;
!
I
J )
QUANTUM
/'
MECHANICS
QUANTUM MECHANICS F.
MANDL,
M.A.,
D.Phil.
Department of Theoretical Physics, ...

Author:
Franz Mandl

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;

!

I

J )

QUANTUM

/'

MECHANICS

QUANTUM MECHANICS F.

MANDL,

M.A.,

D.Phil.

Department of Theoretical Physics, UniversilJ of Manchester

SECOND EDITION

LONDON

BUTTERWORTHS 1 957

ENGLAND:

BU1TERWORTH & co. (PUBUSHERS) LTD. LONDON: 88 Kingsway, W.C.2

AUSTRALIA:

BUTTERWORTH & CO. (AUSTRALIA) LTD. SYDNEY: 20 Loftus Street MELBOURNE: 473 Bourke Street BRISBANE: 24{) Queen Street

CANADA:

BUTTERWORTH & CO. (CANADA) LTD. TORONTO: 1367 Danforth Avenue, 6

NEW ZEALAND: BUTrERWORTH & CO. (NEW ZEALAND) LTD. WELLINGTON: 49/51 Ballance Street AUCKLAND: 35 High Street SOUTH AFRICA: nUTrERWORTH & CO. (SOUTH AFRICA) LTD. DURBAN: 33/35 Beach Grove U.S.A.:

BUTrERWORTH INC. WASHINGTON, D.C.: 7300 Pearl Street, 20014

1st Edition, 1954 2nd Edition, 1957 2nd Impression, 1958

3rd

Impression, 1960

4th Impression, 1961 5th Impression, 1962 6th Impression, 1964 7th Impression, 1966

@

Butterworth & Co. (Publishers) Ltd. 1957

Set in Monotypl1 Baskervill, type

Mau and prinkd in Greal BriltJin b)' WiJlitJ1ll Clowes IUId Sons, Limi", London and Bledu

CONTENTS PAGE viii

PREFACE TO THE SECOND EDITION..

ix

PREFACE TO THE FIRST EDITION

1.

1

MATHEMATICAL INTRODUCTION

1 5 8 14 20

1. 2. 3. 4.

Vectors Vibrating string Mathematical generalizations Continuous spectra Exercises I

22

II. WAVE-MECHANICAL CONCEPTS

5. 6. 7.

The Schrodinger wave equation: derivation The SchrOdinger wave equation: discussion Energy eigenfunctions: theory Exercises II III. ENERGY EIGENFUNCTIONS: EXAMPLES

8. 9.

One-dimensional potential barrier One-dimensional square well potential with rigid walls lO. Free particle wave functions and three-dimensional square well potential II. Linear harmonic oscillator 1 2. Spherically symmetric potentials in three dimensions 1 3. Spherical harmonics 14. Angular momentum Exercises III IV. GENERAL PRINCIPLES OF

QUANTUM

IS. Resume of classical mechanics

16. 17. 18. 19. 20.

MECHANICS

Interpretative postulates of quantum mechanics Operators with continuous spectra Functions of operators Simultaneous measurability of observables Uncertainty principle Exercises IV v

23 25 31 34 36 36 38 40 43 47 49 54 56 58 58 60 70 71 74 82 88

CONTENTS PAGE

V.

90

MATRIX MECHANICS

21. Matrix representation of wave functions and operators.. 90 22. Properties of matrix operators 93 93 (0) Matrix multiplication 95 (6) Adjoint matrices 95 (c) Change of basis 97 (d) Eigenvalue problems 100 23. The equations of motion of operators 24-. The Dirac notation for wave functions and operators .. 101 102 Exercises V VI. SYSTEMS OF MANY PARTICLES

104-

VII. TIME-INDEPENDENT PERTURBATION THEORY

131

25. Angular momentum 26. Spin 27. Addition of angular momenta 28. Systems of identical particles: Pauli principle Exercises VI

29. First-order perturbation: non-degenerate case . . 30. First-order perturbation: degenerate case 31. Simple examples on perturbation theory Example I : Ground state of the helium atom Example 2: Two identical particles Example 3: Normal Zeeman effect Exercises VII ..

104l lO l l6 125 130

131 134 136 136 137 140 141 142

VIII. COLLISION PROCESSES 32. Elastic scattering by a fixed centre of force 33. The integral equation of the scattering problem 34. Born approximation Example 1: Scattering by a square well potential Example 2: Scattering by a screened Coulomb potential 35. Laboratory and centre of mass coordinate systems 36. Inelastic scattering processes 37. Time-dependent approach to collision problems 38. Partial waves Exercises VIII .. vi

142 145 149 151 151 153 158 161 166 175

CONTENTS

IX.

X.

INTRODUCTION TO GROUP-THEORETICAL IDEAS 39. Groups of transformations 40. Rotation operators 41. Invariants of groups of transformations 42. Spin 43. The representation of groups 44. Perturbations 45. Selection rules Exercises IX

RELATIVISTIC THEORY OF THE ELECTRON 46. The Dirac equation for a free particle 47. The spin and magnetic moment of the electron 48. Plane wave solutions of the Dirac equation 49. Relativistic invariance Exercises X

HINTS

FOR SOLVING

THE

EXERCISES

PAGE 177 177 180 181 183 186 191 195 201 203 203 209 212 216 222 224

ApPENDIX A: VECTOR ANALYSIS

261

BIBUOGRAPHY

263

INDEX

265

vii

PREFACE TO THE SECOND EDITION IN this second edition the approach and scope of the first edition have not been altered. In particular, no ,lltcmpt has been made to deal morc comprehensively with standard applications of quantum mechanics which arc treated well in existing textbooks. However, following several suggestions, two new topics arc included. I have adeled a section on partb! wave analysis i n scattering theOl),. This subject is so important in atomic, nuclear and high-energy physics that it seemed desirable to include it. A new chapter at lhe end of lhe book is devoted to the Dirac equation. This contains a discussion of the relativistic invariancc orthe theory which underlies the construction of interactions in beta-decay and field theories generally. In this chapter some knowledge or the special theory or relativity is pre supposed on the part or the reader. In addition, a number of smaller changes and corrections, ror some of which t :Ull indebted to readers, have been incorporated. or these I should mention the redellnition or spherical harmonics (section 13) so that their phases now agree with those used almost exclusively in angular momentum problems. I should again like to express my gratitude to my friend Dr. Handel Davies, of Christ Church, with whom I discussed the proposed changes and who read and criticized the new material; also to my wife, Bett)', ror assisting me throughout the preparation or this edition.

Harwell, Berks Apdl, 1957

VIII

PREFACE TO THE FIRST EDITION

IN the autumn of 1952 I gave a course of lectures on quantum mechanics to experimentalists at A.E.R.E., Harwell, in which I developed the theory in such a way as to bring out the unifYing mathematical scheme underlying quantum mechanics. This book, based on these lectures, uses the same approach. In the first five chapters the mathematical formalism of quantum mechanics is systematically developed, with particular attention to its physical interpretation. Mathematical rigour or generality are not aimed at, the stress being on a strongly geometrical formulation of the mathematics, given in considerable detail and kept as simple as possible. The mathematical methods underlying quantum mechanics are developed in Chapter I, new physical concepts being introduced only in Chapter II. The simple but important applications of Chapter III should make the reader familiar with these concepts which, in Chapter IV, are extended to the general scheme of quantum mechanics. Questions of the measurability of one or more observables are here dis cussed in much greater detail than is customary in all but very advanced textbooks, but I believe that an understanding of these problems is essential to an understanding of quantum mechanics. Matrix mechanics is treated in Chapter V, not using correspondence arguments (or mere formal analogies) from classical mechanics, which are difficult to appreciate, but as a particular representation of the general theory, as it arises in practice. The use of the general theory in solving specific problems is illustrated in the remaining four chapters. Since many existing books deal satisfactorily with such applications, I have restricted myself to a few topics, of particular importance in many branches of applied quantum mechanics. Chapter VI deals with angular momentum, spin and symmetry properties of systems of particles. These questions are so important in atomic and nuclear physics that I have treated them in considerable detail. Perturbation theory, one of the most useful methods for handling complicated problems, is treated in Chapter VII, while Chapter VIII is devoted to scattering problems. The latter form such a vast subject that I have only been able to deal with some principal aspects; some topics, such as the phase-shift analysis of scattering problems, have been omitted, as they are treated well in many existing books. In Chapter IX the basic ideas of group theory are developed and applied to perturbation theory, and the derivation of conservation laws and selection rules. The introduction of group theoretical concepts seemed desirable since they lead to a uniform ix

PREFACE

treatment of many diverse problems, and occur frequently in the literature. At the end of each chapter, examples are given, which form an integral part of the book as reference is made to them and a knowledge of them is assumed later. A section giving detailed hints for their solutions is therefore included, and the reader is advised to study this, even if he does not work through the examples himself. The knowledge presupposed on the part of the reader is kept to a minimum. Apart from some classical physics, calculus and vector analysis, he is assumed to be familiar with the qualitative concepts of quantum theory, such as the wave nature of matter, and their experi mental basis. It is thus hoped that the book is suitable for a wide class of reader, including experimental research worker and student alike, and should enable them to perform quantum-mechanical calculations and read the theoretical literature. In spite of its lack of mathematical rigour, the book should serve the theoretical student as an introduction to more advanced works. In my approach to quantum mechanics, I have been greatly in fluenced by the writings of J. von Neumann and by Dr. J. S. deWet (Oxford) whose lectures first introduced me to some of these ideas, and it is a pleasure to thank him for many discussions since then. I am very grateful to Dr. Handel Davies (Oxford), with whom I discussed many aspects of this book, for a critical reading of the manuscript and for suggesting many substantial improvements, to Dr. J. M. Cassels (Liverpool), for some valuable criticisms of the text from the viewpoint of an experimentalist, and to my wife, Betty, who is responsible for many clarifications and who helped me at all stages of proof reading. I am indebted to many of my colleagues for helpful discussions arising out of my lectures, to Dr. E. Bretscher (A.E. R.E.) for encouragement to write this book and to the Director of A.E. R.E., Harwell, for per. mission to publish it.

Harwell, Berks

F.M.

April, 1954

x

I MATHEMATICAL INTRODUCTION THE physical concepts underlying quantum mechanics are radically different from those of classical physics. It is all the more helpful that the same mathematical methods, developed mainly by mathematicians of the nineteenth century, are used in the two theories. In this chapter we shall introduce, and develop as far as required, the main ideas of these mathematical methods, particularly those con cerning eigenvalue problems, illustrating them by a simple problem of classical physics, that of the vibrating string. We shall not introduce any new physical ideas until Chapter II. We shall not aim at mathematical rigour or generality. What we shall do, is to present the underlying intuitive ideas of this very beautiful branch of mathematics as clearly as possible. We shall see that the mathematical formalism lends itself to a strongly geometrical interpre tation which helps towards its understanding. To formulate this geometrical picture we would recall some of the basic ideas of vector analysis. I.

Vectors

Let us, to begin with, consider vectors in ordinary three-dimensional space. We shall however use a notation slighdy different from the conventional one which will be more suitable for our purposes. We introduce a system of rectanID!lar coordinates Xh X2, X3 (usually denoted by x, y, z). A point P in space is then labelled by giving its coordinates (Xh X2, X3) ' We define a vector as having comp�nents .&U.J-.!(�)'-'1(3), in this cOQrd�nate system The simplest example of a vector is the position vector r of the point P; its components are simply the coordinates of P: r(l) =Xh etc. Other physical quantities which are vectors are velocity, force and acceleration. If and v are vectors, then C.U +C2V where c. and C2 are real numbers, is the vector with com ponents c.u(l) + c2v(I ), etc.

u

• ..

u

We define the norm

N(u) ot the vector u by N(u) = [u( I»)2 + [u(2)] 2 + [u(3»)2 3 =2Ju(r)] 2 ,-1

and we see that the norm is positive for any non-zero vector. length or magnitude of u is defined by

lIuli

=

VN(u)

(1 ) The

(2 )

I.

VECTORS

where the positive square root is to be taken. For the position vector r,

I1rll = ,/(XL 2 + .\'"22 +X)2). We call u a normali;;.ed or normed vector or a unit vector if it is of unit length (3) lIull-J. Since cu (where c is a real number) is a vector parallel to u, of magni. tude cl/ull , we can normalize any vector u by multiplying by a faClOr l/iluli. the normed vector being u/llull. ""'"

Figure I.

DuompoJi/ioTi � n zmil ute/or into its (omponenfs

II

"

"

uflle,yL I,

_ _____

--' v

Given two vectors u and v (components u(r) and v(r), r-l, 2, 3), we define their scalar or inlleT product by (u, v)

�

u( I) ,(I) +«(2),(2) + u(3),(3) (4)

As is weB known, the inner product can also be expressed in terms of lhe magnitudes ofo and v and the angle 0 between the two vectors, (u, v) -llullllvll oos O.

(4*)

From equations (I) and (+) the inner product of a vector u with itself is its norm, N(u)-(u, u). (5) 2

1.

VECTORS

Tw.Q vectorsu andy.are�nDnnaLor (JrthDgD1Yllif�=�/�, i.e. ID�� inn�er product vanishes, (6) (u, v) =0. If we introduce three unit vectors -e, (,= I , 2, 3) in the directions of the coordinate axes x, we can write any vector u in the fonn

u =u( l )el + u(2)ez + u(3)e3 3 = 2u (,) e,. ,-1

(7)

The geometrical interpretation of this equation is shown in Figure 1. But there is nothing special about the three orthogonal unit vectors e,• which distinguishes them from any other set of three vectors a2' (ii) d( -x) "(x). (iii) " ( Ax) =�"(x) ( )' >O) . (iv) x"'(x) = - d(x) . ( ) If r is the vector with components (x, z) we define the three dimensional "-function "(r) "(x) "(l)"(Z). (18) d(r) has similar properties to "(x); in particular �e the domain ofintegration, ff(r) d(r ro)d {f(0, ro), �fro �es insi outsIde " " " , d-r being an element of volume. (i)

If 61

"I

=

y,

e

=

_

-r =

If ro lIes

• A more detailed treatment of the Dirac delta function will be found in the books by Schiffll and Dirac IS, mentioned in the bibliography at the end of this book.

19

EXERCISES

.lIt

tions

Show that for

I

I

I

EXERCISES

1

the fundamental domain - L <x 0 where! (9', '1')9' - (9', phil.) 1.8. Derive and solve the roUmving eigenvalue problem which occurs in the theory or a vibrating square membrane whose sides, orlength L, are kept fixed : iJ2w ,l2w (1.8.) -0, ax2 + �Jlz + ).w ..

w(O,y) -w(L,y) -0, (0

!

I

J )

QUANTUM

/'

MECHANICS

QUANTUM MECHANICS F.

MANDL,

M.A.,

D.Phil.

Department of Theoretical Physics, UniversilJ of Manchester

SECOND EDITION

LONDON

BUTTERWORTHS 1 957

ENGLAND:

BU1TERWORTH & co. (PUBUSHERS) LTD. LONDON: 88 Kingsway, W.C.2

AUSTRALIA:

BUTTERWORTH & CO. (AUSTRALIA) LTD. SYDNEY: 20 Loftus Street MELBOURNE: 473 Bourke Street BRISBANE: 24{) Queen Street

CANADA:

BUTTERWORTH & CO. (CANADA) LTD. TORONTO: 1367 Danforth Avenue, 6

NEW ZEALAND: BUTrERWORTH & CO. (NEW ZEALAND) LTD. WELLINGTON: 49/51 Ballance Street AUCKLAND: 35 High Street SOUTH AFRICA: nUTrERWORTH & CO. (SOUTH AFRICA) LTD. DURBAN: 33/35 Beach Grove U.S.A.:

BUTrERWORTH INC. WASHINGTON, D.C.: 7300 Pearl Street, 20014

1st Edition, 1954 2nd Edition, 1957 2nd Impression, 1958

3rd

Impression, 1960

4th Impression, 1961 5th Impression, 1962 6th Impression, 1964 7th Impression, 1966

@

Butterworth & Co. (Publishers) Ltd. 1957

Set in Monotypl1 Baskervill, type

Mau and prinkd in Greal BriltJin b)' WiJlitJ1ll Clowes IUId Sons, Limi", London and Bledu

CONTENTS PAGE viii

PREFACE TO THE SECOND EDITION..

ix

PREFACE TO THE FIRST EDITION

1.

1

MATHEMATICAL INTRODUCTION

1 5 8 14 20

1. 2. 3. 4.

Vectors Vibrating string Mathematical generalizations Continuous spectra Exercises I

22

II. WAVE-MECHANICAL CONCEPTS

5. 6. 7.

The Schrodinger wave equation: derivation The SchrOdinger wave equation: discussion Energy eigenfunctions: theory Exercises II III. ENERGY EIGENFUNCTIONS: EXAMPLES

8. 9.

One-dimensional potential barrier One-dimensional square well potential with rigid walls lO. Free particle wave functions and three-dimensional square well potential II. Linear harmonic oscillator 1 2. Spherically symmetric potentials in three dimensions 1 3. Spherical harmonics 14. Angular momentum Exercises III IV. GENERAL PRINCIPLES OF

QUANTUM

IS. Resume of classical mechanics

16. 17. 18. 19. 20.

MECHANICS

Interpretative postulates of quantum mechanics Operators with continuous spectra Functions of operators Simultaneous measurability of observables Uncertainty principle Exercises IV v

23 25 31 34 36 36 38 40 43 47 49 54 56 58 58 60 70 71 74 82 88

CONTENTS PAGE

V.

90

MATRIX MECHANICS

21. Matrix representation of wave functions and operators.. 90 22. Properties of matrix operators 93 93 (0) Matrix multiplication 95 (6) Adjoint matrices 95 (c) Change of basis 97 (d) Eigenvalue problems 100 23. The equations of motion of operators 24-. The Dirac notation for wave functions and operators .. 101 102 Exercises V VI. SYSTEMS OF MANY PARTICLES

104-

VII. TIME-INDEPENDENT PERTURBATION THEORY

131

25. Angular momentum 26. Spin 27. Addition of angular momenta 28. Systems of identical particles: Pauli principle Exercises VI

29. First-order perturbation: non-degenerate case . . 30. First-order perturbation: degenerate case 31. Simple examples on perturbation theory Example I : Ground state of the helium atom Example 2: Two identical particles Example 3: Normal Zeeman effect Exercises VII ..

104l lO l l6 125 130

131 134 136 136 137 140 141 142

VIII. COLLISION PROCESSES 32. Elastic scattering by a fixed centre of force 33. The integral equation of the scattering problem 34. Born approximation Example 1: Scattering by a square well potential Example 2: Scattering by a screened Coulomb potential 35. Laboratory and centre of mass coordinate systems 36. Inelastic scattering processes 37. Time-dependent approach to collision problems 38. Partial waves Exercises VIII .. vi

142 145 149 151 151 153 158 161 166 175

CONTENTS

IX.

X.

INTRODUCTION TO GROUP-THEORETICAL IDEAS 39. Groups of transformations 40. Rotation operators 41. Invariants of groups of transformations 42. Spin 43. The representation of groups 44. Perturbations 45. Selection rules Exercises IX

RELATIVISTIC THEORY OF THE ELECTRON 46. The Dirac equation for a free particle 47. The spin and magnetic moment of the electron 48. Plane wave solutions of the Dirac equation 49. Relativistic invariance Exercises X

HINTS

FOR SOLVING

THE

EXERCISES

PAGE 177 177 180 181 183 186 191 195 201 203 203 209 212 216 222 224

ApPENDIX A: VECTOR ANALYSIS

261

BIBUOGRAPHY

263

INDEX

265

vii

PREFACE TO THE SECOND EDITION IN this second edition the approach and scope of the first edition have not been altered. In particular, no ,lltcmpt has been made to deal morc comprehensively with standard applications of quantum mechanics which arc treated well in existing textbooks. However, following several suggestions, two new topics arc included. I have adeled a section on partb! wave analysis i n scattering theOl),. This subject is so important in atomic, nuclear and high-energy physics that it seemed desirable to include it. A new chapter at lhe end of lhe book is devoted to the Dirac equation. This contains a discussion of the relativistic invariancc orthe theory which underlies the construction of interactions in beta-decay and field theories generally. In this chapter some knowledge or the special theory or relativity is pre supposed on the part or the reader. In addition, a number of smaller changes and corrections, ror some of which t :Ull indebted to readers, have been incorporated. or these I should mention the redellnition or spherical harmonics (section 13) so that their phases now agree with those used almost exclusively in angular momentum problems. I should again like to express my gratitude to my friend Dr. Handel Davies, of Christ Church, with whom I discussed the proposed changes and who read and criticized the new material; also to my wife, Bett)', ror assisting me throughout the preparation or this edition.

Harwell, Berks Apdl, 1957

VIII

PREFACE TO THE FIRST EDITION

IN the autumn of 1952 I gave a course of lectures on quantum mechanics to experimentalists at A.E.R.E., Harwell, in which I developed the theory in such a way as to bring out the unifYing mathematical scheme underlying quantum mechanics. This book, based on these lectures, uses the same approach. In the first five chapters the mathematical formalism of quantum mechanics is systematically developed, with particular attention to its physical interpretation. Mathematical rigour or generality are not aimed at, the stress being on a strongly geometrical formulation of the mathematics, given in considerable detail and kept as simple as possible. The mathematical methods underlying quantum mechanics are developed in Chapter I, new physical concepts being introduced only in Chapter II. The simple but important applications of Chapter III should make the reader familiar with these concepts which, in Chapter IV, are extended to the general scheme of quantum mechanics. Questions of the measurability of one or more observables are here dis cussed in much greater detail than is customary in all but very advanced textbooks, but I believe that an understanding of these problems is essential to an understanding of quantum mechanics. Matrix mechanics is treated in Chapter V, not using correspondence arguments (or mere formal analogies) from classical mechanics, which are difficult to appreciate, but as a particular representation of the general theory, as it arises in practice. The use of the general theory in solving specific problems is illustrated in the remaining four chapters. Since many existing books deal satisfactorily with such applications, I have restricted myself to a few topics, of particular importance in many branches of applied quantum mechanics. Chapter VI deals with angular momentum, spin and symmetry properties of systems of particles. These questions are so important in atomic and nuclear physics that I have treated them in considerable detail. Perturbation theory, one of the most useful methods for handling complicated problems, is treated in Chapter VII, while Chapter VIII is devoted to scattering problems. The latter form such a vast subject that I have only been able to deal with some principal aspects; some topics, such as the phase-shift analysis of scattering problems, have been omitted, as they are treated well in many existing books. In Chapter IX the basic ideas of group theory are developed and applied to perturbation theory, and the derivation of conservation laws and selection rules. The introduction of group theoretical concepts seemed desirable since they lead to a uniform ix

PREFACE

treatment of many diverse problems, and occur frequently in the literature. At the end of each chapter, examples are given, which form an integral part of the book as reference is made to them and a knowledge of them is assumed later. A section giving detailed hints for their solutions is therefore included, and the reader is advised to study this, even if he does not work through the examples himself. The knowledge presupposed on the part of the reader is kept to a minimum. Apart from some classical physics, calculus and vector analysis, he is assumed to be familiar with the qualitative concepts of quantum theory, such as the wave nature of matter, and their experi mental basis. It is thus hoped that the book is suitable for a wide class of reader, including experimental research worker and student alike, and should enable them to perform quantum-mechanical calculations and read the theoretical literature. In spite of its lack of mathematical rigour, the book should serve the theoretical student as an introduction to more advanced works. In my approach to quantum mechanics, I have been greatly in fluenced by the writings of J. von Neumann and by Dr. J. S. deWet (Oxford) whose lectures first introduced me to some of these ideas, and it is a pleasure to thank him for many discussions since then. I am very grateful to Dr. Handel Davies (Oxford), with whom I discussed many aspects of this book, for a critical reading of the manuscript and for suggesting many substantial improvements, to Dr. J. M. Cassels (Liverpool), for some valuable criticisms of the text from the viewpoint of an experimentalist, and to my wife, Betty, who is responsible for many clarifications and who helped me at all stages of proof reading. I am indebted to many of my colleagues for helpful discussions arising out of my lectures, to Dr. E. Bretscher (A.E. R.E.) for encouragement to write this book and to the Director of A.E. R.E., Harwell, for per. mission to publish it.

Harwell, Berks

F.M.

April, 1954

x

I MATHEMATICAL INTRODUCTION THE physical concepts underlying quantum mechanics are radically different from those of classical physics. It is all the more helpful that the same mathematical methods, developed mainly by mathematicians of the nineteenth century, are used in the two theories. In this chapter we shall introduce, and develop as far as required, the main ideas of these mathematical methods, particularly those con cerning eigenvalue problems, illustrating them by a simple problem of classical physics, that of the vibrating string. We shall not introduce any new physical ideas until Chapter II. We shall not aim at mathematical rigour or generality. What we shall do, is to present the underlying intuitive ideas of this very beautiful branch of mathematics as clearly as possible. We shall see that the mathematical formalism lends itself to a strongly geometrical interpre tation which helps towards its understanding. To formulate this geometrical picture we would recall some of the basic ideas of vector analysis. I.

Vectors

Let us, to begin with, consider vectors in ordinary three-dimensional space. We shall however use a notation slighdy different from the conventional one which will be more suitable for our purposes. We introduce a system of rectanID!lar coordinates Xh X2, X3 (usually denoted by x, y, z). A point P in space is then labelled by giving its coordinates (Xh X2, X3) ' We define a vector as having comp�nents .&U.J-.!(�)'-'1(3), in this cOQrd�nate system The simplest example of a vector is the position vector r of the point P; its components are simply the coordinates of P: r(l) =Xh etc. Other physical quantities which are vectors are velocity, force and acceleration. If and v are vectors, then C.U +C2V where c. and C2 are real numbers, is the vector with com ponents c.u(l) + c2v(I ), etc.

u

• ..

u

We define the norm

N(u) ot the vector u by N(u) = [u( I»)2 + [u(2)] 2 + [u(3»)2 3 =2Ju(r)] 2 ,-1

and we see that the norm is positive for any non-zero vector. length or magnitude of u is defined by

lIuli

=

VN(u)

(1 ) The

(2 )

I.

VECTORS

where the positive square root is to be taken. For the position vector r,

I1rll = ,/(XL 2 + .\'"22 +X)2). We call u a normali;;.ed or normed vector or a unit vector if it is of unit length (3) lIull-J. Since cu (where c is a real number) is a vector parallel to u, of magni. tude cl/ull , we can normalize any vector u by multiplying by a faClOr l/iluli. the normed vector being u/llull. ""'"

Figure I.

DuompoJi/ioTi � n zmil ute/or into its (omponenfs

II

"

"

uflle,yL I,

_ _____

--' v

Given two vectors u and v (components u(r) and v(r), r-l, 2, 3), we define their scalar or inlleT product by (u, v)

�

u( I) ,(I) +«(2),(2) + u(3),(3) (4)

As is weB known, the inner product can also be expressed in terms of lhe magnitudes ofo and v and the angle 0 between the two vectors, (u, v) -llullllvll oos O.

(4*)

From equations (I) and (+) the inner product of a vector u with itself is its norm, N(u)-(u, u). (5) 2

1.

VECTORS

Tw.Q vectorsu andy.are�nDnnaLor (JrthDgD1Yllif�=�/�, i.e. ID�� inn�er product vanishes, (6) (u, v) =0. If we introduce three unit vectors -e, (,= I , 2, 3) in the directions of the coordinate axes x, we can write any vector u in the fonn

u =u( l )el + u(2)ez + u(3)e3 3 = 2u (,) e,. ,-1

(7)

The geometrical interpretation of this equation is shown in Figure 1. But there is nothing special about the three orthogonal unit vectors e,• which distinguishes them from any other set of three vectors a2' (ii) d( -x) "(x). (iii) " ( Ax) =�"(x) ( )' >O) . (iv) x"'(x) = - d(x) . ( ) If r is the vector with components (x, z) we define the three dimensional "-function "(r) "(x) "(l)"(Z). (18) d(r) has similar properties to "(x); in particular �e the domain ofintegration, ff(r) d(r ro)d {f(0, ro), �fro �es insi outsIde " " " , d-r being an element of volume. (i)

If 61

"I

=

y,

e

=

_

-r =

If ro lIes

• A more detailed treatment of the Dirac delta function will be found in the books by Schiffll and Dirac IS, mentioned in the bibliography at the end of this book.

19

EXERCISES

.lIt

tions

Show that for

I

I

I

EXERCISES

1

the fundamental domain - L <x 0 where! (9', '1')9' - (9', phil.) 1.8. Derive and solve the roUmving eigenvalue problem which occurs in the theory or a vibrating square membrane whose sides, orlength L, are kept fixed : iJ2w ,l2w (1.8.) -0, ax2 + �Jlz + ).w ..

w(O,y) -w(L,y) -0, (0

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