 April, 29 April, and 2 May 1930.

15 Faraday Lecture 1930 lypewritten with carbon copy (12 + 11 pp.), and one handwritten page [H.B.G. Casimir], 24 pp., English and Danish, mf. 12. Probably typescript of shorthand report of the Faraday Lecture, given to the Chemical Society in London on 8 May 1930, on the award of the Faraday Medal. The handwritten sheet is dated 3 October 1930.

16 Philosophical Aspects of Atomic Theory 1930 Typewritten with carbon copy (25 + 24 pp.), and one handwritten page [Margrethe Bohr], 50 pp., English, mf. 12. Probably typescript of shorthand report of lecture to the Royal Society of Edinburgh, given on 26 May 1930, on the award of the James Scott Prize. The handwritten page is an outline for the lecture.

17 Das Wirkungsquantum 1930 Typewritten, carbon copy, and handwritten [N.Bohr, 0. Klein and Betty Schultz], 75 pp., German and Danish, mf. 12. Typescript of shorthand report and elaboration of lecture, “Zum Problem des Wirkungsquantums” (On the Problem of the Quantum of Action), given to the Deutsche Physikalische Gesellschaft on 20 June 1930, on the auard of the Planck Medal. Many pages are dated, carrying dates between 30 June and 3 September 1930.

IKVENTORY OF RELEVANT MANUSCRIPTS IN T H E NIELS BOHR ARCHIVE

18 Solvuy Conference [October 19301 Typewritten, carbon copy, and handwritten [N. Bohr and J.E. Verschaffelt], 42 pp., English, French, and German, mf. 12. Records of the discussion at the 6th Solvay Conference in 1930. However, part of the material seems to belong to the Solvay Conference of 1933. [36 I1-[370]

19 Bristol Lecture 1931 Typewritten with handwritten figures, formulae and corrections, 14 pp., English, mf. 12. Manuscript of H.H. Wills Memorial Lecture, entitled “Space-Time-Continuity and Atomic Physics”, given at the University of Bristol o n 5 October 1931.

INDEX

Subjects which appear throughout the volume, such as causality, complementarity, quantum mechanics and uncertainty principle, are not listed (but, e.g., matrix mechanics and wave mechanics are). Parentheses have been used to indicate a cursory reference t o a subject which is otherwise treated more extensively. This will often, though not necessarily, coincide with a case where the subject appears in a footnote or has been listed under a heading which differs from the term actually used in the text. It is hoped that the many cross references will also help the reader identify such subjects. A n italicized page number indicates a biographical note or a reference to a biographical note in another volume.

IKDEX

absorption of charged particles 76, 82, 131, 156, 375 of radiation (13), 40, 140, 382, 395 f., 417, (460) action at a distance 102 classicaVmacroscopic 205, 214, 233, 250, 261, 282, 381 principle 204, 210 action-angle variables 47, 49, 165 f . , 178, 463-464, 465 actor/spectator XXII, 196. 235, 253 adiabatic principleiprocess 127, 154, 345, 389, 423 Adler, H . (24), (424), (449), 451 alkali specrra 385, 429 f . alkaline earths, spectra of 385 f., 395 Allen, H.S. 51 wdecay 230-231, 246, 376, 378, 405 f f . wrays, scattering of 376, 379, 398, 403, 405 amplitude function, see wave function analogy X X I V , X X V I , 35, 49, 59, 61, 63,93, (95), 123, 141, 153, 155, (157), 158, 167, (310), 316, (320), 397, 406, 434, 466, 470 analogy between material particles and radiation 28, 70-71, 92, ( 1 16), 149, 264, 287, 394, 423 between quantum theory and biology/psychology 189. 196, (204), 206, (212), 216 f . , 272, 298, 352 between quantum theory and thermodynamics 328 f . , 400, 453, 469, 472 f . Hamilton’s, between mechanics and optics 1 1 1 , (150), 204, 210, 264, 287 angular momentum (see also spin) 52, 145, 158, 307, 310, 348 f . , 392, 394, 404 Arrhenius, S. 374 Aston, F . W . 145, 378 Aston’s rule 404 atomic constitution/structure 56, 75, 80, 82, 88, 94, 123 f . , 136, 152, 155, 158,204,211,224ff., 237, 239, 242 f., (262), 263, 266, (274), (284), 285, 290, (299), (317), 359, (371), 373 f f . , 381, 383, 385 f . , 388, 392, 395 f . , 399 f . , 407, 457 f . model/picture 52, 226, 239, 359, (376), 377, 379, 381, 383 ff., 392

number 224 f . , 228, 237 f . , 243, 379, 385, 387 f f . stability, see stability of atoms srationar) states weight 375, 377 f.. 387, 404 attenuation of radiation, see absorption of radiation Balmer, J . J . 228, 242, 382 Balmer formula 383 f f . , 388 band spectra 395-398, 404, 439 f . Barkla, C . G . 386 barrier penetration 230-231, 246, 405 Berlin ( X I I ) , 24, (46), 48, 317, 421, (456), (463), 465 Berlin ;\cadem! (xi]), 21 C., 317, 419 f.. 480 Lecture, Bohr’s, 1930 317, 480 Besso, R.1. 8 &decay 25, 309, 376, 378, 404, 406 f . . 425. 442 f., 445, 446-447 biology ( X X I ) , 196, (206), (216), 234-235, 251-253, 272-273, 274-275, 298-299, 300-301, 352 Bjerrum, N . 308, 396, 442, 444 black-body radiation (lo), 203, 208, 227, 231, 382, 398, (423) Bloch, F. 25, 425 Bohr, Aage V I I , 330, 414, 449, 469 Bohr, Christian (Niels Bohr’s father) x\ I I ff. Bohr, Christian (Niels Bohr’s son) 24, 324 Bohr, Ellen X V I I I , 451 Bohr, Hans 24, 424 Bohr, Harald X V I I , (xx), X X I I I , X X I V . 28, 3 2 , 308, 435, 436 Bohr, Margrethe XXII, xxv, 7, 12, 23, 5 8 , 90, 422, 424, 459, 461, 469, 478, 480 Bohr-Kramers-Slater theory 37, 39, (60). ( 6 2 ) , (65), (70), (115), 139 f . , (148), (191). 317, (43 1) Boltwood, B.B. 377 Boltzmann, L . 320, 322 ff., 327, 359, 400, 352 Boltzmann statistics 143 Boltzmann’s H-theorem (see also entrop:, . irreversibility nnd second la\\ of thermodynamics) (320 f f . ) , 438, 440 Bonhoeffer, K.F. 397 Born,M,3f.,Xf., 1 2 , 3 0 , 3 6 , 3 7 f f . , 5 2 , 5 1 . 9 ~ f ~ . .

INDEX

103, 124f., 127, 137-139, 142, 153, 164f., 170, 230, 245, 393, 415, 417, 422 f., 459 Bose, S.N. 144 Bose-Einstein statistics (50), 144, (397), 398, 404 f . , (466) Bothe, W. 139, 185 Bothe-Geiger experiment 17, 102, ( 1 15), 139 f . , (148), 185 Bragg, W . H . 387 Bragg, W . L . 36, 37, 100, 387, 415 Breit, G. 181 ff. Brillouin, L. 36, 37, 318, 333, 336, 339, 347, 415 Bristol Lecture, Bohr’s, 1931 317, 319, 361370, 481 Bristish Association for the Advancement of Science 7, 317, 355-356, 374 Broglie, L. de (see also de Broglie wave) 3 1, 36, 37,39, 5 3 , 6 0 , 6 2 , 6 4 , 70 f . , 77 f . , 84 f . , 92, 96, 102, 116 f., 126, 137, 149, 151, 153, 184,229,245,264 f., 287 f., 333-334,359, 393 f., 415, 417, 439, 441 Brussels 35, 37, 50, 99, 191, 310 ff., 338, 415, 43 1 , 467 Buddha xx Bury, C.R. 390 Cambridge (XII), (xv), 44 f . , 277 f . , 317, 357, 359 f . , 377, 438, 449 f . , 457 f . , 480 Cambridge Lectures, Bohr’s, 1922 458 Lecture, Bohr’s, 1925 458 Lectures, Bohr’s, 1930 317, 480 Lecture, Bohr’s. 1947 449 f . Campbell, N . R . 4 , 2 6 , 2 8 f., 31 f., 69, 131, 156, 478 canonical distribution/ensemble 325 f . , 467, 469 f . capture of electrons by 0-particles 376, 458 Casimir, H . B . G . 304, 307, 332, 426 f., 430, 479 f . cathode rays 224, 237, 386 Cavendish Laboratory 317, 360, 405, 458, 480 Chadwick, J . 330, 379 Chemical Society, London (XII), (xv), 317, 371, 373, 480 Chievitz, 0. 442, 444, 449, 451 choice of nature, see free choice

Chou, P.-Y., 304, 308 Christiansen, C . xx f . coherence, see phase collisions 3 4 , 4 0 , 7 1 , 9 7 , 105, 112, 119 f . , 127 f . , 131, 135, 138, I50 f . , 154 f f . , 163 f . , 166, 204, 21 1 , 228, 242-243, 3 1 I , 322, 324, 382-383, 407, 417, 434, 449 collisions, inverse/of the second kind 383, 405 combination principle (see also spectral l a w ) 153, 382 f . , 388 f . , 391, 396 commutation relation 95 f . , 125, 153, 161, 163, 165 f . , 168, 179 Como Conference, 1927 5, 29 f f . , (37), (44), 98, 109 f . , 113, 147 f . , 264, 287 Lecture, Bohr’s, I927 V I I , 4, 7 f., 21, 26-35, 41-51, 57 f . , (67 ff.), (73 f.), 98, 109-136, 147-158, 197,264-268,287-293, (362), 432-441, 478 Compton, A.H. 36, 37, 100, 136, 139, 158, 415 Compton effect/recoil 17, 19 f . , 37, 39 f . , 69, 7 9 f . , 8 7 f . , 115, 120ff., 139f., 148, 150f., 162-163,165,173,175,186,359,394,416f., 422 Compton-Simon experiment ( 1 15), 139, (148) Condon, E.U. 405 conduction, electric 144, 235, 252, 349, 398 configuration space 32, 34, (98), 102-103, 128, 154, 174, 181, 394, 434 Confucius xx conjugate variables (see also action-angle variables)22,49 f . , (80), (88), 95 f f . , 125 f . , 153 f., 158, 160, 167,204, 210f., 348,420, 466 consciousness 189 f . , 205, (212), 215, 235, 253, (259), 272, (279), 298, 431 constitution of atoms, see atomic constitution/structure of nuclei, see nuclear constitution/structure coordinate space, see configuration space Copenhagen Conference, 1929 304-309 University XXIV, 110, (186), 196 f . , 202, 255 ff., 262, 278, 283, 425, 427 correspondence with classical physics (see also transition from micro- to macromechanics) 13 ff., 59 ff., 65, 80, 88, 94-95, 97, 98, 1 1 1 , 124 f . , 127, 133-134,

INDEX

135 f . , 152-153, 154, 156 f f . , 229, 244, 263-264, 268, 27 1, 275, 286, 292, 296, 301, 307, 314, 384, 389, 391, 393, 396, 398, 400, 402 f., 406, 424, 426 f . , 429 f., 458, 460, 462 Coster, D. 390 Courant, R. 42, 436 covalent bond 392-393, 397 Crookes, W . 374 Curie, M . 36, 37, 318, 375, 415, 423 Dalton, .I.374 Danish Physical Society 10, 316, 479 Darwin, C.G. 4, 30, 32, 90, 112, 157 f . , 304, 305, 308, 310-316, 318,348-349,412,415, 478 Davisson, C . J . 4, 7, 58, 60, 62, 64, 70, 77, 84, 394 Davisson-Germer experiment (see also reflection of electrons) 4, 7, 70, 77, 84, 394 de Broglie wave (see also Broglie, L. de and phase wave) 23, 39, 70 f . , 101-102, 132, 139, 142, 156, 163 f., 167, 177, 229, 245, 264 f., 287 f., 393, 417, 421 Debye, P . 36, 37, 144, 318, 394, 415 Dennison, D.M. 145, 158, 397 Dieke, G.H. 37, 412, 415 diffraction of atoms 132, 156, 167 of electrons 29, 101-102, 163, 190, 334 f f . , 339, 346, 360, 363-364, 367, 394, 431 of light 17 f f . , 23, 137-138, 190, 420, 431 of X-rays 387, 394 dilemma, see paradox/dilemma Dirac, P.A.M. 3 f . , 7 , 9 , 12, 17, 19, 23, 36, 37, 39,44,45, 52,95,97, 100, 103-105, 124 f . , 138, 144, 146, 153, 155 f . , 158, 160, 162, 164 ff., 175, 191, 267 f . , 291 f . , 318, 333, 339, 347, 348, 359, 393 ff., 402, 412, 415, 417, 421 ff., 426 f f . , 432, 439 f., 459, 461, 463 direction of time (see also irreversibility) 46, 50, 327, 438, 440, 452, 466 discontinuity 10, 14, 17, 19, 21, 23, 33, 52, 56, 59, 61, 63, 69, 75, 80 f., 88, 91, 93 f., 97, 114, 124, 127, 142, 144, 148, 151 f., 154, 160 f., 163, 166, 178, 185, 227, 241, 261, 282, 419, 421, 423, 433, 462

disintegration of nucleus, see nuclear transmutations dispersion 3, 13, 17 f., 76, 82, 94, 124, 138, 153, 156, 179, 359, 393, 395, 460 disturbance of object by measurement 40, 91, 104, 114, 148, 165, 171, 181-182, 206, 216, 233, 249, 266, 289, 349, 352, 418 Doppler effect 18, 23, 29, 31, 37, 120 f . , 150 f . , 165, 173, 316, 420 double-slit experiment 15, 365-366 duality/dualism 8, (9), 27, 59, 61, 63, 108 dynamical theory of gases, see statistical mechanics Edinburgh Lecture, Bohr’s, 1930 312, 317, 351-352, 480 Edlen, B. 385, 395 Ehrenfest, P . 36, 37-41, 44, 127, 154, 181 ff., 304, 305, 306 f . , 308, 389, 412, 415-418, 426 f . , 438 eigenfunction/eigenvalue, see proper vibration Einstein, A . xx, 4, 8, 10, 16 f . , 20, 21 f., 29 f., 32, 36, 37 ff., 44, 46, 50 f . , 60, 62, 64 f . , 69, 75 f . , 81, 83, 91 f., 94, 100, 101-103, 114f., 124, 144, 148, 153, 158, 160, 162ff., 178, 184, 191, 205, 213, 227 f., 230, 233, 241, 243, 245, 250, 260, 281, 308. 318, 319, 328, 359, 363, 367 ff., 380, 382, 393, 396, 402, 405, 412, 415 f f . , 418-421, 423, 431, 452 f., 455, 467 Einstein weighing experiment 3 19, 368-369 “Ekliptica” discussion circle XXIV electron capture 376, 458 diameter, classical 401 f f . , 445 f f . diffraction, see diffraction of electrons groups 387-391, (402), (458) interference (see also reflection of electrons) 105, 270, 295, 365-366 orbit/trajectory 13,52,104,112,155, 161 ff., 172 ff., 311, 339 ff., 347 f . , 366, 379 f f . , 384, 388, 392 ff., 422, 460 polarization 309, 347, 349 reflection, see reflection of electrons scattering (see ako reflection of electrons) 105, 309 spin (see also magnetic moment of electron) 3, 7, 52, 158, 305, 307, 310 f . , 337-349,

INDEX

391 f . , 394, 398, 404, 422, 457 theory, classical 136, 157 f . , 268, 292, 348, 359, 395, 401 theory, relativistic (see also relati\istic quantum mechanics) 4, 7, 268, 292, 330, 333, (339), 347-348, 356, 395, 402, 426 f . elementary particle 136, 146. 157-158,209, 231 f . , 247 f . , 265, 267 f . , 288 ff., 401-403, 407 process, see individual/elementary process Elsasser, W .138 emission of radiation 40, 52, 127, 133, 140, 154,156,173,177,226,240,360,380,382ff., 388, 402, 405, 417, 445, 447 ensemble (see also identical systems) 101, 320, 325, 469 f . entrop!, 320 ff., 326 ff., 400, 438, 440, 449 f f . , 467 f f . equicalence between matrix and wave mechanics 3, 8 f . , 96, 126, 422 erzodic hypothesis 320 Eucken, A . 397 Evans, E . J . 385 exchange relation, see commutation relation excitation of atoms/spectra 94, 97 f . , 123 f . , 133, 138, 143, 152, 157, 180, 385 of nuclei 405 exclusion principle 24, 136, 144, 146, 158, 391, 394-395, 397-398, 404, 425 E m e r . F. 192 Fajans, K . 378 Faraday, kl. 360, 373 f . , 407 f. Faraday Lecture, Bohr’s, 1930 316 f . , 319, 328, 330, 371-408, 454, 480 Medal 37 1 , 480 Favrholdt, D. xxii Feld, B.T. 330 Fermi, E. 142-144, 318 Fermi-Dirac statistics (50), 144 f . , 349, (397), 398, 404 f . , (466) ferromagnetism 439 f . Festskrift, Copenhagen Unicersity, 1929 110, 196-197, 202, 255-273, 278, 283, 425-426, 427 field theory 23, 136, 157, 421

fine structure 349, 389, 396, 457 fine-structure constant 3 14, 402, 328 f . Fowler, A . 385 f . Fowler, R . H . 8-9, 14, 36, 37, 312, 415, 421-424, 457 f . Franck, J . 71, 80, 88, 138,228, 242. 382 f.,395 Franck-Hertz experiment 10, 71, (80), (88), (94), (123), 138, 166, 228, 242, 382 Frank, P. 452 free choice 45 f., 104 ff., 205, 212, 228, 233, 261, 267, 272, 282, 291, 297, 370, 382, 300, 471 energy 326 f . , 449, 453 will 206, 215-217, 220 f . , 234. 250-251, 275-276, 302 frequency condition, Bohr’s, see postulates, Bohr’s Fues, E. 304. 462 f . -,-ray microscope, see microscope Gamow, G . 25, 304, 308, 405 Geiger, H . 139, 185, 379, 405, 342, 444 Geiger-Bothe experiment 17, 102, ( 1 15), 139 E.. (148), 185 generalrelativity 136, 157,260,281,(328),368f.. (453) generalization 46, 59, 61, 63, 78, 80, 85. 88, 105, 116, 125, 335, 149, 152, I57 f . . 164-165, 168, 203, 208, 229, 244, 261, 271, 282, 297, 359, 379, 469 Gerlach, W . 318, 333 f . Germer, L . H . 4, 7, 77, 84, 394 Gibbs, J . W . 320, 324-325, 400, 370 g-factor 310, 349 Goethe, J . W . vi, xxi Gottingen 9, 30, 42, 325, 436 Goudsmit, S.A. 3, 37, 52, 136, 158, 304, 305, 392, 412, 415, 457 f . gravitation 157 f . , 260, 281, 368-369 group structure of electron configuration 387-391, (4021, (458) velocity 23, 29, 31, 76 ff., 84 f . , I17 f., 149, 42 1 Gurney, R . W . 405 gyromagnetic factor 310, 349 hafnium 390-391

Hamburg (32), 35, 41 f f . , 326 f . , (432), 435 f f . , 449, 452 Hamilton, W . R . 204, (210), 264, 287 Hamiltonian 128, 154, 162, 348 Hamilton’s analogy betmeen mechanics and optics 11 1 , (150), 204, 210, 264, 287 equations 95, 125, 153, 174 f. harmonic oscillator in quantum mechanics 134, 157, 173 Hartree, D.R. 395 Harvard 452 heat radiation, see black-body radiation Hecke, E. 194, 445, 447 Hegel, G.U’.F. xx Heisenberg, E . 25 Heisenberg, W . v f . , xv, xx, 3 f . , 7 f f . , 10-12, 15-20, 21 f f . , 29, 32 f . , 36, 37, 39, 44, 45, 47,51 f.,60,62,65,71,75,80,82,88,93ff., 100, 103, 105-106, 111, 116, 120, 123 f f . , 132ff., 136f., 140,141-142,144,145,149f., 152 f . , 155 ff., 159-186, 204, 211, 229, 232, 244, 248, 263 ff., 285 f., 289, 314 f., 318, 324-326, 359, 364, 393, 395, 397, 399, 402, 412, 415 f f . , 424-425, 429 f., 433, 439 f . , 446 f f . , 457, 463 Heisenberg resonance 136, 158, (395), (397) Heitler, W .304, 305, 397, 439 f. helium ionization potential of 392, 395 metastable state of 395 ortho- and para- 145, 395, 397 spectrum 52, 145, 385 f . , 392, 395, 398 Helmholtz, H. von 374, 408 Hermann, A . 10, 41 Hertz, G . 71, 138, 228, 242, 382 Hevesy, G . de 378, 390, 458 Herffding, H . xx-XXI, xxii, xxm, X X I V , 24, 27, 424 Holst, H . 308 homopolar bond 392-393, 397 H o m e r , J . 195 H-theorem (see also entropy, irreversibility and second law of thermodynamics) (320 ff.), 438, 440 Huckel, E . 144, 304 Humboldt, W.von VI Hund, F. 9, 145, 154, 158, 397, 423, 463

Huygens, C . 360 hydrogen ionization potential of 393 molecule 145, 392-393, 397-398 spectrum 3, 52, (138), 379 f . , 383 f f . , 389, 392 Hylleraas, E . A . 395 hyperfine structure 392, 404 ideal gas 47, 50, 143 f., 438, 440, 464, 166 identical atoms/nuclei 145, 397-398, 406-407 particles (see also Bose-Einstein 5tatiStics and Fermi-Dirac statistics) 102, (136), 144 f . , (I%), 193, (392 f . ) , 394 f . , (397 f . ) , 441, 443 systems (see a h ensemble) 13, 460 individual/elementary process 41, 75 f . , 81, 83, 94 f . , 101 f . , 116, 123-124, 134 f . , 139, 149, 152 f . , (156), 157, 228, 242 f . . 261, 263, 282, 285 f . , 327, 359 f . , 381 f . , 384, 388, 393 f . , 400, 405, 453 individual wave, see phase wave interference of electrons (see also reflection of electrons) 105, 270, 295, 365-366 of light 21 f . , (60), (62), (64), 78, 85-86, 92, 115, 117 f . , 148, 150 of wave functions 60,62,64,78,85-86,96 f f . , 104-105, 106, 117 f . , 126, 129, 141, 139 f.. 153, 155 internal conversion 405 intra-nuclear electrons 319, 378, 403 f . , 406 introspection (see also consciousnei5 crnd psychology) 275, 301 inverse collisions 383, 405 ionization potential 392 f . , 395 irrationality 96, 127, 148, 154, 158, 263, 265, 271, 285, 288, 297 irreversibility (see also direction of time, entropy and second law of thermodynamics) 319, 326-329, (382), 400, 449, 452-454, 467-472 isotopes 377-378, 404, 458 Jacobsen, J.C.G. 44, 304, 438, 458 James, W. 27 Jammer, M. X X I I , 27

INDEX

Joffe, A.F. 307 Jordan, P. 3 f., 8,26,28 f., 31,52,69,71,95, 97,124 f., 138,153, 155,158,160,164 ff.,

171,175, 184,393,423 Kaluza, T. 158 Kaluza-Klein theory (14),112, 157-158,422,

(439), (441),(463)

Kapitza, P.318,348,349 Kennard, E . H . 112,157 Kepler’s laws (379),384,(388) Kierkegaard, S.X X I I , 27 kinetic theory, see statistical mechanics Kirchhoff’s law 382 Klein, 0. 14,16,24,26,28,35,43 ff., 58,60, 62,97,127, 155 ff., 191, 194,256, 273, 304,305, 307, 308, 383,422 f., 425 ff., 430,432,435,437 ff., 446 f., 462,478 f f . Klein-Kaluza theory (14), 112, 157-158,422,

(4391,(4411,(463)

Knauer, F.344,347 Knudsen, M.36,37,415 Koch, P.P.41,435 Kossel, W. 387 f., 392 Kramers, H.A. 3,35, 36,37,39,44,80,88,94, 100, 124, 139-140, 153, 196, 197, 256,

273, 304, 305, 308, 318, 393, 412, 415, 417,422,425-430,438 ff., 457,478 Kronig, R. de L. 16,304,305,404,439 f. Kuhn, T.S. VI, 9,324-325 Ladenburg, R. 124,153, 390 Lande, A . 391 Langevin, P.36,318,359 Langmuir, I . 36,37,415 language, rble of xx, 41,(52), 158, 195,205,

214,262,271,283,297, 360,418,448,452 Lao-Tse xx Larmor, J . 359,375 Laue, M.von 387 Leiden 21,44,(415),418 Lemmerich, J. 1 1 Lenard, P.374 Lenz, W. 41,344,435 letters from and to Niels Bohr 412-414 Lewis, G . N . 46,387,393 lifetime 13, 133, 138, 143, 156,405,458,460

line spectra 52, 94, 124, 138, 143, 152 f., 173,

226,228,240,242,311 f.,(333),(349),379f., 382 f f . , 392,395 f . , 457 width 13,(133), 138, 143,156,173,396,460 London, F.397 Lorentz, H . A . 35, 36,37 f., 50, 359,375,393, 415 Lorentz force 309,334 ff., 339,341 f f . Jubilee Meeting, 1925 21,418 Ludwig, K.F.W. XXI Maar, Mrs 24, 194,425,446 f magnetic moment of atom 166,267,291,311,333, 341,348 f.,

392 of electron (see also electron spin) 24, 52, 136,144,158,267,291,307,(309),310 f., 331-336, 337-349, 392, 394-395, 425, 442 ff., 446,479 of nucleus 392,404 of proton (see also spin of proton) 136,158 Main Smith, J . D . 391 Manchester, 320,377 f., 385 Marsden, E.379 Massey, H.S.W. 347 matrix mechanics 3, 8,14 f., 29,31, 33, 39 f . , 52,71,95 ff., 125 ff., 137,152 ff., 157, (180),(184),(393),417 f., 424,433,462 f. matter wave, see wave function Maxwell, J.C. 317,357,359 f., 374 f. Maxwell Centenary Celebrations, 1931 317,357,359 distribution (143),322 ff., 400 Lecture, Bohr’s, 1931 316 f . , 357-360 McLaren, S.B. 52 mechanical theory of heat, see statistical mechanics Meitner, L. 445,447 Mendeleev, D.1. 375,387 Meyenn, K. von 10 Meyer-Abich, K.M. 27 microscope (see also optical instruments)

16-20,29, 31, 37,40, 121, 142, 151 f., 162-163,186,(224),(236),316,417 Millikan, R.A. 3 1 1 , 384,386

INDEX

molecular chaos/disorder 323 f., 439 f . formation/structure (9), 154, 376, 392-393, 394, 395-398, 423, 463 spectra (see also band spectra) 395-398 Maller, C. 47 f., 195, 304, 463 f f . , 479 Maller, P.M. xx, XXII Moseley, H.G.J. 379, 387 f., 390 f. Mott, N.F. 305, 309, 341, 346 f., 398 Munich 10, 17, 325 Neumann, J. von 138, 327, 453 Newton, I . 359 f . , 374 Nielsen, J . 308 Nielsen, J . Rud 195, 278 Nobel Lecture, Bohr’s, 1922 262, 284 Nordheim, L. 304, 306, 308 normal state 177 f f . , 311 f., 383 f . , 386, 395 nuclear constitution/structure 158, 3 19, 330, 356, 360, 377 f . , 381, 403-407, (458) stability 145, 225, 238, 403, (406 f.) statistics 330, 404, 406 transmutations (see also 0-decay) 225, 228, 230-231,238-239,243,246,375-376, 378, 403-407, 445, 447 Nuttall, J.M. 405 objectivity (see also subjectivity and subjectobject distinction/relation) 190, (192), 196,204 f., 209,213,233,249 f., 259,279, 431, (479) observability (21 ff.), 24, 32, 37, (40), 47, 49, (52), (70), 80, 88, 92, (95), (97 f.), 104, 116, 125, (127), (129 f.), 132, (148), 149, (153 ff.), 156, (157), (204), (209), (224), (236), (260), (281), 305, 310, (315), (326), (333), (339), 343, 346 f., (418 f.), (421), 425, (450), 464, 466 operator 128, 138, 154, (169), 175, 461 Oppenheimer, J.R. 138, 446 f . optical instruments (see also microscope) 22, (71), 77,84,93,98, 118, 120f., 132, lSOf., 156, 224, 236, 419 Orsted, H.C. 374 Oseen, C.W. 189, 192, 322, 324, 413, 430-432 paradox/dilemma

XXI,

15, 21 f., 45 f., 50, 52,

69 ff., 76, 80, 82, 88, 92, 96, 98, 116, 119, (125),126,131,139,(148), 149f., 153,155f., 182 f . , 190, 204, 211, 227, 232, 242, 248, 265:269,288,294,322,324f., 327 f f . , 365 f . , 368 f., 419 f., 430, 453 ff., 466, 473 paramagnetism 34, 144, 349, 434 partition between subject and object, see subject-object distinction/relation Paschen, F. 386 Paschen-Back effect 391 Pauli, W. V I , 3, 8 ff., 16 f., 19, 24, 30, 31, 32-35, 36, 37, 41 ff., 46, 53, 96, 125, 136, 144, 146, 153, 158, 162, 164, 166, 177, 181, 191 ff., 304,305,306,307,308,309f., 318, 326-330, 337-347, 391 f., 402, 411, 413, 415, 422, 425, 431, 432-456, 467 f., 472 Pauli principle, see exclusion principle Peierls, R.E. 330 Pei-Yien Chou 304, 308 penetration of charged particles 76, 82, 131, 156, 375 periodic system of elements 136, 158, 224, (228), 237, (243), 333, 375, 378 f . , (383), 386 f., 390 f f . permanent state 177 ff., 311 f., 383 f . , 386, 395 perturbation, see disturbance of object by measurement Petzoldt, J. 192 phase 20, 37, 49, 60, 62, 64, 79, 86, 98, 106, 118, 132,(133), 149f.. 156, 164ff., 170ff., 179ff., l86,193,270,295,439ff.,443,466 phase space 143, 163, 172, 320, 467, 470 velocity 29, 31, 76 ff., 84 f., 118, 149 wave (see also de Broglie wave) 77 f., 84 f., 92,96,117,126, 129,132, 149f., 153, 155f. photoelectric effect 10, 23, 69, 115, 148, 162-163, 227, 241, 270, 295, 420 Pickering, E.C. 385 Planck, M. (21), 36, 37, (39), 47, 50, 69, 113, 178, 189, 192, 193, 194, 203 f f . , 208 ff., 214, 217, 227, 233, 241 f., 250, 261, 268, 282,293, 359, 374, 380 f . , 413,415,442 f . , 456-457, 463, 467 Planck Jubilee Issue, 1929 189-195, (196 f.), 201-217,268,293,309,441 ff.,445, 447 f., 456, 479

INDEX

Medal 317, 480 Planck-Einstein relations, see q u a n t u m relations Planck’s formula/la\v 10, (178), (227), (2411, (363), 382, 398, 423 polarization of electrons 309, 347, 349 postulates Bohr’s 70, 80, 88, 143, 162, 178, 263, 285, 381-382, 383 f . , 393 of q u a n t u m mechanics (see also q u a n t u m postulate) 14 f . , 70 f., 423 f . Preunische Akademie, Berlin (XII),21 f., 317, 419 f . , 480 probability amplitude/function (34), (l05), 139, (142), 164 f f . , 168, 171, 175 f . , (178), 181, 270, 295, (393 f.), (423), (434) proper vibration 12, (13),(33), 34,49,70f., 96f., 126-127, 129, 132, 134 f., 154 f f . , 171 f . , 174, 177 f . , 181, (326), (423), (433), 434, 439 f . , (450), 460, 466 Proust, J . L . 378 Przibram, K . 50 psychology (27), 189, (194), 204-206, 212-217, 220-221,233-234,250-251,259, 269,272, 275-276, 279, 293, 298, 301-302, 352, ( 4 4 9 , (447)

224 f . , 228, 230, 237 f f . . 213. 246. 375-376, 377 f f . , 404, 405-407 R a m a n effect 393 Ramsauer effect 163 Ramsay, W . 376 Rayleigh, Lord 76. 82, 1 1 I , 150, 396 reality of stationary stater 13 f . , 32, 35, 49, 98, 130-135, 355-157, 267, 290, (423), 424, 434 reciprocity 192-194, 202, 204 f . , 210 Ef., 214 f . , 271, 296-297, 441, 343 f f . , 479 reduction of wa\e packet 33, 93, 112, 174, 333 reflecrion of electrons (see also Da: iison-Germer experiment, diffraction of electrons and interference of electrons) 70, 77, 84, 92, 116, 149, 176-177, 230, 245, 265, 288, 334 of X-rays (see also diffraction of X-rays) 7 7 , 84 relativistic electron theorq, see electron rheory, relatibistic many-body problem 35, 435 q u a n t u m mechanics (see also electron theory, relativirtic) (190 f,), (233), (249), (267-268), (292), 314, 333,339,302,426 f., (4301, (432) relativity,generall36,157,260,281, (3283,368f.. quantum electrodynamics 4, 95, 146, (155), 156, 359, (453) (402), 439 f . , 446 f . resolving power, 121, 150 f . resonance (see also Heisenberg resonance) 13. mechanics, relativistic, see relativistic q u a n 97, 127, 154, 460 t u m mechanics resonance fluorescence/radiation 20, 133, 156. number 26, 35, 49, 126, 134 f . , 154 f . , 157, 179-181, 186 173, 228, 230, 243, 245, 388 f . , 391, 394, reversibility, see irre\ erribilit) 397, 435, 466 Richardson, O . W . 36, 37, 311, 318, 349, 415 postulate (4 f.), (7), (30), (44),48, 52, 70, 75, Riemann, G . F . B . X X I V 81, (89), 91, 93 f f . , (107), 108, (109 f.), Riemann surface X X V I (113), 114,116, 119f.,122ff.,130f.,134f., Ritz, W . 228, 242, 382 (147), 148 f f . , 157 f . , 190, 200, 229, 232, R o m e Conference, 1931 319 244, 248, 265, 288, (393), 431, 465, (478) Rontgen rays, see X-rays relations21 f.,39,59,62,64,69f.,76ff.,83f., Rosenfeld, L. v, X V I I , X X V I , 27 f . , 30, 304, 305, 86,92,117,119,126,129,132,149f.,154ff., 306, 307, 309, 310, 320 363, (382), 388, 392 f . , 416, 419 Rosseland, S . 304, 383, 405 statistics 50, 144 f., 349, 397 f . , 404 f . , 466 Rossi, R . 377 rotator, quantized 182-183, 396 f . , 426 f. radiation damping/reaction 13, 133, 138, Royal Danish Academy V I I I , (XII), X X I ,4, 41, 142-143, 156, 177, 401 f . , 460 55-56, 107-108, 199-200, 317, 353-354 radioactivity (see also nuclear transmutations)

Rohal Society of Edinburgh 317, 351-352, 480 Rozental, S . X I X Rubin, E . xxvi Rud Nielsen, J . 195, 278 Ru5sel1, A.S. 377 f. Rutherford, E. 8, 52, 76, 82, 138, 225, 238, 373, 375 f f . , 379, 381, 387, 405, 413, 422, 424. 457-458 Rydberg, J . R . 228, 242, 382, 385, 387 Saha, \ I . 16 S a m p i o n , R . 4 . 312 Scandinai ian hlathematici Congress, 1925 ( 3 ) , ( 7 ) , 263, 285 Scientist