Advances in
IMAGING AND ELECTRON PHYSICS VOLUME
154 DIRAC ’S DIFFERENCE EQUATION AND THE PHYSICS OF FINITE DIFFERENCES
EDITORINCHIEF
PETER W. HAWKES CEMESCNRS Toulouse, France
HONORARY ASSOCIATE EDITORS
TOM MULVEY BENJAMIN KAZAN
Advances in
IMAGING AND ELECTRON PHYSICS VOLUME
154 DIRAC ’S DIFFERENCE EQUATION AND THE PHYSICS OF FINITE DIFFERENCES H ENNING F. HARMUTH Retired, The Catholic University of America Washington DC, USA
BEATE MEFFERT HumboldtUniversität Berlin, Germany
AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier
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10 9 8 7 6 5 4 3 2 1
To the memory of Max Planck (1858–1947) Founder of quantum physics and distinguished participant of the Morgenthau Plan, 1945–1948.
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CONTENTS
Preface Future contributions Foreword List of frequently used symbols
1 Introduction 1.1 1.2 1.3 1.4 1.5
Maxwell’s Equations with Magnetic Dipole Currents Lagrange Function Basic Concepts of the Calculus of Finite Differences Finite Differences and Spatial Dimensions Curved Space in a Difference Theory
2 Modified Dirac Equation 2.1 2.2 2.3 2.4 2.5 2.6
3
Differential Equation with Magnetic Current Density Modified Dirac Difference Equation Solution of the Difference Equation for :x0 Time Variation of :01 ([, R) Hamiltonian Formalism and Quantization Finite Limit of the Period Number L
ix xi xv xvii
1 1 9 15 22 37
47 47 58 68 79 91 99
Inhomogeneous Dirac Difference Equation
108
3.1 3.2 3.3 3.4
108 115 132
Inhomogeneous Equation (2.233) Resolution %x h/m0c Quantization of the Solution Evaluation of the Energy Û for Small Distances %x
140
Equations are numbered consecutively within each of Sections 1.1 to 6.11. Reference to an equation in a different section is made by writing the number of the section in front of the number of the equation, e.g., Eq.(1.145) for Eq.(45) in Section 1.1. Illustrations are numbered consecutively within each section, with the number of the section given first, e.g., Figure 1.11. References are listed by the name of the author(s), the year of publication, and a lowercase Latin letter if more than one reference by the same author(s) is listed for that year.
vii
viii
Contents
4 Dirac Difference Equation in Spherical Coordinates 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8
5
158
Electron in an Electromagnetic Field Relativistic Lagrange Function Quantization of the Homogeneous Equation Unbounded Electron in a Coulomb Field AntiParticles With s0 Near s = –1 Solutions for s0 Near s = i Solutions with s0 in the Neighborhood of s = –i Energy or Mass Ratios for E/m0c 2 >1 and %rmin
158 163 171 179 184 190 201 210
Inhomogeneous Equations for Coulomb Potential
222
5.1 5.2 5.3
222 232 235
Quantization of the Inhomogeneous Term Separation of the Functions :1j (S, R) Solutions for v(R) and w(S)
6 Appendix 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11
Calculations for Section 2.3 Calculations for Section 3.2 Inhomogeneous Difference Equations Further Elaboration of Eq.(2.128) Quantization of Lcr2 to LcK5 Polynomials as Solutions of Difference Equations of Second Order Separation of Variables in Section 4.3 Solution of D1(S) and D4(S) Calculation of C1v and C4v in Section 6.7 Slanted Coordinate Systems With Finite Differences Riemann Manifolds and Bended EigenCoordinates
241 241 243 247 258 260 262 271 279 293 299 301
REFERENCES AND BIBLIOGRAPHY
310
INDEX
316
PREFACE
It is a pleasure to welcome this ninth contribution to these Advances, or their forerunner Advances in Electronics & Electron Physics, by Henning F. Harmuth, of which the most recent have been written in collaboration with Beate Meffert, a professor in Berlin, whom Harmuth met some 35 years ago when she was a PhD student in East Germany. Harmuth, now retired, reminds me that his student days coincided with the early development of information theory in electrical engineering; information theory revealed the lack of transient solutions of Maxwell’s equations, with which much of Harmuth’s earlier – and at the time, controversial – work was concerned. The replacement of infinitesimal elements, dx, dt, by finite intervals, %x, %t, eliminated the need for unphysical infinite information from physics. When I wrote prefaces for those early articles or books, I was obliged to adopt a defensive tone, for Harmuth’s early work was severely criticized and it is gratifying, looking back, to recall that much of his highly original work first appeared in these pages. A special issue of Electromagnetic Phenomena (Vol. 7, No. 1, 2007, see www.emph.com.ua), dedicated to Harmuth, contains a full curriculum vitae. The present volume is the third and (according to the author) last of a trilogy, of which the two preceding volumes form volumes 129 and 137 of these Advances. I have no doubt that it will be widely appreciated. Peter W. Hawkes
ix
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FUTURE CONTRIBUTIONS
S. Ando Gradient operators and edge and corner detection V. Argyriou and M. Petrou (Vol. 156) Photometric stereo: an overview W. Bacsa Optical interference near surfaces, subwavelength microscopy and spectroscopic sensors C. Beeli Structures and microscopy of quasicrystals C. Bobisch and R. Möller Ballistic electron microscopy G. Borgefors Distance transforms Z. Bouchal Nondiffracting optical beams F. Brackx, N. de Schepper and F. Sommen (Vol. 156) The Fourier transform in Clifford analysis A. Buchau Boundary element or integral equation methods for static and timedependent problems B. Buchberger Gröbner bases T. Cremer Neutron microscopy N. de Jonge (Vol. 156) Carbon nanotube electron sources for electron microscopes A. X. Falcão The image foresting transform R. G. Forbes Liquid metal ion sources B. J. Ford The earliest microscopical research C. Fredembach Eigenregions for image classification
xi
xii
Future contributions
A. Gölzhäuser Recent advances in electron holography with point sources D. Greenfield and M. Monastyrskii (Vol. 155) Selected problems of computational charged particle optics M. I. Herrera The development of electron microscopy in Spain J. Isenberg Imaging IRtechniques for the characterization of solar cells K. Ishizuka Contrast transfer and crystal images A. Jacobo Intracavity type II secondharmonic generation for image processing L. Kipp Photon sieves G. Kögel Positron microscopy T. Kohashi Spinpolarized scanning electron microscopy R. Leitgeb Fourier domain and time domain optical coherence tomography B. Lencová Modern developments in electron optical calculations H. Lichte New developments in electron holography M. Matsuya Calculation of aberration coefficients using Lie algebra S. McVitie Microscopy of magnetic specimens P. G. Merli and V. Morandi Scanning electron microscopy of thin films M. A. O’Keefe Electron image simulation D. Oulton and H. Owens Colorimetric imaging N. Papamarkos and A. Kesidis The inverse Hough transform K. S. Pedersen, A. Lee and M. Nielsen The Scalespace properties of natural images E. Rau Energy analysers for electron microscopes
Future contributions
xiii
E. Recami and M. ZamboniRached (Vol. 156) Localized waves: a review R. Shimizu, T. Ikuta and Y. Takai Defocus image modulation processing in real time S. Shirai CRT gun design methods T. Soma Focusdeflection systems and their applications I. Talmon Study of complex fluids by transmission electron microscopy M. E. Testorf and M. Fiddy Imaging from scattered electromagnetic fields, investigations into an unsolved problem N. M. Towghi Ip norm optimal filters E. Twerdowski Defocused acoustic transmission microscopy Y. Uchikawa Electron gun optics K. Vaeth and G. Rajeswaran Organic lightemitting arrays M. van Droogenbroeck and M. Buckley Anchors in mathematical morphology M. Yavor Optics of charged particle analysers
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Foreword
The development of information theory has alerted us to the fact that infinite information is as impossible in physics as infinite energy. We cannot observe, process or display infinite information. The conservation law of energy ended centuries of attempts to build a perpetuum mobile and in the process eliminated the concept of infinite energy. Today we use the concept of infinite information whenever we use nondenumerable many infinitesimal intervals dx or dt. It may take some time before the similarity with infinite energy is generally understood. Infinitesimal and nondenumerable have always been outside the realm of the observable. They strictly belong to mathematics, a science of the thinkable. How was it possible that these concepts became so important in physics, a science of the observable? The answer is, of course, the success of differential calculus. The development of quantum theory, the Compton effect, and Heisenberg’s uncertainty relation were not strong enough to shake our believe in differentials. It was probably the most important effect of information theory to make us think of replacing dx, dt by arbitrarily small but finite differences %x, %t. The concept of infinite had to be replaced too, but this was less dramatic. An infinite distance requires only denumerable finite intervals %x. Denumerable may be used as limit for large numbers, which can be observed. But nondenumerable is not a limit for anything observable. Nondenumerable and infinitesimal are like the two sides of a coin, since an infinitesimal distance dx is obtained by dividing a finite distance %x into nondenumerable intervals. That the substitution of finite differences %x, %t for differentials dx, dt could bring new results has been known since Hölder (1887), who proved that differential equations and difference equations define different classes of functions. The calculus of finite differences is as old or older than differential calculus, but their state of development differs enormously. A typical collection of tables of integrals fills about 900 pages (Gradsteyn and Ryzik 1980) while a corresponding table of sums is readily accommodated on a single page. The lack of development of the calculus of finite differences means that much effort has to be devoted to the development of mathematical methods before one can apply them to physics. We regret the resulting difficulty of seeing the physical results obscured by all the mathematics but there is no way around this problem. xv
xvi
Foreword
Two previous books applied the calculus of finite differences to quantum electrodynamics and the KleinGordon equation (Harmuth and Meffert 2003, 2005). Here we extend the theory to Dirac’s equation. As in the case of the KleinGordon equation we obtain solutions for particles with negative mass –m0 that are completely equivalent to the solutions with positive mass +m0. But in addition we obtain solutions for nuclear distances of the order of 10–13 m and less rather than for the usual atomic distances. There are a number of other deviations from the differential theory. For instance, the eigenvalues of an electron in a Coulomb field deviate slightly from those of the differential theory. The deviations are similar to the Lamb shift (Lamb and Retherford 1947). Sections 1.4 and 1.5 show some surprising results for our concept of space caused by the replacement of dx by %x. The beginnings of these results go back to a book in 1989 but they were never published in detail to avoid fruitless controversies. Advancing age called for a decision to publish now or never. The authors want to thank HumboldtUniversität in Berlin for help with computing and library services. We further want to take this last opportunity to express our appreciation for scientific contributions, help, or support to some fellow scientists: K.L. Lukin, S.A. Masalov, G.P. Pochanin, Academy of Sciences of Ukraine, Kharkiv; N.N. Kolchigin and V.A. Katrich, Karazin Kharkiv National University, Kharkiv, Ukraine; T.W. Barrett, Washington DC, USA; the late K.G. Beauchamp, University of Lancaster, Great Britain; V. Bolotov, Institute for Electromagnetic Research, Kharkiv, Ukraine; V. Borisov, St. Petersburg University, Russia; the late Chang Tong, Tsinghua University, Beijing, PR China; P. Hillion, Le Vesinet, France; M.G.M. Hussain, University of Kuwait; I.Ya. Immoreev, Moscow Aviation Institute, Russia; A.G. Luk’yanchuk, Sewastopol National Technical University, Ukraine; A. Patashinski, Northwestern University, Evanston IL, USA; F. Pichler, JohannesKeplerUniversität, Linz, Austria; V. Sugak, Academy of Sciences of Ukraine, Kharkiv; R. Yelf, Georadar Research Pty Ltd, Coffs Harbor NSW, Australia; and Zhang Qishan, Beihang University, Beijing, PR China. Henning F. Harmuth and Beate Meffert *
*
*
This is the sixteenth and last book that I wrote either as single author or – after age 65 – with coauthors. The large number was made possible by the support of four scientific editors: the late Ladislaus L. Marton (Academic Press), the late Richard B. Schulz (IEEE Transactions on Electromagnetic Compatibility), Myron W. Evans (retired from World Scientific Publishers), and Peter W. Hawkes (Elsevier/Academic Press). Scientific editors are as important as authors for the publication of new ideas. Science not published is no better than science not done. Henning F. Harmuth
LIST OF FREQUENTLY USED SYMBOLS Ae Ae Am Am0, Am1 Ãe Ãm1[ B c D E, E E e F ([) g, g[ , gR ge gm g H, H HcL(R) H, H h = = h/2Q I Ir1, I+1 , IK1 Ir , I + , IK IT (L/N) Jr , Ji JR , JI J1 –J8 L, L
As/m As/m Vs/m Vs/m – – Vs/m2 m/s As/m2 V/m VAs As – – A/m2 V/m2 – A/m – – Js Js – – – – – – – –
L, L
–
m0 m m N p1
kg kg – – –
absolute value of the electric vector potential electric vector potential magnetic vector potential Eq.(2.229) Eq.(3.26) Eq.(3.210) magnetic flux density 299 792 458; velocity of light (definition) electric flux density electric field strength Euler Roman Medium E, energy electric charge Eqs.(2.313), (2.324) Eq.(3.426) electric current density magnetic current density Euler Script Medium G, Eq.(3.426) magnetic field strength Eq.(6.329) Euler Script Medium or Bold H; Eqs.(2.11), (1.132) 6.626 075 5 × 10 –34, Planck’s constant 1.054 572 7× 10 –34 Eqs.(5.114), (5.119) Eqs.(5.116), (5.118) Eq.(5.120) Eq.(2.433) Eq.(3.313) Eq.(3.317) Eq.(6.375) Euler Roman Medium or Bold L; Eqs.(2.215), (2.228) Euler Script Medium or Bold L; Eqs.(1.123), (1.124) rest mass variable mass Euler Roman Medium m; Eq.(6.79) T/%t, Eq.(2.26) Eq.(3.27) ( Continued ) xvii
xviii
List of frequently used symbols
pC Q %r s S T t %t U UcL(L) v Z = N/c Z
– – m V/Am – s s s VAs – m/s V/A –
Eq.(3.26) Eq.(1.145) arbitrarily small but finite space interval magnetic conductivity Euler Script Medium S; Eq.(6.615) arbitrarily large but finite time interval time variable arbitrarily small but finite time interval Eq.(2.53) Eq.(2.61) velocity 376.730 314; wave impendance of empty space Euler Roman Medium Z; 1, 2, …; charge number
B Be B C CL, C^L Hr, H+, HK = 1/Zc % % [ R L > –L0 < L0 M1, M2, M3 MC Mc Mr, M+, MK 2 N = Z/c Se Sm T Ge Ge0 , Ge1 Gm Ge1 8 Xm Xp
– – – – – – As/Vm – – – – – – – – m m – – Vs/Am As/m3 Vs/m3 A/Vm V V A – – – –
Ze2/2h 7.297 535×10 – 3, Eq.(1.145) ZecA e/m 0c 2, Eq.(1.145) Bx, By, Bz, matrices, Eqs.(2.18)–(2.110) matrix, Eq.(2.19) Eqs.(2.413), (2.438) Eqs.(5.124), (5.134), (5.143) 1/Nc 2; permittivity symbol for difference quotient: %F/% x, Eq.(1.35) symbol for finite difference: x + %x xj /c%t, normalized distance; Eqs.(2.26), (2.31) t/%t, normalized time; Eq.(2.26) wave number, Eq.(2.352) smallest integer larger than –L0 largest integer smaller than L0 Eq.(2.32) h/m0 c, Compton wavelength =/m 0c = M C/2Q Eq.(4.322) Eq.(2.411) 4Q × 10 – 7; permeability electric charge density magnetic charge density electric conductivity, Eq.(1.17) electric scalar potential Eq.(2.229) magnetic scalar potential Eq.(3.29) Eq.(5.211) Eq.(6.85) Eq.(6.86)
1 Introduction 1 . 1 MAXWELL EQUATIONS WITH MAGNETIC DIPOLECUR.R.ENTS Maxwell's equations always permitted electric dipole currents caused by ind11c:ed or rotating electric dipoles. Most molecules form rotating electric dipoles. T h e large dielectric constant of bariumtitanate is due to rotating dipoles. Without electric dipoles no electric current could flow through capacitors since their dielectric is an insulator for electric monopole currents. T h e existence of electric dipole currents is obscured in Maxwell's equations by having onc clcctric current density term that stands for monopole, dipole, or higher order multipole currents. Dipole currents were not understood in Maxwell's days. For instance, we read much about Ohm's law for monopole currents but never about Ohm's law or some equivalent for dipole c~irrents.This explains why the similarity between electric: dipole currents due to rotating electric dipoles and magnetic dipole currents due t o rotating magnetic dipoles was riot recognized. The lack of a magnetic (dipole) current, density in Maxwell's equations was never fully accepted. Dirac (1931) attempted to add magnetic monopole currents to Maxwell's equations for very convincing theoretical reasons. But the lack of reliably observed magnetic rrlonopoles or charges proved to be an insurrnouritable obstacle. A new stimulus for the irivestigation of a magnetic current density term was provided by the development of electromagnetic communication with deeply submerged submarines (Merril 1974). T h e original Maxwell equations led to communicatiori systems that permitted a transmission rate of information of about 5 bit/s (=one teletype letter per second). The inability of Maxwell's equations to describe the propagatiori of signals through seawater with its high olirriic losses was recognized. The addition of a magnetic current density term overcame the problem and showed that the transmission rate of information could be increased significantly by largerelativebandwidth techniques1 (Harmuth 1986a, b, c, d; Harniuth, Boules, Hussain 1999). It was known that one obtained different differential equations as well as different solutions if one assumed no niagnetic current density a t the 'The term "ultra wide bandwidth" is frequently used for large relative bandwidth, which shows the principle is still not generally understood. The bandwidth for submarine communication is in the order of 50 Hz, which is hardly ultra wide.
1
1 INTRODUCTION
FIG.1.11. Monopole currents carried by independent positive and negative electric or magnetic charges (a). Dipole current due to an induced dipole (b). Dipole current due to orientation polarization of inherent dipoles (c); the change of the signs of the charges makes the currents ih and i, flow in the directions shown.
beginning of the calculation, or made an initially assumed current density zero a t the erld of the calculatio11. This was always taken as a warning sign that something was not understood. But it required several years before it was recognized that this strange result was caused by not distinguishing between monopole, dipole, and higher order multipole currents (Anastasovski et al. 2001). At about the same time a mathematician i ill ion^ 1991, 1992a, b; 1993) realized that Maxwell's equations belong to a class of differential equations that do riot permit independent initial and boundary conditions. If initial and boundary conditions cannot be chosen independently one cannot get solutions that satisfy the causality law, since a boundary condition a t the time t > 0 could affect an initial condition a t the time t = 0. The addition of a magnetic current density term to Maxwell's equations, either for monopole or dipole currents, yields equations that generally can satisfy the causality law and thus yield signal solutions. Signal solutions are zero before a certain finite time and have finite energy. Matllenlaticians call such solutions quadratically integrable causal functions. For examples of rrionopole, induced dipole, and inherent dipole currents refer to Fig.l.11. In Fig.l.1la we see independent negative arid positive charges that are pulled up or down by a field strength denoted by and . This produces monopole currents. Only electric charges and field strengths are accepted to have been reliably observed to produce this effect. Figure 1.1lb shows an induced dipole that nlay represent a hydrogen atom. An electric field strength polarizes the atorn by pulling the electron up and the proton down. A restoring force is depicted by a coil spring. A dipole
+
2 ~ i l l i o nobtained his results earlier than implied by the dates of his publications. However, he could not overcome the peer review which implied the causality law could not be important in electrodynamics if Maxwell's equations had worked without it for a
century.
3
1.1 MAXWELL EQUATIONS WITH MAGNETIC DIPOLE CURR.ENTS
current flows as long as electron and proton move either apart or back together again. The resulting induced dipole current comes to an end if the applied field strength is below the ionizing field strength; otherwise the dipole currcnt bccomes a monopole current. Only elcctric charges and field strengths are currently known to produce induced dipole currents. Figure 1.1lc shows two inherent dipoles represented by charges and  at the cnds of a rigid rod. Most molecules arc inhcrcnt clcctric dipoles. All known magnetic dipoles are inherent dipoles. An applied field strength makes the dipoles rotate to produce orientation polarization. Vertical currents i,, and horizontal currents ih are produced by the rotation. The horizontal currents ih will cancel in a random mixture of inherent dipoles but the vertical currents will add. If the field strength is suddenly reduced to zero the dipole current will not be reversed as in the case of the induced dipole of Fig.l.1lb but stop. This creates an observable difference between induced and inherent dipoles. We note that molecules that are inherent electric dipoles are also inducible dipoles since the electrons can still be pulled in one direction and the nuclei in the other. Very strong field strengths will not turn a current due to orientation polarization into a monopole current. This is a second observable difference between the two types of dipoles. T he addition of a magnetic dipole current density term to Maxwell's equations removed the problem with the causality law3. The modified equations can be written in the following form with international units in a coordinate system at rest:
+
aD dt dB
+
(1)
+ g m
(2)
c u r l H =  g, curlE=
at
(3)
div D = p, div B = 0
or div B = p, for magnetic monopoles
(4)
An old fashioned notation is used here but it is the notation with which the problem of Maxwell's equations with the causality law was found. It is quite possible that the operators grad, curl, and div are better for physical understanding than V and 0.In Eqs.(l)(4) E and H stand for thc clcctric and magnetic field strength, D and B for the electric and magnetic flux density, g, and gm for the electric and magnetic current density, p, and p, for the electric and a hypothetical magnetic charge density. Thc magnetic current density g, does not depend on the existence of a charge density p,. Equations ( 1 ) (4) are augmented by constitutive equations. In the simplest case we have 3See Harmuth, Barrett, Meffert 2001, Sec.l.1 for details and many references.
1 INTRODUCTION
D = cE, B = pH,
[ ~ s / m= ~ [As/Vm] ] [V/m] [vs/m2] = [Vs/Am][A/m]
(5)
g, = u E ,
[ ~ / r n= ~ [A/Vm] ] [V/rn]
(7)
(6)
g, = sH, [ v / m 2 ]= [V/Am][A/m] (8) where E, p , u , and s are scalar constants called permittivity, permeability, electric conductivity, and magnetic conductivity. We note that a and s may be monopole, dipole, or higher order niultipole conductivities. An electric monopole current cannot flow through a capacitor, which makes the monopole conductivity zero, but a dipole current can flow since the dipole conductivity is not zero. This difference becomes very evident for alternating currents. If E , p , 0 ,s vary with location, time, and direction one must replace the scalar variables in Eqs.(5)(8) by timevariable tensors. In more general cases the equations may be replaced by partial differential equations. If one is satisfied with a periodic sinusoidal time variation of El H, D, B, g,, and g, one may use functions of frequency ~ ( w )p(w), , u(w), and s(w) but this produces a theory outside the conservation law of energy and the causality law. We list a number of relations derived from the modified Maxwell equations (1)(4). Their derivation was publislied in a book3. The electric and magnetic field strengths are related to vector potentials A, and A~ as well as scalar potentials 4, and 4,, where 4, is zero if there are no magnetic monopoles:
For A, = 0 arid 4, = 0 one obtains the equations derivable from Maxwsell's original equations. However, these equations contain a contradiction that calls for A, # 0, while 4, = 0 is acceptable (Harmuth, Barrett, Meffert 2001, Sec. 3.1). T h e vector potentials A, and A, are not completely specified since Eqs.(9) and (10) only define curl A, and curl A,. Two additional conditions can be chosen that we call the extended Lorentz convention:
1.1 MAXWELL
EQUATIONS WITH MAGNETIC DIPOLE CURRENTS
5
The potentials of Eqs.(9) and (10) then satisfy the following inhomogeneous differential equations:
024,
C 1 d24,  ,2 at2  04, = zP,,
4,
= O for p,
=O
(16)
Particular solutiorls of these partial differential equations may be represented by integrals taken over the whole space:
4,
= O for p,
=0
(20)
Here r is the distance between the coordinates E , 7 , C of the current and charge densities and the coordinates x, y, z of the potentials: r = [(x 
E ) ~+ ( ~  l ) )+~(Z  s ) ~ ]112
(21)
We note that only Eqs.(l)(4) are needed t o derive Eqs.(9)(21), the constitutive equations (5)(8) are not used. T h e modification of Maxwell's equations implies a modification of the Lagrange function and the Haniilton function. The general relativistic form of these rnodified functions is quitc complicatcd. To help with undcrstanding we start with the nonrelativistic equations. The Lorentz equation of motion for a mass m , a charge e , and a velocity v
yields from the original Maxwell equations for v LM (Euler Script Medium L):
> ZecA, and ZA, >> A, we obtain:
+
Maxwell's original equations yield the term (1/2m)(p  eAm)2 e4,. The terms LC,, LC,, LC, are correcting terms. If the simplifying assumption leading to Eqs.(33)(35) are not made one obtains the exact nonrelativistic Harnilton function (Harmuth, Barrett, Meffert 2001, Eq. 3.244):
Terms milltiplied by ( Z e ~ A , / r n c ~or) ~(Ze~A,/rn,c')~have been added to the sirnplified terms of Eqs.(33)(35). The correcting terms LC,, LC,, LC, have riot been changed.
8
1 INTRODUCTION
It all becomes much more complicated when the restriction v > ZecA, and ZA, >> A, we get:
Without the correcting terms LC,, LC,, LC, we have the convetltional relativistic Hamilton function for a charged particle in an electromagnetic field, written with three components rather than one. We call them the zero order approximation in a, = a , ( r , t ) = ZecAe(r,t)/moc2, where a, is a dimensionfree normalization of the magnitude Ae(r, t ) . A first order approximation in a, is provided by the following equations:
The factor Q stands for the expression
and a, rnay be written in the following forms:
1 . 2 LAGRANGE
Ze2 2h,
N =
FUNCTION

3 h 7.297 535 x 10 fine structure constant, Xc = mo c
2.210 x 1 0 5 ~ , ( rt), for electron, A, in As/m a, 1 1.204 x 102Ae(r,t) for proton a,
(45)
We shall use the following simplc and uniquc rulc for thc rcplaccmcnt of noncornmuting factors ab: 1 ab + (ab 2
+ ba)
It was not necessary yet to distinguish between noncommuting and commuting factors but our equations show that this problem will have t o be addressed eventually.
I11 order to use the Hanlilton formalism of Eqs.(l.l41)(1.143) we have to rewrite LC,, LC,, LC, according to Eq.(l.l27) into functions of the moment p rather than the time derivatives x, ?j, i of the coordinates. This requires much effort and leads to very long equations. We refer to the literature (Harmuth, Barrett, Meffert 2001, Sec. 3.2). The equations are simplified by expanding them in powers of a and use only the first order approxiniation as required for Eqs. (1.141)(1.143).Even this simplification requires that LC, of Eq.(l.l27) is broken into five components to produce manageable equations:
Lcx
= Lcz1
+ Lc:r2 + Lcz3 + Lcz4 + LC25
(1)
From Eq.(l.l45) we derive the relations
that we need to show that the components of LC, vary with a in first approximation, as one would expect from Eqs.(l.l41)(1.143). We write them in the following form (Harmuth, Meffert 2005, Eqs. 2.120 t o 2.124):
1 INTRODUCTION
The last three lines of this equatiorl make use of Eq.(l.l46) to prepare for noncommuting factors. The second term LcX2in Eq.(l.l27) becomes:
The third term LczSin Eq.(1.127) becomes:
1.2
LAGRANGE FUNCTION
11
The fourth corriporierit LcrL.4 i11 Eq(l.127) remains uncliariged since there are 110 time derivatives of r , y, or z:
The fifth corriponent Lc,s in Eq.(l.l27) is rather long:
We shall need LCriot only in Cartesian coordinates but also in spherical coordinates. This presentation is found in the literature (Harmuth, Meffert 2005, Secs. 1.3, 6.5). Wc havc now thc matrix
1 INTRODUCTION
c
0
0
with the following elements:
/'
Ze . LC, =  r ( r d ~ , ~ r s i n 6 +Aes)
+Ze c
Ze LC. = rd(r
+
c
C
sin 6 +A,,  +A,,)
+ r sin 19 @A,,
 FA,,
84,
[sin il+=
+ (ir: + +):

d 84, G z
( r sin 6 +A,,  +A,,)
The term L C ,of Eq.(9) is broken into five components, just as LC, was, to obtain manageable equations:
+
Lcr = kcrl kc1.2 + Lc1.3 + Lcr4 + Lcr5 (12) To write these components in terms of the momentum p rather than the time derivatives i, r d , r sin t9+ of the coordinates we need three equations that hold in first order approximati011 of either a, or a (Harmuth, Meffert 2005, Eqs. 5.243 to 5.245):
1.2 LAGR.ANGE FUNCTION
13
With the help of these equatiorls we obtain the first terms of Eqs.(9)(11) using the rrlonlerlt~imp:
&I
Ze .
= r19(r c
sin 6 @A,,  PA,,)
The secor~dterrrls of Eqs.(9) (ll)becorrie in this riotatiorl in first order of a:
Ze
r
C
 eAm),
r sin 19 x (1
+ (P
;GF)~) Ii2?
d19 (20)
14
1 INTRODUCTION
The third terms in Eqs.(l.l33)(1.135) require that one expresses F, 6, (ij from Eqs.(13)(15): Ze
LC,, = /(TBA., C
 T sin 19 (ijAer)dr
X
[l
Ze LCs3 =  / ( r sin 6 +Ae, c .
COV3 =

+
(P  eAm)v dr 2 2 112 ( p  eAm)2/m.oc 1
iAe,)r dt9
2J ( Y A . ~  r I ? ~ , , sin ~ )6 r dp C
The fourth set of terrns in Eqs.(9)(11) does not require Y, easier t o calculate:
C
.
(22)
/
Ac m,oe .
= 2a
I?, + and is thus
(TBA~, r sin 6 @A.o)dr 1d + + (P r d6
((p  e A m ) ~

e~rn),
1 ap
1.3
BASIC CONCEPTS O F T H E CALCULUS O F FINITE DIFFER.ENCES
15
(?Aeu  r 8 ~ . , ~ sin ) r 6 dp
The fifth and last set of terms in Eqs.(9)(ll) remains unchanged since there are no tirne derivatives of r , 6, p:
 ~ C I X ~ ~ O C ~
e
1.3 BASICCONCEPTS OF
.
r sin 6
THE
8 A e 0 ) ~ ~ (28) ~ C P
CALCULUS O F FINITE DIFFER.ENCES
There is a variety of ways to derive difference operators from differential operators. In mathematical books (MilneThomson 1951, Ch. 11; Norlund 1924, 5 1) one typically finds the substitutions
which we refer to as the "right" difference quotient. The corresponding "left" difference quotient has the form
1 INTRODUCTION
0
1
N2
2
1
N1
N
X
FrG.1.31.The symmetric difference quotient of first order needs to be supplemented at the limits of an interval by the right and the left difference quotient.
while a symmetric difference quotient is defined by
Since it is hard to work out three parallel physical theories based or1 these three difference quotients and to see at the end which is the best, we must decide from the beginning which one to use. The right and the left difference quotient were discarded since they led in simple cases to ever increasing or decreasing solutions while the symmetric difference quotient yielded stationary solutions (Harmuth 1989, Sec. 12.4). However, the right and the left difference quotient are sometimes required to supplement the symmetric difference quotient at the limits of an interval as shown by Fig.1.31. The symmetric difference quotient exists only for the points z = 1, 2, . . . , N  1. The right difference quotient remedies the problem for the left bol~ridaryz = 0, the left difference quotient for the right boundary z = N. The typical case where this becomes important is in connection with difference equations of second order that require two initial conditions. We shall discuss this at the end of Section 2.3. The symbol 2 is used for difference operators while A is used for a finite difference, e.g., At?. The notation can be simplified if one uses the substitution
arid t,herl drops the prime. Equation (3) assumes the following form:
The higher order difference operators are riot obtained as in differential calculus by repeated application of the first order operator, but we maintain the formal notation as shown in the followirig Eqs.(6) and (7). The three choices of Eqs.(l)(3) occur for all difference operators of odd order. We list here a few of the higher order operators but refer to the literature for their derivation (Harmuth, Meffert 2005, Secs. 1.2, 6.1):
1 . 3 BASIC
CONCEPTS O F THE CALCULUS O F FINITE DIFFER.ENCES
17
An important rule for the differenciation of a product u(Q)v(0)is very similar to the rule for its differentiation: ~,(e),,(e)
de
=
dv(e)
~(e)A0 + v(e)
du,(e) A0
+O(AO)~
Differential calculus permits us to define the inverse operation of differentiation by means of the differential equation
arid its formal solution
A corresponding process leads in the calculus of finite differences from the difference quotient of first order to summation rather than integration. In order to use the results of Norlund (1924, Ch. 3) and MilneThomson (1951, Ch. VIII) we follow closely their derivation. This requires to use the notation
For the connection of Eq.(13) with our notation in Eq.(3) we choose first w = 2Az
1
INTRODUCTION
and make then the substitutiori x = z'
+
 U.(Z/

AX)
2Ax

Az:
= p(xl  A x ) ,
x' = x
+Ax
(15)
Consider the function f ( x )
arid the shifted furlctiori f ( x
+ w)
A formal solution F ( z l w ) of Eq.(13) is obtained by substituting f ( z arid f ( z ) for u.(x w ) and u ( x ) in Eq.(13):
+
F ( r l w ) = Co  w 'y p ( x s =o
+ sw) =
/
p(u)dv  w s=o
0
p(x
+ sw)
+ w)
(19)
This is the Hauptlosung or principal solution of Eq.(13). It is also called the sum of the function cp(x). The integral over p ( v ) in Eq.(19) is written instead of the constant Co because a divergency of this integral may compensate a divergency of the sum over cp(x sw); a constant Co could not do that. Norlrind introduced t,he following notation for this summation:
+
The function F(xlw) is said to be obtained by summing p ( u ) from c to x , in analogy to integrating cp(xl) by the operation SCxp(x1)dx'. Norlund
1.3 BASIC CONCEPTS O F THE CALCULUS OF FINITE DIFFERENCES
19
generalized F(xlw) beyond Eq.(20) t,o functions p(z) that can be made summable by multiplication with an exporlential function epX("):
+
Note that X(x SW)means a function X of x + sw (Norlund 1924, § 2; MilneThomsorl 1951, Ch. VIII). The more general Eq.(21) is needed to obtain the sum of the constant a in Table 1.31. We write F(x1w) for the symmetric difference quotient on the left side of Eq.(15), substituting first w = 2A2, x = x'  Ax and then w = 2, x = x'  1: for w = 2Ax, x = x'  Ax
Au = Au for w = 2Ax
(22)
For the choice w = 2, x = x'  1 we get the following result that will be used from here on unless Eq.(22) is specifically pointed out: for w = 2, x = x' 1
Since we shall generally use the values w = 2Ax or w = 2 we derive the summatiorl of Eq.(22) directly from Eq.(15) by the substitutions
20
1 INTRODUCTION
TABLE 1.31 SUMS v(xl) = v(x) OF CERTAIN FUNCTIONS cp(xl) = cp(x) ACCORDING TO Eg.(23) FORw = 2. THEINTEGRALS OF cp(x) ARE SHOWN FORCOMPARISON.
eYx
eY(c+l)
eYx
shy 7 y complex, lyl
>
A x = A x ( x i , yk:), A y = A y ( ~ iyk), , a=LY(zi,gk), Asxy= Asxy( x , ~Yk) , (7) Let US niakc thc transition from finite differences to infinitesimal differences in Eq.(7): 30ne could also have concluded that the transition from arbitrarily small but finite differences to differentials should not be made, but it took 150 years before this conclusion became thinkable. 4 ~ h o s ewho did not grow up with information theory may reflect that infinite information is no better. than infinite energy. In the days of the perpetuum mobile the concept of infinite energy was not as objectionable as it is today. Infinite information may require a comparable time to become fully appreciated.
1.5 CURVED SPACE IN A DIFFER.ENCE THEORY A,111 ,
,,,'
\
r
\\\
'
, X,,y3,zk=ijk ,i.j, k=l,O,l
v> ~ c1~rather than ( p  e ~ , ) ~ / , r n ; c
.........
10, I O O I T ( ~ / ~ )
0.02
KIN
,.' .._.... 0 .; _.' .. . _. ... ..__
._
1. I T ( A / N )
0.04 +
.
'' 3
.. ..
LW.2 .
F1G.2.45. Plot of I T ( K / N ) ,1 0 I ~ ( n / N )and , 1 0 0 1 ~ ( ~ / for N ) As = 0.1, 1, 10 and N = 100, A 2 = 0 according to Eq.(33) in the interval 0.05 5 n / N 5 0.05.
A1 (6)= A2 (&) =
e i ~ 2 / 2=
2 sin /3,
" 2 ' ( " I N ) (cos hl + i 2 sin /3, 2
sin 3 ) (34) 2
Equation (25) becomes:
u,([, 0)
2
= _ ~ X ~ ~ " X Z / ~ ~ " ( X I C  X Z ~ ~ ) / ~IT(^/^) sin ,>KO
sin ,b,
sin
N
(35)
We note that /3, does not become zero for cp, = n = 0 according to Fig.2.42 if the exact formula of Eq.(13) is used for P,, only the approximation of Eq. (14) yields zero. The substitution of F ( < )of Eq.(2.324) and of Eq.(35) into Eq.(2.313) produces the solution qol( 4  (A:  2Xi Xi)/4. It seenis that a new solution of Eq.(l) is required to close the gap between N/2 and KO or no and N/2. This is riot so. According to Eq.(2.32) the coefficients A:, A;, A: are of the order O ( A X ) or ~ O(At)2. Hence, I K 0 ! in Eq.(36) is of the order N/2  O(At)'. The sum of Eq.(36) represents the area under the step function shown in Fig.2.46. If the area runs from K = N/2 to K = +N/2 the first and the last amplitude needed are for n = N/2 1 and K, = N/2  1. Hence, the summation signs in Eq.(36) may be replaced:
+
and there is no need for a solution for I K , ~ 2 N/2. To extend our solution from V5,6 of Eq.(lO) to 717,s of Eq.(l8) we define a constant that permits to write Eqs.(l9) and (20) in a simpler form:
pK
2.4 TIME VARIATION OF Qol( h/moc. This requires that
The horrlogeneous part of Eq.(4) may be written with the help of Eqs.(2.232) and (2.32) in the following form:
3
INHOMOGENEOS DIR.AC DIFFERENCE EQUATION
The second and third line in Eq.(4) yield:
If Ax is significantly larger than the Compton wavelength Xc = h;/fmocwe may ignore the second term in brackets on the right side of Eq.(9). We then get a second order difference equation according to the first term on the right side of Eq.(9), otherwise we get a third order difference equation according to the whole right side of Eq.(9). With the help of Eq.(2.216) we may rewrite the first term on the right side of Eq.(9) as follows:
An: > h h o c = Xc
3.1 INHOMOGENEOUS EQUATION (2.233)
111
For the other extreme, Ax ~O / ~ , L'ZX2, 21,K,
X20
(76)
6(0,0) = 0 for 0 C(
nFIG.3.44. Plot of x i , ( & ) according to Eq.(38) for N = 100, Gal = 1, $04 = 1, $el0 = 0, ArnL> ZecAesr sin 6 $17:
(25)
rd and r sin I9 @ are not one obtains a different condition
>> ZA,,rd/c
and
A,,
>> ZAesr sin I9 $/c
(26)
The meaning of Eqs.(25) and (26) is that the energy due t o the potential A, is small compared with mc2 or the energy due to the potential A,. Corresponding results may be derived from Eqs.(7) and (8). We may simplify Eqs. (6)(8) as follows:
p, = m?
+ eA,,
, po
= mrd
+ eAm0, p,
D
= fm3
The sirriplified Eqs.(13)(15) assume the form
= rnr sin 19 cC,
+ eA,,
(27) (28)
4.2
R.ELATIVISTIC LAGR.ANGE FUNCTION
163
The three simplified terms X,,, X o , X, of the Hamilton function assume the following form:
One recognizes the terms of the conventional Hamilton function of an electrically charged particle in an electromagnetic field with potential A, and 4, augmented by the correcting terms LC,, Leo, and LC,. The term mu2/2 in Eq.(l) shows that our Lagrange function is not relativistic. This simplification was accepted since it permits to obtain X.,., Xs, X, of the Hamilton function as well as the correcting terms L,,, Lc9, LC, explicitly. The relativistc Lagrange function will be introduced in the following section. The Hamilton function can then be represented by series expansions only. The substitution of Eqs.(12)(15) into Eqs.(21)(23) yields the Hamilton function X without the approximations made to derive Eqs.(30)(32):
We still have to show that the vector LCcan be written explicitly. Its three components LC,, Lcs, LC, are shown in Eq.(l). Each of them can be broken up into five subcomponents. They are very long and they will not be used in this book. Hence, we refer to the literature [Harmuth and Meffert 2005, Eqs. (5.128)(5.132)].
The Lagrange function of Eq.(4.11) is not relativistic but it led to the explicit Hamilton function of Eq.(4.133). The modified Maxwell equations can be combined with a nonrelativistic constant mass or a relativistic variable mass. The different results appear worth showing even though in the end we will use the relativistic variable mass only. We proceed in this historically correct but slow way since we must introduce three concepts: 1. The modification of Maxwell's equations by the inclusion of a magnetic (dipole) current density. 2. The replacement of infinitesimal distances by arbitrarily
164
4 DIR.AC DIFFERENCE EQUATION IN SPHER.ICAL COOR.DINATES
short but finite distances. 3. The replacement of infinite distances by arbitrarily large but finite distances. We want to emphasize the need for three different concepts and to separate them, in order to show the effect of each concept. We turn to the relativistic Lagrange function. This irnplies giving up the explicit representation of the Hamilton function for a representation in terms of series expansions. The introduction of the rest mass yields the equation of motion (Harmuth and Meffert 2005, Sec. 5.2)
as well as the conservation law of energy:
A fourvector p can be defined with three spatial components p,, ps, p, P=
mov (1 v2/C2)1/2
+ r8es +
[email protected], for spherical coordinates P = p ~ e+' p ~ e + s p,e, v = re,
(3)
and the corriponerlt p4 :
Here E denotes an energy rather than the magnitude E of an electric field strength. Momentum p and energy E are connected by the formula
Leaving out the unit matrix we obtain the relativistic generalization of the conventional part of the Lagrange function C of Eq.(4.11):

+ e(4, + Am,?: + ~ , s r 8+ A,,r
moc
2
1 + m,0v2 +e(6, 2
+ A m . v ) for v2/c2 mjoc2 and 2 < g < 2: l T h e energy levels of the hydrogen atom according t o Dirac's equations are usually treated surprisingly brief in text books. Becker (1963, 1964, vol. 2, Ch. F I I ) , who was for decades a standard for thoroughness, avoids the topic completely.
180
4 DIRAC DIFFERENCE EQUATION IN SPHERICAL COORDINATES
For small values of Ar or for g = 2 square roots to obtain:
+ O(Ar) one may readily extract the
Substitution of ss for so into Eq.(6.842) and the use of a complex value p = p~ ipI bring
+
Equations (6.85) and (6.86) yield
Substitution of Eqs.(4) and (5) into Eq.(3) yields two equations for p~ and p~ that yield for m.ocAr/ti
1 (9)
Three corresponding values of sz are obtained with the help of Eq.(6):
192
4 DIR.AC DIFFERENCE EQUATION IN SPHERICAL COORDINATES
s2=i+Asal=i
[ (mOiAr) $1 ] 1+
(1 
complex
on imaginary axis
(10)
(12)
The choice of the signs +Aszl, +Asz2, Asas and particularly the use of the sign f will be explained presently. To make s 2 located close to +i we must exclude s 2 of Eq.(lO) since the magnitude of the real part will always be at least equal to 1:
Only Eqs.(ll) and (12) for E > rn,0c2 can yield Asz2, Asz3 0, s2 on imaginary axis
(16)
1/2
4.6
SOLUTIONS WITH
So
NEAR S = i
193
which is not a very interesting result since it can be satisfied by the limit A r + d r . But Eq.(16) requires a mininluni value for Ar, wliicli is an important result since it cannot occur in a differential theory:
In the first line of Eqs.(17) and (18) Ar is still connected to the Compton wavelength h/moc of the electron, but in the second lines Ar is a function of the energy E and the constants of nature hc; the mass m o of the electron only adds a correction. We develop the solution for s2 on the imaginary axis according to Eq.(16) with E > ,moc2 and s2 = i ( l First we derive from Eqs.(6.85) and (6.86) the relations
Substitution of

s2
= so into Eq.(6.842) brings
Substitution of Eqs.(l9) and (20) produces a real equation. We use the notation Xe = Xel Xe4 with Xe = Xel, Xe4 = 0 or Xe = Xe4, Xel = 0 and write sg (I  AS^^):
+
194
4
DIRAC DIFFERENCE EQUATION IN SPHERICAL COORDINATES
With Xe = d 
we obtain for p:
Figures 4.62 to 4.67 show plots of p as function of rnocAr/?i for various values of the parameters ~ / r n and ~ c 1.~ The sign is used for f everywhere. We see from Figs.4.62, 4.66, and 4.67 that 1 does not have much effect on the plots. The same holds true for the terms in Eq.(23) multiplied with 8.rrZa. Consider the solutions xl(p) and x4(p) of Eqs.(6.876) and (6.877). Written for sz = i(1  Asz3) = i exp(Asz3) rather than 33 we get
+
For a negative integer value of p the factorial series can terminate long before p reaches N,which in essence means a polynomial replaces a series and the question of convergence does not arise. Figure 4.62 shows that p can equal any negative integer. Let us choose p = 2 and the parameter ~ / r n= ~ 5. c ~We obtain mocAr/fi = 0.306. Equations (24) and (25) become
FrG.4.62. Plots ofp according to Eq.(23) for ~ l r n o c '= 5, . . . , 11. The parameter 1 equals 1; Z = 1; a = 7.2975 x lop3.
n~ocAr/h
+
F1G.4.63. Plots of p according to Eq.(23) for ~ l r n o c '= 2, . . . , 5. The parameter 1 equals 1; Z = 1; a = 7.2975 x
FIG.4.64. Plots of p according to Eq.(23) for ~ l r n o = c ~1.7, . . . , 2. The parameter 1 equals 1; Z = I; a = 7.2975 x
196
4
DIRAC DIFFERENCE EQUATION IN SPHERICAL COORDINATES
m o c A r / h +
FIG.4.65. Plots of p according to Eq.(23) for E/rn,oc2 parameter 1 equals 1; Z = 1; a = 7.2975 x lov3.
=
1.67,
... ,
1.7. The
FlG.4.66. Plots of p according to Eq.(23) for ~ / r n , o c=~5, . . . , 11. The parameter 1 equals 2; Z = 1; a = 7.2975 x lop3.
mocAr/h
+
FIG.^.^7. Plots of p according to Eq.(23) for E/rnoc2 = 5, . . . , 11. The parameter 1 equals 3; Z = 1; a = 7.2975 x
The variable a0 is choosable, do follows from Eq.(6.826) with so = sz, while a1 and dl follow from Eqs.(6.823), (6.824). The factor r(p)/I'(p 1) yields
The factor
1:P
is eliminated in analogy to (1)P in Eq.(4.527):
The factor exp(Asz3p) cannot be reduced to 1 if we exclude the limit Asa3 + 0 to avoid the special point where the unit circle intersects with the imaginary axis. Hence, we obtain
These solutions decrease exponentially with increasing p. If we multiply Eqs.(30) and (31) with vl or vz of Eq.(6.717) we obtain sinusoidal oscillations f~ ( pexp(iE8Atlti) ) that decrease exponentially with increasing p. Different choices of p and ~ / , r n , in ~ cFigs.4.62 ~ to 4.67 do not change this result qualitatively. Let 11s recall that Eqs.(16) arid (18) limit A r to a finite value. Hence, p = r / A r has an upper limit for any finite value of r, which implies that xl(p) and ~ 4 ( p do ) riot decrease arbitrarily. We derive sorne numerical values. From Fig.4.68 we take the value ,mocAr/h = 0.306 for E/,rnoc2 =.5, or E = 5moc2 = 4.09 x [J],arid p = 2; the scale of mocAr/h was considerably expanded to yield 0.306. This yields Asz3 = 1.12 from Eq.(16). To make exp(Aszsp) in Eqs.(30) and (31) drop to 0.01 we need
which yields po.ol= 4.11. The value of A r follows from the value rnocAr/fi= 0.306 obtained from Fig.4.68: ti mo c We obtain A r = 1.18 x
FIG
[m] from mocAr/fi = 0.306. The distance ro.01
= p0.01Ar
(34)
follows from Eqs.(32) and (33) as ro.01 = 4.86 x 10l3 [m]. The nunlbcrs derived for E/moc2 to ~ 0 . 0are ~ listed in the first line of Table 4.61. Also listed there are the corresponding values for E = 10, . . . ,
198
4
DIRAC DIFFERENCE EQUATION I N SPHERICAL COORDINATES
mocAr/h
t
FIG.^.^8. Plots of p according to Eq.(23) for E/moc2 = 5 , . . . ,11. The scale of p is expanded compared with Fig.4.62. The parameter 1 equals 1; Z = 1.
mocAr/ti . F1G.4.69. Plots of p according to Eq.(23) for E/moc2 = 20,. . . ,200. The parameter 1 equals 1; Z = 1.
mocAr/h + FIG.4.610. Plots of p according to Eq.(23) for E/moc2 = 500, 1000, parameter 1 equals 1; Z = 1.
The
4.6 SOLUTIONS WITH
So NEAR S
=i
199
TABLE 4.61 ENERGYIN NORMALIZED FORME / ~ O ~ ~IN NJOULE; D Ar/(ii/moc) FROM F I G S . ( ~ . ~ TO 8 ) (4.610) WITH EXPANDED SCALEOF mocAr/h FOR p = 2; Aszs ACCORDING TO E ~ . ( 1 6 )po.01 ; = lnO.Ol/Asz3 ACCORDING TO E ~ . ( 3 2 ) ; Ar ACCORDING TO E4.(33); ro.01 = po.olAr ACCORDING TO E9.(34); mo REST MASSOF ELECTRON. m,o = 9.109 389 7 x [kg], c = 299 792 458 [m/s], h = 1.054572 7 x [Js] moc2 = 8.187 1112 x lo'* [J], moc = 2.730926 3 x [Js/m] h/moc = 3.861 5934x 10l3 [m]
2000 that were derived with the help of Figs.4.68 to 4.610 using a larger scale for mocAr/h. Let us observe that the values of A r and ro.01 for the energies ~ l r n o = c~ 5 . . . 2000 are typical nuclear distances. Hence, the eigensolutions in the point s g = 1  O ( A r ) in Fig.4.61 yield the solutions known from Dirac's differential equations, those in the point s:, = 1 + O ( A r ) yield equivalent solutions for antiparticles with ,mo + mo, but those in the point s 2 = i(l  AT)) yield eigensolutions at nuclear rather than atomic distances. We turn t o a pecularity of our plots and investigate how much smaller the parameter ~ l r n o c=~ 1.67 in Fig.4.65 can become and how large ,rn.ocAr/h can become for the jump from +cc to a. To derive an equation for the location of the jumps we look at Eq.(23) and see that only B = 0 can produce such a jump:
One derives an equation for x:
4
200
DIRAC DIFFERENCE EQUATION I N SPHERICAL COORDINATES
+
(q  1)[4  9(q  1)2]52 8(q2  1)'123: + 13(q  1) = 0 (36) If the factor 4  9(q  1)2 is zero one obtains negative values of z for any q = ~ / r n o c>~ 1 as required by Eqs.(l5) and (16). There can be no jump. We obtain for this distinguished case q = E/moc2 = 513 = 1.66.. . (37) which is slightly smaller than the smallest value E/moc2 = 1.67 shown in Fig.4.65. It may be readily verified that one obtains plots for p like those in Figs.4.62 to 4.65 for E/moc2 < 513. As ~ / r n . = ~ c513 ~ is approached the jumps in Fig.4.65 move to ever larger values of *mocAr/h.w i t h E2/rn$c4 1 always larger than (5/3)2  1 = 1619 and rnocAr/ti increasing arbitrarily we obtain arbitrarily large values of of Eq.(16). The distance po.01 at which exp(Asz3p) drops to 0.01 approaches zero and both the normalized distance po.01 and the absolute distance ro,ol = po.olAr approach zero. We turn to the solutions on the unit circle defined by the choice s2 = i As22 of Eq.(ll). Equation (15) yields
+
No lower limit is imposed on Ar as in Eq.(16). Substitution of s2, s i = 1 2iAs22, p = p~ ipI and Eqs.(l9), (20) into Eq.(6.842) yields two equations for p~ and p ~ :
+
+
The solution of p~ and p~ yields:
4.7
SOLUTIONS WITH Sg IN THE NEIGHBORHOOD O F 1:
20 1
Since p is complex it cannot be chosen equal to a negative integer riunlber to terminate factorial series as shown by Eqs. (24) and (25). We must use Eqs. (6.876), (6.877) and obtain convergence by conformal mapping or by asyrnptotic termination. However, the knowledge of s 2 = iAsz2 and p = p ~ .i p I permits us to rewrite the terms in front of the summation sign of Eq.(6.876):
+
With
s 2 = 1:
+ Asa2 = i ( l  iAsz2)we get for small values of Asz2
S2 = jeiAszz , a
, s~ 2 ~.P , iAsaap e
= fiei~Ar.p,
r ; , ~ T=
asz2
(43)
F'rorn Eq.(4.524) we obtain for large values of p the equation
'(PI
1
eppp
= 1
r ( ~ + ~ + l )p
1
p
.
1
e p e  i ~ ~ b ~
p ,,,PR+~PI pl+pR
(44)
which is sirnilar t,o Eq.(4.525). Equation (42) assurries the form
The product of xl(p) with vl(0), v2(B) of Eq.(6.717) yields a t great distances expressions of the forrri
This equation is similar to Eq.(4.528) but pl+"R as well as KATneed elaboration.
According to Fig.4.61 the equivalent of
s2
in the neighborhood of s =
1: is s4 =  s 2 as shown by Eq.(6.835). Hence, Eqs.(4.61)(4.65)derived
for .s2 apply again. We obtain for
s4
202
4 DIR.AC DIFFERENCE EQUATION IN SPHER.ICAL COORDINATES
The three conditions of Eqs. (4.67)(4.69) remain unchanged. Equations (4.610)(4.612) are replaced by
E < moc2, complex (2) sq=(i+As42)=i*
[
1 (m O~Ar)
E > m,oc2,
E > moc2,
(
(&
(T) (&
)(
 1)
 I)
]
 1 ) < 1, complex (3)
> 1, on imaginary axis (4)
Following Eqs.(4.610)(4.612) we have given Assl and a positive sign since s 4 in Fig.4.61 rotates from s 4 = i to s 4 = 1 in the mathematically negative direction, while As43 has a negative sign because s 4 on the negative imaginary axis is always in the interval 0 > s 4 i. The magnitude of the real part of Eq.(2) is always larger than 1. Hence, (i  AS^^) can never be close to i; we exclude the case E < moc2. Only Eqs.(3) and (4) can yield values of s 4 close to i:
>
As42 = [I 
=
mocAr
m,ic4  1) (1
[ + (F)~($

1
2 0,
s4
on unit circle
(5)
2 0, s4 on imaginary axis (6)
These equations equal Eqs.(4.615) and (4.616). We obtain once more Eqs.(4.617) and (4.618). Substitution of s 4 = s o into Eq.(6.842) brings
4.7
SOLUTIONS WITH So IN THE NEIGHBOR.HOOD O F i
203
A real equation is obtained if Eqs(4.619) and (4.620) are substituted. Again we write Xe = Xel Xe4 with Xe = Xel, Xe4 = 0 or Xe = Xe4, Xel = 0. We obtain with s4 = i(1  Asq3) and si = (1  2As43) in first order of Ar:
+
2  Xe f 8xZamoc2 We substitute Xp = J 
(m324
E2  1)li2]
(8)
and obtain:
Figures 4.71 to 4.76 show plots of p as functions of m,ocAr/li with the parameter ~ / r n ~ The c ~ .positive sign is used everywhere for f. The plots differ significantly from the ones of Figs.4.62 to Fig.4.67. We note that the plots of Figs.4.72 and 4.73 look quite similar to those of Figs.4.63 and 4.64, but the scale for p differs by a factor 5. The solutions xl(p) and x4(p) of Eq~(6.876)and (6.877) are written for ~4 = i(l  i e x p (   A ~ ~rather ~) than for ss:
+
204
4 DIRAC
DIFFERENCE EQUATION IN SPHERICAL COORDINATES
0.1
0.15
0.2
mocAr/ h
0.25
0.3
0.35
+
F1G.4.71. Plots of p according to Eq.(9) for ~ / r n o c=~5, . . . , 11. The parameter i! equals I; Z=1; a = 7.2975 x
mocbrlh

FlG.4.72. Plots of p according to Eq.(9) for E/moc2 = 2, . . . , 5. Note the difference of the scale of p compared to Fig.4.63. The parameter 1 equals I; Z=1; a = 7.2975 x
FIG.4.73. Plots of p according to Eq.(9) for ~/rnoc' = 1.7, . . . , 2. Note the difference of the scale of p compared to Fig.4.64. The parameter 1 equals 1; Z=1; a = 7.2975 x
4.7
SOLUTIONS WITH
So
IN THE NEIGHBORHOOD O F
i
205
F1G.4.74. Plots of p according to Eq.(9) for ~ / r n . o c= ~ 1.67, . . . , 1.75. The parameter 1 equals 1; Z=1; a = 7.2975 x lop3.
0.1
0.15
0.2
0.25
0.3
0.35
mocAr/h + FIG.^.^5. Plots of p according to Eq.(9) for ~ / r n , o c=~ 5, . . . , 11. The parameter 1 equals 2; Z=1; a = 7.2975 x
mocAr/h

FrG.4.76. Plots of p according to Eq.(9) for ~ / r n , o c=~5, . . . , 11. The parameter 1 equals 3; Z=1; cr = 7.2975 x lop3.
206
4
DIRAC DIFFERENCE EQUATION IN SPHERICAL COORDINATES
FIG.4.77. Plots of p according to Eq.(9) for ~ 1 r n . = o ~5 ~, . . . ,11. The scale of p is expanded compared with Fig.4.71. The parameter 1 equals 1; Z = 1.

mocAr/h FIG.4.78. Plots of p according to Eq.(9) for ~ / r n o c=~20,. . . ,200. The parameter 1 equals 1; Z = 1.
mocAr/h
+
F1G.4.79. Plots of p according to Eq.(9) for E/rnoc2 = 500, 1000, 2000. The parameter 1 equals 1; Z = 1.
4.7 SOLUTIONS
WITH
so IN THE NEIGHBORHOOD OF i
207
TABLE 4.71 ENERGYIN NORMALIZED FORM ~ / m , o cAND ~ IN JOULE;Ar/(fi/rn.oc) FROM SCALEOF mocAr/fi FORp = 2; As43 F I G S . ( ~ . ~ TO 7 ) (4.79) WITHEXPANDED ACCORDING TO E&.(G); po.01 =  ln0.01/As43 ACCORDING TO E ~ . ( 4 . 6 3 2 )A ; r ACCORDING TO E&.(4.633); ro.01 = po.olAr ACCORDING TO E&.(4.634); mo REST MASSOF ELECTRON.
As in the case of Eqs(4.624) and (4.625) a negative integer value of p can terminate the factorial series long before p reaches N, which means the question of convergence does not arise. Figure 4.71 shows that p can equal any negative integer. Let us choose p = 2 and the parameter ~ / r n o c=~5. We obtain mocAr/ti = 0.3011. Equations (10) and (11) become:
The variable a0 can be chosen, do follows from Eq.(6.826) with .so = 54, while a1 and dl follow from Eqs.(6.823), (6.824). The factor r(p)/J?(p 1) yields p  1. The factor (i)P is eliminated with the help of Eqs.(4.527) and (4.629): =
(1)PiP = hip = h f i = f i = 1, i, 1, i
(14)
xl (p) = fi eAs43p[a~(P  1) + all
(15)
x4(p) = $'ieAS43p[d~(P  1)
(16)
We obtain:
+ dl]
208
4 DIR.AC DIFFER.ENCE EQUATION IN SPHERICAL COORDINATES
The comments in the two paragraphs following Eq.(4.631) apply again, since Asn3 of Eq.(4.616) and As43 of Eq.(6) are equal. We derive numerical values from Figs.4.77 to 4.79 in accordance with Eqs(4.632) to (4.634) and Table 4.61. The new values are listed in Table 4.71. Again we obtain typical nuclear distances. The results obtained in Section 4.6 from Eqs.(4.635) to (4.637) apply again since B in Eq.(4.623) is equal to B in Eq.(9). Let us consider the limit case defined by Eq.(6) for As43 = 0
which yields a minimum value for Ar:
Extracting the square roots on both sides yields two solutions:
Normally one would not pay any attention to the difference between these two equations, but the results of Section 4.5 make us cautious to look out for results that may hold for antiparticles, mo + rn,o. Equation (18) holds both for mo and m,o, and Eqs.(l9) and (20) call attention to this fact. Consider Eq.(8) for Ass3 = 0 according to Eq.(17) and m,ocAr/li substituted from Eqs.(l9) or (20):
The first line of this equation can be simplified
+
E/moc2 for E/m,oc2  1  E/moc2  2 for E/moc2  1 
(22) (23)
4.7 SOLUTIONS WITH
So
IN THE NEIGHBORHOOD O F
i
209
FIG.^.^10. Plots of pl according to Eq.(24) for 1 = 1 and Z = 1. The plot holds for ic in Eq.(24) replaced by the plot  for f replaced by .
+,
~/rnoc~ +
F1~.4.711.Plots of pz according to Eq.(25) for 1 = 1 and Z = I . The plot holds for f in Eq.(25) replaced by +, the plot  for f replaced by .
+
+
and we obtain two quite different results pl and pa from Eq.(21):
Plots of pl for 1 = 1 are shown in Fig.4.710. The solid line holds for +87rZa in Eq.(24), the dashed line for 87rZa. Since pl is never negative we cannot terminate Eqs.(lO) and (11) by choosing a negative integer value for p l . Figure 4.711 shows plots of p2 for 1 = 1 and Z = 1. The solid line holds for +87rZa in Eq.(25), the dashed line for 87rZa.
210
4 DIR.AC DIFFER.ENCE EQUATION IN SPHERICAL COOR.DINATES
We extend the analysis to the results of Section 4.6. Instead of Eq.(17) we get now frorri Eq. (4.616)
which equals essentially Eq.(17). Equation (18) remains unchanged and Eq.(4.622) becomes:
Equations (22) and (23) apply again and we get
These are Eqs. (24) and (25) with reversed sign. We get again the plots of Fig.4.710 and 4.711 but pl, p 2 are replaced by pl,  p 2 . 4.8 ENERGY O R MASS RATIOSFOR.
E/rn0c2
> 1 AND
Armin
We had obtained in Eqs(4.616) and (4.76) a limit for the possible resolutions and A s 4 3
which yields a minimum value for the distance resolution Ar = A r m i n :
(
,mOcArmin
h
12= I& )+(
(2)
This equation holds for particles with mass m , o as well as for antiparticles with mass 7no. We substitute so = s 4 = i, S: = 1 into Eq.(6.842). The notation Xe = Xel .Ae4 = is used as explained in connection with Eq.(4.622). The following equations are readily obtained:
+
4.8 ENERGY OR MASS RATIOS FOR. E/moc2 > 1 AND Armin
kp
1J
[
 i(w,
+ w,)p + 4 i n ~ (a   1) = O
'I2]
(J~(TTT~ + 4irrZa
112
+
p = 4  i(wP w,)
211
(z)

(wpwm)' I 2
)
(3) (4)
The terms containing w, and w, are worked out with the help of Eqs.(6.85) and (6.86) using Eq.(2) for ( ~ n ~ c A r , ~ , / h We ) ~ . are only interested in the case E/moc2 > 1:
We note that ( w , w , ) ~ / ~ is always imaginary since W,W, For w,/w, we obtain:
Further we get
is negative definite.
212
4 DIRAC DIFFER.ENCE EQUATION IN SPHERICAL COORDINATES
We have worked out Eqs.(7) to (14) in some detail to avoid more ambiguous signs f than necessary. Substitution of Eqs.(12) and (14) into Eq.(4) yields:
We rewrite this equation for the four possible combinations of the two ambiguous signs f.
Plots of pol and po2 are shown in Fig.4.81. Since pol and po2 are never negative we cannot terminate Eqs(4.710) and (4.711) by choosing a negative integer value for pol or po2. Figure 4.82 shows plots of pas and pod . There are negative values in the interval 1 5 E/,moc2 5 2. Since we can generally not choose a negative integer value for pol to Po4 we try the method of asymptotically approached polynomials developed for E < m o c 2 in Section 6.8 from Eq.(6.847) to (6.852). We substitute so = s4 =  i , S ; = 1 into Eq.(6.852). With Xel Xe4 = X e = d m and either Xel = 0 or Xe4 = 0 we obtain:
+
From Eq.(7) we get wpwm = 4 while Eqs.(l3) and (2) yield
Equation (20) is reduced to
4.8
ENERGY OR MASS RATIOS FOR 0.4~
I
armin
213
_ _ ~ ~ _ . _................................ __.___.__.___.
,....... .......
0.3/
.: ;/
~ / r n> ~1 c AND ~
PO 1 Po2

*..
,I.
I
0.2 ;!'
i'
4
i
0 . 1j
i
I
20
40
60 E / m g c 2 +
80
100
F1G.4.81. Plots of p o l and pon according to Eqs.(16) and (17) for Z = 1 and I = 1 in the interval 1 5 E/moc2 5 100.
FIG.4.82. Plots of PO3 and PO4 according to Eqs.(l8) and (19) for Z = 1 and 1 = 1 5 10.
in the interval 1 5 E/moc2
5
10

15
20
E/moc2 FlG.4.83. Plots of pll and plz according to Eqs.(23) and (24) for p = 1, Z = 1, 1 = 1 in the interval 1 5 E/moc2 5 20.
2
1.5
1
2.5
3.5
3
4
E/moc2 i FIG.^.^4. The functions pol, poz, p n , plz according to Eqs.(l6), (17), (23), (24) for Z = 8, p = 1, 1 = 1 in the interval 1 5 ~ / m o c5 ~4. The apparent intersection of p12 and pol slightly to the right of E/moc2 = 1 disappears when the resolution is increased. Q,\>,~"
1.5
I* , a .
,/*'
1. ,,sr
0.5 
0.5

Po 1
,J'
./< ..................................
......... ............ ._.. ,a', __.. ..' ,,,' 1.5
Po2
,'*5
3.5
3
2.5
,,'
4
,,,/'

1,'
1 t'
Pll
E/moc2 FIG.^.^5. The functions pol, po2, pll, plz according to Eqs.(l6), (17), (23), (24) for Z = 8, p = 2, 1 = 1 in the interval 1 ~ / r n . ~ c4.~
1 AND Armin
215
TABLE 4.81 AS
FUNCTION OF I ACCORDING TO E&.(25).
We write p l l and pl2 for p to resolve the ambiguity of the sign f :
Figure 4.83 shows plots of pll and pl2 for p = 1, 1 = 1, and Z = 1. These plots intersect with the plots of Fig.4.82 in the interval 1 < E/,moc2 < 2. We are here interested in larger values of E/,moc2, but no intersections of the plots of Figs.4.81 and 4.82 with the plots of Fig.4.83 can occur for E/?noc2 > 2. Since the four plots of pol to PO4 in Eqs.(l6) to (19) can be combined with the two plots of pll and pl2 in Eqs.(23) and (24) in eight ways we need some guidance which combinations to choose. We observe that the sign of the terms in brackets of po2, p04, and pl2 changes for
a t a sufficiently large value of ~ l r n ~ Table c ~ . 4.81 shows for which values of Z this will be the case. For an overview of how intersections between pol, p02 and p l l , pl2 can be produced by choosing Z and p refer to Figs.4.84 t o 4.86. For Z = 8 , p = 1 in Fig.4.84 the intersections are all a t E/*moc2 = 1. A more interesting result is shown in Fig.4.85 for Z = 8, p = 2. The plot pl2 irltersects the plotspol a t about E/rnoc2 = 2.5 and p02 at about E/rnoc2 = 2. More precise values of E/moc2 are listed in the first two rows of Table 4.82. Also show11 are ~rr,ocAr,i,/h according to Eq.(2) and Armin:
Figure 4.86 shows the intersection ofp12 with po2 for Z = 14, p = 2 a t about ~ / r n= ~ 100; c ~ this is the only intersection for Z = 14, p = 2. Again a more
TABLE 4.82
LIST OF FIGURES, INTERSECTIONS OF PLOTSSHOWN,CHARGENUMBER Z, QUANTIZATION NUMBER p , WITH T H E RESULTING VALUESO F ~ l r n o cFROM ~ Fl~s.4.85A N D 4.86, m.ocArmin/hACCORDINGT O E Q . ( ~A)N D ArminACCORDING T O E ~ . ( 2 6 ) .
Figure intersec Z
p
E/m.oc2
m,ocAr,i,/fi
Armin
precise value of ~ / , r n . is ~ cshown ~ in Table 4.82 together with mlocArmin/h and Armin.The values shown for Arminin Table 4.82 are typical nuclear rather than atomic distances. The problem of finding intersections between pol, po2 and p l l , pl2 is more complicated than suggested by Figs.4.84 to 4.86. There are intersections for Z = 9, 10 and p = 2, but not for Z = 11, 12, 13. Intersections start again for Z = 14 and continue for Z = 15, 16 but with decreasing values of ~ / m . for ~ c the ~ intersections. To get some overview of the periodic system of elements we observe that the intersections between pl2 and po2 in Figs.4.85 and 4.86 are determined by the equation P12 = Po2
(27)
For large values of E2/,moc2we write p12 and Po2. Equation (27) becomes
Certain elements with their charge number Z are listed in Table 4.83 together with pO2and p12 p according to Eq.(28) for 1 = 1. We then choose the value of p that yields the smallest value of p12 that is larger than 1002. This value of p12 is listed in Table 4.83. The resulting value of E/moc2 is obtained by making plots like the one in Fig.4.86 and using some numerical approximation t o obtain ~ / accurate ~ to ~six decimals c as ~ shown in Table 4.83. The values of E/moc2 of Table 4.83 are used in Table 4.84 to derive mocArmin/fi according to Eq.(2) and Arminaccording t o Eq.(26). We see
+
4.8
ENERGY OR MASS RATIOS FOR
~ l r n >~ 1c A~N D Armin
217
TABLE 4.83 SELECTEDELEMENTS FROMT H E PERIODIC SYSTEMO F ELEMENTS SHOWING T H E CHARGENUMBER Z, Po2 AND pi2 p ACCORDING T O E ~ . ( 2 8 ) T, H E QUANTIZATION NUMBERp ACCORDINGT O E~.(24), p 1 2 , AND ~ l r n o cFROM ~ T H E RELATION p o ~= plz FOR1 = 1.
+
Element
Z
that Arminhas for all elements typical nuclear distances of 10l3 t o 10l5 m. No clear pattern is recognizable. Osmium (Os), Iridium (Ir), and Silicon (Si) yield the largest values for ~ / r n , o c Sodium ~, (Na) and Carbon (C) yield 1, which is useless for Eq. (1). Let us consider some charge numbers Z much larger than those in Table 4.84. Figures 4.87 to 4.89 show plots of p02 and pl2 for Z = 10 000, 15 000, and 20000. The intersections of poz and plz are at ~ l r n = ~ 786.0296, c ~ 1486.672, and 2680.501. These large values are the purpose of showing
218
4
D I R A C D I F F E R E N C E EQUATION IN S P H E R I C A L COORDINATES
TABLE 4.84 ELEMENTS FROMT H E PERIODIC SYSTEMO F ELEMENTS, THEIR CHARGE NUMBER Z,QUANTIZATION NUMBER p , ~ l m o FROM c ~ TABLE4.83, AND T H E RESULTING VALUESOF mocAr,i,/li ACCORDINGTO E&.(2) AS WELLAS O F ArminACCOORDINGTO E ~ . ( 2 6 ) 1; = 1. Element
Z
p
Ne Ar Kr Xe Rn Na Ka Rb 0s Fr F C1 Br I At
10 18 36 54 86 11 19 37 55 87 9 17 35 53 85
2 2 3 4 6 1 2 3 4 6 2 2 3 4 6
4.584780 2.993256 3.553579 3.573578 4.637487 1 2.548919 3.134755 3.228624 4.202668 2.715467 3.723851 4.476241 4.106520 5.037681
0.223 0.354 0.281 0.291 0.221 0.427 0.337 0.326 0.245 0.396 0.279 0.229 0.251 0.203
8.63 x 1.37 x 1.09 x 1.13 x 8.53 x 1.65 x 1.30 x 1.26 x 9.46 x 1.53 x 1.08 x 8.85 x 9.70 x 7.82 x
Fe Ru
26 2 44 3 76 6
1.522218 1.965205 46.23822
0.871 0.591 0.0216
3.36 x lo13 2.28 x lo13 8.35 x 10l5
2 3 6 0 2 3 4 6
1.468844 1.885190 22.724234 1 103.4024 13.848867 6.493666 6.933404
0.929 0.626 0.044
3.59 1 0  1 ~ 2.42 x lo13 1.70 x 10l4
0.00967 0.0724 0.156 0.146

92 7 94 7
22.955720 12.832754
0.0436 0.0782
1.68 x 3,02 x 10l4
0s
Co Rh Ir C Si Ge Sn Pb U P1
27 45 77 6 14 32 50 82
~ l m o c rn~cAr,~,/li ~
Armin[m]
3.73 x 2.80 x 6.02 5.63 x
lo14
10l3
lo13 lo13
lo14
lo13
10l3 lo=13 10l4 lo13 lo13 10l4
lo14
10l4
lo15 lo14 lo14 10l4
illustrations for unrealistic large values of Z. A value ~ / , r n , = ~ c1836.152 ~ is well within the range shown, but we would interpret it as the mass ratio ,m,/m, of proton and electron. ~ cwell ~ ,as ~ m o c A r m i n / h , arid Arminfor Table 4.85 lists Z, p, ~ / r r ~ as Figs.4.87 t o 4.89. The electromagnetic Coulomb force is not strong enough t o produce the values ~ / m ~ cand ' Arminin Table 4.85, but the strong interaction forcc seems to be right for the required force and distance A r m i n . Hence,
4.8
ENER.GY OR. MASS R.ATIOS FOR
~ / m o> c ~1 AND
Armin
219
~lrnoc' 4 20Q
456.25
400
600
800
1000
\, \, \,
456.5 456.75
'.*'
457
'. "... ,.Po2
.I. . ..
457.25
...
PI2
457.5
.__ ._
,.._......~
1
457.75 458
f
F1G.4.87. The functions poz, pl2 according to Eqs.(l7) and (24) for Z = 10000, 5 1000. The intersection of poz and
p = 688, 1 = 1 in the interval 1 5 E/moc2 p12 is at E/rnoc2 = 786.0296.
\ 685.5

1000
1500
2000
\$
\. \,
.' '.
686
',....?2 2J
686.5
. _
Pl2 ._ ,,>= :.
 68
687.5
,
500
'
I
688
F1G.4.88. The functions po2, p12 according to Eqs.(l7) and (24) for Z = 15000, p = 1032, 1 = 1 in the interval 1 5 ~ l m o c ' 5 2000. The intersection of poz and p12 is at ~/m.oc' = 1486.672.
500
915.25
\*
915.5
1 0 0 0 1 5 0 0 2000 2 5 0 0 3000
'* '*
915.75
'.
"..
. %Po2 ___ P12 .._ ._  9 1 6 . 2 5 ,pz= 916
916.5 916.75 917
1' 3
:
F1G.4.89. The functions poz, pl2 according to Eqs.(l7) and (24) for Z = 20000, 5 ~ l r n o c ' 5 3000. The intersection of po2 and p12 is at ~ l r n o = c ~2680.501. p = 1376, 1 = 1 in the interval 1
220
4 DIR.AC DIFFERENCE EQUATION IN SPHERICAL COORDINATES
TABLE 4.85 CHARGE NUMBER Z, QUANTIZATION NUMBER p, AND T H E LISTOF FIGURES, RESULTING VALUESOF E/rn0c2 FOR THE INTERSECTIONS O F pon AND p l a , mocAr,i,/h ACCORDING TO E Q . ( ~ AND ) AT,;, ACCORDING TO E ~ . ( 2 6 )1=1. ; Figure
Z
p
~ l r n . 0 ~ ~r n o c ~ ~ , ; , / hAT,;, [m]
the replacement of the electromagnetic Coulomb force in Dirac's difference equation by a centrally symmetric strong interaction force may lead to the theoretical derivation of mass ratios as an eigenvalue problem. Due to the lack of a strong interaction force that is as simple and universally accepted as the Coulomb force, only experts of the strong interaction force can expect to solve this problem in a satisfactory and generally acceptable way. We turn to the solution in the point s = s2 = i in Fig.4.61. Equations (1) and (2) hold again. From Eq.(6.842) we obtain with so = s 2 = i, si = 1 with @ written rather than p to distinguish it from p in Eqs.(4) and (15):
We recognize that @ is the negative of p in Eqs.(4) and (15) if ~ / r n , o cis~ replaced by ~ / (  m ~ c ~ ) :
d
Next we substitute so = s2 = i into Eq.(6.852) and obtain with m = Xe and @ = p the relation
~ c ~ by El(moc2). This is Eq.(22) with a reversed sign and ~ / m replaced The sign reversal of moc2 is trivial here due to the ambiguity of the sign f in front of 87rZa:
4.8 ENER.GY OR MASS RATIOS FOR E/moc2 > 1 AND Armin
221
Equations (30) and (33) suggest to associate the solution at s = i in Fig.4.61 with the solution at s = 1 for antiparticles, while the solution at s = i is associated with the solution at s = +1 for particles. The results in Tables 4.82, 4.84, and 4.85 hold both for particles and antiparticles if we replace moc2 by moc2. In Table 4.83 we must also reverse the signs of P o 2 , P l 2 l and p.
5 Inhomogeneous Equations for Coulomb Potential 5.1 QUANTIZATIONOF THE INHOMOGENEOUSTERM We turn to the quantization of the inhomogeneous term of Eq.(4.39). The term Q defined in Eq.(l.l44) makes this term very complicated:
Since we do not know how to quantize ( p  eA,)2 in a derlominator we use a series expansion to remove the denominator for small values
( p  e ~ , ) ~ / r n ; c Xo. To obtain the required value of Xo we proceed in the following way. Let 6 run frorri 0 to n. The variable x then runs from 1 to 1, and X runs from l/Axto l/Ax.The difference Ax is arbitrarily small but finite. Hence, we need factorial series with abscissa of convergence Xo < l/Ax. This is possible if the factorial series of Eqs.(44) and (45) terminate, which calls for negative integer values of pl or p4:
The series expansions of Eqs.(44) and (45) become polynomials and the abscissa of convergence Xo becomes w:
Equations (42), (43), and (46) yield the eigenvalues
The requirement stated in the text following Eq.(33) is satisfied. The next step is the solution of the functions xl(p) and x4(p) on the left side of Eqs.(21) and (22). This task is carried out in the following Section 6.8. We need there only Xel and Xe4 according to Eqs.(42) and (43) but not the coefficients C1, and C4" of Eqs.(47) and (48).
6.8
SOLUTION O F
~'(p)
AND
6.8 SOLUTION O F xl(p)
~4(p)
AND
279
x4(p)
The left equations for xl(p) and x4(p) in Eqs.(6.721) and (6.722) are written in the following form:
For a Coulomb potential 4,(p) we get from Eq.(2.32):
Equations (1) and (2) become in formal notation
and in explicit notation:
We use the Laplace transform to solve these equations. This requires rewriting the factor p according to the following scheme:
280
6 APPENDIX
Equations (9) and (10) are rewritten:
[(P+ 1)  lIxi(p
+ 1 )  2iXeixi(p)  [ ( p 1 ) + 1 ] ~ 1 (p 1 ) i(wPp+8nZa)x4(p) = O
(13)
The Laplace transform of these two equations requires the following six expressions:
Substitution into Eq.(12) yields:
We leave out the integration and multiply with equation for y4 and yl is obtained:
sP+~.
A differential
+ + 2iXe4~+ l ) y 4 + iwms2y', + 8niZasyl = 0
(s2 1 ) s ~ ; (s2
(16)
Froni Eq.(13) we get in analogy:
We represent yl and y4 by series expansions in an arbitrary point S
= so:
6.8
y4 =
SOLUTION OF
C d,, (s  SO)^"',
We shorten s  s o to x and

yi
xl(p) AND x4(p)
=
1 dv(l)f
V)(S SO)^+"^
for these equations only

Xel, Xe4 to Xe:
Substitution of Eqs.(l8) and (19) into Eqs.(l6) and (17) yields:
Consider the terms with the power xp+u in Eq.(20):
(18)
6 APPENDIX
We replace v by p  1 for a reason that will be recognized in Eq.(24) below and we rewrite Eq.(22) in the following form:
From Eq.(21) we obtain in analogy with ~ ~ , , ~ c l ,  ~= 0:
We write the first few equations of the system defined by Eqs.(23) and (24) in detail:
6.8
SOLUTION OF
xl(p)
AND
x4(p)
From the first equation of Eq.(25) we obtain d o as function of a o :
We want to keep a0 as a choosable constant. This permits us to use the second equation of Eq.(25) for the determination of s o
We recognize the relation
that will be used frequently. Then we derive s o :
284
6 APPENDIX
The singular point so of the series expansions of Eq.(18) can have the four values s 2 , 33, s 4 , and ss:
We may write s: in a form that connects us to the singular points sz and for the series expansion for the KleinGordon equation [Harmuth and Meffert 2005, Sec.5.5, Eqs.(l3), (14), Fig.5.1011. From Eqs.(5) and (6) we obtain s3 obtained
With the definition
we may rewrite Eq.(28):
Hence, s: is equal to the old s:! and s3 of the KleinGordon equation while s o of Eq.(29) is equal to their square roots:
6.8
SOLUTION OF
xl(p)
AND
~4(p)
285
TABLE 6.81 VALUES O F THE SINGULAR POINTS5'2, Ss, FOR VARIOUS VALUES OF g.
Sq,
AND ss
OF
E Q S . ( ~AND ~ ) (36)
F1G.6.81. Loci in the splane of sz(g) (solid line), ss(g) (dashed line), s4(g) (dotted line), and ss(g) (dasheddotted line) according to Eqs.(35) and (36) as well as Table 6.8 1.
Table 6.81 lists certain values of s2, ss, s4, and ss as functions of g. Figure 6.81 shows the loci of s2(g) to s5(g). We return to Eq.(25). If a0 and so are choosable we obtain do from Eq.(26). The third and fourth equation of Eq.(25) may then be written as inhomogeneous equations. Eqliation (26) is rewritten with the help of Eq. (27):
These two equations have a unique solution if the rank of the coefficient matrix equals the rank of the extended coefficient matrix. The coefficient matrix yields with the help of Eqs.(23) and (24):
Since the rank of the coefficient matrix is zero the rank of the extended coefficient matrix must be zero too. The factor a0 on the right side of Eqs.(37) and (38) can be ignored:
Substitution from Eqs.(23), (24), and (27) yields for both equations the same result:
+ +
+
+
( W ~ / W ~ ) ~ /~ [l )~p ( ~25; S: Xe4)so 21  so(2wpp  8nZawm/wp)  so(2wmp 87rZa) = 0
+
(42)
At this juncture we must decide which value of so to choose from Fig.6.81. To this end consider Fig.6.82. It shows so = s3 = 1/2/2 on the left (a). The circle of convergence for series expansions according to Eq.(18) reaches the point s = 0, which is necessary for the application of the Laplace transform of Eq.(14). A smaller value of ss according to Fig.6.82b would be acceptable too, but a larger value would limit the circle of convergence by the singular point s = s 2 rather than s = 0. There are, however, methods that permit larger values of s3 close to 1 or g close to 2. In Section 6.7 we terminated the factorial series of Eqs.(6.744) and (6.745) by choosing negative integer values for p l and p4. A second method is based on conformal mapping and a third method by approaching polyriomials asymptotically (Harmuth and Meffert 2005, Secs. 6.9, 6.10 and text following Eq. 5.521). Substitution of so = s3 = 1  O(Ar), wm,wp = O(Ar), and Xel, Xe4 frorri Eqs(6.749), (6.750) into Eq.(42) yields an equation for p:
We are interested in solutions for which either ll or l4 but not both are zero. This permits us to simplify writing by using l for ll or l4 and add the subscript 1 or 4 when needed. Equation (43) becomes:
6.8 SOLUTION
OF XI(/)) AND ~ 4 ( p )
b
a
F1c.6.82. Largest real value of ss that permits the circle of convergence to be limited by the singular point s = 0 as well as by s2 (a). A smaller value of ss yields a circle of convergence that is only limited by s = 0 (b), while a larger value of s3 makes sz limit the circle of convergence.
+
We substitute p = p~ ipl. The real and the imaginary part of Eq.(44) are zero individually. Using Eq.(32) we get:
Once more we return to Eq.(25). Since a0 is choosable and do is derived from a 0 and so we may leave out the first two equations. We write the remaining equations schematically as follows:
In order t o obtain an asymptotically approached polynomial .we demand that the last two variables a,, d, be zero. We have then 21, variables ao, do, . . . a,1, d,1 and 21, equations starting with 61,4, a 1 , 4 and ending with 6,,0 = 0, Q,,o. T he determinant of the coefficients must be zero:
/ 61,4
5
61,6
61,7
Q1,4
1
Q1,6
a1,7
62,2
62,3
62,4
62,s
62,6
62,7
Q2,2
a2,3
a2,4
a2,5
a2,6
a2,7
=0
0 DL1,0
b~l,l 0
611,2
6'1,3
611,4
611,5
611,6
611,7
QL1,2
a'l,3
@'1,4
a~1,s
a b  1 , ~
a~1,7
0 Q.,o
1
0
6L,2
663
6L,4
645
2
Q L , ~
QL4
QL5
1
(48) For the solution of this equation we multiply the second column from the right with 6L,5/6L,4and subtract it from the last column on the right. The lower right corner of the determinant assumes the following form:
Next we multiply the last row with
and subtract it from the third from last row. Finally, we multiply the last row with
6.8 SOLUTION OF
xl (p) AND x4(p)
289
and subtract if from the fourth from last row. Equation (49) assumes the following form:
The terms 8Ll,6and (1.L1,6 may readily be calculated but their value is of no interest here. Equations (50) and (48) are satisfied for SL,4aL,5  aL,4SL,5 =0 Substitutiorl from Eqs.(23) and (24) brings:
+ (8: + 2ihe4so + l)(sa + 2 i k l s o + 1)) = 0
(52)
We substitute s o = SQ = 1O(Ar) as well as .Ael = Xe, Xe4 = 0 or Xe4 = Xe, Xel = 0 according to Eq.(44), and p = p~ + ipI according t o Eq.(45). The real and the imaginary part of the equation are zero individually. The real part yields
and the imaginary part brings
This equation is more complicated than it looks due to the ambiguity There are three different solutions: of the signs of Xe = fJ.
6
290
APPENDIX
Equations (56) and (57) hold only for 1 = 0 and 1 = 1, while Eqs.(55) and (58) have no such restriction. We turn to Eq.(53). The first two terms may be rewritten into the following form (PR
+

2
1)
+ 2(pR + L  1) = (PR + L)2 1
(59)
and Eq. (45) yields:
In order to keep track of signs we write some intermediate steps of the solution for E:
The signs of the terms of {. . .
are reversed
and we obtain
In analogy to Eqs.(55)(58) we check what p; and p~Xeyield for the sign anlbiguity At = f using Eqs.(44) and (46):
6.8
SOLUTION OF
xl(f)
AND
~4(f)
For pIXe one obtains the same result but 114 is replaced by 112:
prhe =
(fad)
(~m)
Equations (69) or (70) added to any one of Eqs.(64)(67) yields a negative value and Eq.(63) yields complex solutions for the energy E. Only two pIXe are possible: positive surrls of pf
+
Using Eq.(72) arid k 2 we obtain from Eq.(63)
Comparison of Eq.(25) with Eq.(48) shows that L 1 = n' is the number of pairs of equations for a l l d l , to a,1, d,l before the termination a, = 0, d, = 0. The series exparlsion of Eq.(74) yields:
For the inverse Laplace transform let the path of integration in Eq.(14) begin a t the origin, run around ss and return to the origin as shown by the line t in Fig.6.83a. The point s2 remains outside the loop. More details of the integratiori may be found in the publications of Guldberg and Wallenberg (1911), Nijrlund (1910, 1915, 1924, 1929), and MilneThomson (1951, p. 485). One obtains two factorial series for xl(p) and ~ 4 ( p ) :
6 APPENDIX
F1c.6.83. Location of the singular points sz = sz(g) and se = ss(g) in the complex plane according to Eqs.(35) and (36) for g < 2; (a) shows the integration path l from 0 around ss; (b) shows the circle of convergence around ss that is limited by sz and does not reach 0.
Equations (23), (24), (36), (45), and (46) define up, dp, ss, and p. The fraction ( p 1). . . ( p p)/(p p 1). . . (p p p ) shall always be replaced by 1 for p = 0. This convention avoids the need of writing a special term for p = 0. The upper limit N of the sums is usually replaced by oo since the conventional calculus of finite differences eliminates only the infinitesimal but not the infinite. But we have, according to Eq.(4.317), N intervals Ar = cT/N in the range 0 5 r 5 cT or N intervals Ap = 1 in the range 0 5 p N , which implies N 1 points 0, 1, 2, . . . , N. Hence, the series in Eqs.(76) and (77) can have only N + 1 orthogonal or linearly independent functions, and an upper limit p = N must be used. Unfortunately, this does not mean that the concept of divergence is eliminated. In the case of Fig.6.83b the power series does not converge everywhere along the path l of integration in Fig.6.83a. As a result the factorial series of Eqs.(76) and (77) may diverge for all values of p. To obtain convergence one may decrease the value of sg to ss = 1 1 4 as shown in Fig.6.82a. The resulting value of g = 2.5 yields with the help of Eqs.(31), (32), (5), and (6):
+