QUANTUM MECHANICS

D. I. BLOKHINTSEV

QUANTUM

MECHANICS

D. RElDEL PUBLISHING COMPANY

DORDREC HT - HOLLAN 0

OSNOVY KVANTOVoi MEKHANIKI

Gosudarstvennoe izdatel'stvo tekhniko-teoreticheskoi fiteratury

Moskva-Leningrad, 1944

4. izd., Gosudarstvennoe izdatel'stvo Vysshaya Shkola, Moskva, 1963

Translated from the third and fourth Russian editons by J. B. Sykes and M. J. Kearsley

ISBN-13: 978-94-010-9713-0 e-ISBN-13: 978-94-010-9711-6

DOl: 10.1007/978-94-010-9711-6

© 1964 by D. Reidel Publishing Company, Dordrecht, Holland Softcover reprint oftbe hardcover 1st edition 1964

No part of this book may be reproduced in any form by print, photoprint, microfilm, or any other means without permission from the publisher

CONTENTS

PREFACE TO THE SECOND RUSSIAN EDITION XI

PREFACE TO THE ENGLISH EDITION XIII

INTRODUCTION XV

CHAPTER I. FOUNDATIONS OF QUANTUM THEORY

1. Energy and momentum of light quanta 2. Experimental test of the laws of conservation of energy and

momentum for light quanta 3 3. Atomism 7 4. Bohr's theory 12 5. The elementary quantum theory of radiation 15 6. Black-body radiation 18 7. De Broglie waves. The group velocity 20 8. Diffraction of microparticles 24

CHAPTER II. FOUNDA TIONS OF QUANTUM MECHANICS

9. Statistical interpretation of de Broglie waves 31 10. The position probability of a microparticle 33 11. The principle of superposition of states 35 12. Momentum probability distribution of a microparticle 37 13. Mean values of functions of co-ordinates and functions of momenta 39 14. Statistical ensembles in quantum mechanics 41 15. The uncertainty relation 44 16. Illustrations of the uncertainty relation 49 17. The significance of the measuring apparatus 55

CHAPTER III. REPRESENTATION OF MECHANICAL QUANTITIES BY

OPERATORS

18. Linear self-adjoint operators 60 19. The general formula for the mean value of a quantity and the mean

square deviation 63 20. Eigenvalues and eigenfunctions of operators and their physical

significance. 'Quantisation' 65

VI QUANTUM MECHANICS

21. Fundamental properties of eigenfunctions 68

22. General method of calculating the probabilities of the results of measurement 71

23. Conditions for a simultaneous measurement of different mechanical quantities to be possible 73

24. Co-ordinate and momentum operators of a micro particle 74

25. The angular momentum operator of a micro particle 76

26. The energy operator and the Hamilton's function operator 80

27. The Hamiltonian 82

CHAPTER IV. CHANGE OF STATE WITH TIME

28. Schrodinger's equation 86 29. Conservation of number of particles 90 30. Stationary states 93

CHAPTER V. CHANGE OF MECHANICAL QUANTITIES WITH TIME

31. Time derivatives of operators 95 32. Equations of motion in quantum mechanics. Ehrenfesfs

theorems 97 33. Integrals of the motion 99

CHAPTER VI. THE RELA nON BETWEEN QUA NTUM MECHANICS,

CLASSICAL MECHANICS AND OPTICS

34. The transition from the quantum equations to Newton's equations 102

35. The transition from Schrodinger's time-dependent equation to the classical Hamilton-Jacobi equation 106

36. Quantum mechanics and optics 109 37. The quasiclassical approximation (the Wentzel-Kramers-Brillouin

method) 112

CHAPTER VII. BASIC THEORY OF REPRESENT A nONS

38. Different representations of the state of quantum systems 115 39. Different representations of operators of mechanical quantities.

Matrices 116 40. Matrices and operations on them 118 41. Determination of the mean value and spectrum of a quantity

represented by an operator in matrix form 123 42. Schrodinger's equation and the time dependence of operators in

matrix form 125 43. Unitary transformations 44. The unitary transformation from one instant to another 45. The density matrix

128 130 132

CONTENTS

CHAPTER VIII. THEORY OF THE MOTION OF MICROPARTICLES IN A

FIELD OF POTENTIAL FORCES

46. Introductory remarks 47. A harmonic oscillator 48. An oscillator in the energy representation 49. Motion in the field of a central force 50. Motion in a Coulomb field 51. The spectrum and wave functions of the hydrogen atom 52. Motion of an electron in univalent atoms 53. Currents in atoms. The magneton 54. Quantum levels of the diatomic molecule 55. Motion of an electron in a periodic field

CHAPTER IX. MOTION OF A CHARGED MICROPARTICLE IN AN

ELECTROMAGNETIC FIELD

VII

136 137 143 145 152 156 165 167 170 176

56. An arbitrary electromagnetic field 185 57. Motion of a free charged particle in a uniform magnetic field 190

CHAPTER X. INTRINSIC ANGULAR MOMENTUM AND MAGNETIC

MOMENT OF THE ELECTRON. SPIN

58. Experimental proofs of the existence of electron spin 193 59. The electron spin operator 196 60. Spin functions 199 61. Pauli's equation 202 62. Splitting of spectral lines in a magnetic field 205 63. Motion of the spin in a variable magnetic field 209 64. Properties of the total angular momentum 212 65. Labelling of atomic terms having regard to the electron spin.

Multiplet structure of spectra 216

CHAPTER XI. PERTURBATION THEORY

66. Statement of the problem 221 67. Perturbation in the absence of degeneracy 223 68. Perturbation in the presence of degeneracy 227 69. Splitting of levels in the case of twofold degeneracy 231 70. Comments on the removal of degeneracy 234

CHAPTER XII. SIMPLE APPLICATIONS OF PERTURBATION THEORY

71. The anharmonic oscillator 237 72. Splitting of spectral lines in an electric field 239 73. Splitting of spectral lines of the hydrogen atom in an electric

field 242 74. Splitting of spectral lines in a weak magnetic field 246

VIII QUANTUM MECHANICS

75. A diagrammatic interpretation of the splitting of levels in a weak magnetic field (the vector model) 250

76. Perturbation theory for the continuous spectrum 252

CHAPTER XIII. COLLISION THEORY

77. Statement of the problem in collision theory of microparticles 258 78. Calculation of elastic scattering by the Born approximation 262 79. Elastic scattering of fast charged microparticles by atoms 266 80. The exact theory of scattering. The phase shift of the scattered waves

and the cross-section 272 81. The general case of scattering 277 82. Scattering of a charged particle in a Coulomb field 281

CHAPTER XIV. THEORY OF QUANTUM TRANSITIONS

83. Statement of the problem 284 84. Transition probabilities under a time-dependent perturbation 287 85. Transitions due to a time-independent perturbation 290

CHAPTER XV. EMISSION, ABSORPTION AND SCATTERING OF LIGHT

BY ATOMIC SYSTEMS

86. Introductory remarks 87. Absorption and emission of light

292 294

88. Emission and absorption coefficients 297 89. The correspondence principle 300 90. Selection rules for dipole radiation 303 91. Intensities in the emission spectrum 307 92. Dispersion 307 93. Raman scattering 314 94. Allowance for change of phase of the electromagnetic field of the

wave within the atom. Quadrupole radiation 95. The photoelectric effect

CHAPTER XVI. THE PASSAGE OF MICROPARTICLES THROUGH

POTENTIAL BARRIERS

96. Statement of the problem and simplest cases 97. The apparent paradox of the 'tunnel effect' 98. Cold emission of electrons from a metal 99. A three-dimensional potential barrier. Quasistationary states

100. The theory of IX decay 101. Ionisation of atoms in strong electric fields

CHAPTER XVII. THE MANY-BODY PROBLEM

102. General remarks on the many-body problem

317 320

328 334 335 337 343 346

349

CONTENI'S IX

103. The law of conservation of the total momentum of a system of microparticles 353

104. Motion of the centre of mass of a system of microparticles 354 105. The law of conservation of the angular momentum of a system of

microparticles 357 106. Eigenfunctions of the angular momentum operator of the system.

Clebsch-Gordan coefficients 363 107. The relation of the conservation laws to the symmetry of space

and time 365

CHAPTER XVIII. SIMPLE APPLICA TIONS OF THE THEOR Y OF

MOTION OF MANY BODIES

108. Allowance for the motion of the nucleus in an atom 370 109. A system of microparticles executing small oscillations 372 110. Motion of an atom in an external field 376 111. Determination of the energy of stationary states of atoms

from their deflection in an external field 379 112. Inelastic collisions between electrons and atoms. Determination of

the energy of the stationary states of atoms by the collision method 383 113. The law of conservation of energy and the special significance of

time in quantum mechanics 388

CHAPTER XIX. SYSTEMS OF IDENTICAL MICROPARTICLES

114. The identity of micro particles 391 115. Symmetric and antisymmetric states 395 116. Bose particles and Fermi particles. The Pauli principle 398 117. Wave functions for a system of fermions and bosons 403

CHAPTER XX. SECOND QUANTISATION AND QUANTUM STATISTICS

118. Second quantisation 407 119. The theory of quantum transitions and the second-quantisation

method 414 120. The collision hypothesis. A Fermi-Dirac gas and a Bose-Einstein gas 415

CHAPTER XXI. MULTI-ELECTRON A TOMS

121. The helium atom 422 122. Approximate quantitative theory of the helium atom 428 123. The exchange energy 434 124. Quantum mechanics of the atom and Mendeleev's periodic system 437

of the elements

CHAPTER XXII. FORMA nON OF MOLECULES

125. The hydrogen molecule 446

x QUANTUM MECHANICS

126. The nature of chemical forces 127. Dispersion forces between molecules 128. Nuclear spin in diatomic molecules

457 460 462

CHAPTER XXIII. MAGNETIC PHENOMENA

129. Paramagnetism and diamagnetism of atoms 130. Ferromagnetism

CHAPTER XXIV. THE ATOMIC NUCLEUS

131. Nuclear forces. Isotopic spin 132. Systematics of states of a system of nucleons 133. Theory of the deuteron 134. Scattering of nucleons 135. Polarisation in the scattering of particles which have spin 136. The application of quantum mechanics to the systematics of

elementary particles

CHAPTER XXV. CONCLUSION

465 467

472 475 476 478 482

484

137. The formalism of quantum mechanics 488 138. The limits of applicability of quantum mechanics 491 139. Some epistemological problems 494

APPENDICES

1. The Fourier transformation 503 I I. Eigenfunctions when there is degeneracy 505

Ill. Orthogonality and normalisation of eigenfunctions of the continuous spectrum. The t5-function 506

IV. The significance of commutability of operators 509 V. The spherical harmonic functions Y'm (0, ¢) 510

VI. Hamilton's equations 513 VII. Schrodinger's equation and the equations of motion in curvilinear

co-ordinates 516 VIII. Conditions on the wave function 519

IX. The solution of the oscillator equation 520 X. An electron in a uniform magnetic field

Xl. Jacobi co-ordinates

REFERENCES

INDEX

524 525

528 531

PREFACE TO THE SECOND RUSSIAN EDITION

The second edition of Osnovy kvantovolmekhaniki, like the first (published in 1944 under the title Vvedenie v kvantovuyu mekhaniku [Introduction to quantum mechanics]), is essentially a series of lectures on quantum mechanics given by the author for a number of years in the Department of Physics at the Lomonosov Moscow State University.

The inevitable changes in these lectures have led me to make a number of corrections and additions in the second edition. The chapter concerning the concept of states in quantum mechanics and the uncertainty relation has been considerably altered and clarified. The new edition includes also a treatment of methodological problems in quantum mechanics, and a criticism of idealistic views on quantum theory which are now widely held in other countries. Some additions have also been necessitated by the further development of applications of quantum mechanics in recent years.

In this book, as in the first edition, I have striven to provide the student beginning quantum mechanics with a correct understanding of its physical basis and mathemati­cal formalism, and to indicate the value of the subject by means of some important applications.

The improvement of this book has been greatly assisted by many useful comments from my colleagues; I am very grateful to them, and especially to S. 1. Drabkina, M. A. Markov, A. A. Sokolov, S. G. Suvorov and E. L. Feinberg. The writing of the last section of the book was considerably helped by discussions at the philosophy seminar of Moscow State University and with theoreticians at the USSR Academy of Sciences' Institute of Physics.

I am also obliged to physics students at Moscow State University who have helped to remove misprints and other errors in the first edition.

Xl

PREFACE TO THE ENGLISH EDITION

The English translation of Osnovy kvantovol mekhaniki has been made from the third and fourth Russian editions. These contained a number of important additions and changes as compared with the first two editions. The main additions concern collision theory, and applications of quantum mechanics to the theory of the atomic nucleus and to the theory of elementary particles. The development of these branches in recent years, resulting from the very rapid progress made in nuclear physics, has been so great that such additions need scarcely be defended. Some additions relating to methods have also been made, for example concerning the quasiclassical approxi­mation, the theory of the Clebsch-Gordan coefficients and several other matters with which the modern physicist needs to be acquainted.

The alterations that have been made involve not only the elimination of obviously out-of-date material but also the refinement of various formulations and statements. For these refinements I am indebted to many persons who at different times have expressed to me their critical comments and suggestions.

Particularly important changes have been made regarding the definition of a quantum ensemble in Section 14.

The underlying idea and spirit of the book remain as in the first two editions: to provide the student beginning quantum mechanics with a correct understanding of its physical basis and mathematical formalism, and to indicate by simple examples the ways in which it can be applied in various branches of atomic physics: the theory of the solid state, atomic and molecular physics, optics, magnetism, the theory of the atomic nucleus, and so on.

I have also attached great importance to the use of correct methods; without a mastery of methods, even the loftiest intellect betrays some touch of the labourer. In consequence, the materialistic methodology, explicitly or implicitly, pervades the whole of the book.

In recent years this book has been published in many countries, and I am glad that it has helped in the diffusion of knowledge of and interest in modern atomic physics among many nations.

1 am now deeply indebted to Mr. A. Reidel, the publisher of the English translation, and to Dr. J. B. Sykes and Dr. M. J. Kearsley, the translators, for making this book accessible to a much wider public.

Finally, I am grateful also to those who helped me improve this book, and

XIII

XIV QUANTUM MECHANICS

to my colleagues and students, in particular M. A. Markov, who read the revised manuscript and made a number of useful suggestions and comments, and S. I. Drabkina for her enthusiastic help in preparing the corrections and additions. I am also obliged to the staff of the 'Vysshaya shkola' publishing house, who gave much assistance in connection with the publication of the book in the original Russian.

D. I. BLOKHINTSEV

INTRODUCTION

In recent decades the science of atomic phenomena has not only formed one of the most important branches of modern physics but also found many practical appli­cations. Even the most superficial examination of the field of atomic phenomena reveals features considerably differing from those of the macrouniverse.

The first novel aspect of the microuniverse is its atomism. The elementary particles have entirely definite properties of charge, mass, etc., which are the same for all particles of a given kind. No such atomism occurs in the macrouniverse. Macro­scopic objects are assemblies of large numbers of elementary particles, and the laws of macroscopic phenomena are those appropriate to such assemblies.

This shows that it would be incorrect to regard micro particles as being analogous to macroscopic bodies. Even the point mass of classical mechanics is an abstract idealised picture not of a microparticle but of a macroscopic body whose dimensions are small compared with the distances occurring in a given problem.

The atomism of the micro universe is not restricted to the definiteness of the proper­ties of the microparticles; it also leads to the existence of an absolute measure of mechanical motion, namely Planck's constant n = 1.05 x 10- 27 erg sec. This is of prime importance in the mechanics of microparticles. Physicists were for long un­aware that quantitative changes can become qualitative ones and attempted to under­stand atomic phenomena on the basis of classical macroscopic theories. The dis­covery of Planck's constant was the first real indication of the invalidity of mechani­cally applying large-scale laws to small-scale objects.

In the 1920's further experimental facts were discovered which finally forced the abandonment of this approach. It was shown that electrons possess wave properties: if a beam of electrons is passed through a crystal they are distributed on a screen in the same manner as the intensity of waves of an appropriate wavelength. This is the diffraction of micro particles, a phenomenon unknown to classical mechanics. Later it was shown that not only electrons but all microparticles exhibit this behaviour. In this way a fundamentally new and completely general law was revealed.

The motion of micro particles was found to be in many respects more akin to the motion of waves than to that of point masses along paths. The phenomenon of diffraction is incompatible with the supposition that the particles move in paths. Hence the principles of classical mechanics, where the concept of the path is funda­mental, cannot be used to examine the motion of micro particles.

xv

XVI QUANTUM MECHANICS

The word 'particle' itself, when applied to individual entities of the microuniverse, creates the idea of an analogy with the point masses of classical mechanics much closer than that which actually exists. This should be borne in mind whenever the word 'particle' is used in this book, for brevity, in place of ' micro particle'.

Classical mechanics is only a certain approximation suitable for the discussion of the motion of bodies of large mass moving in fields which vary sufficiently smoothly (macroscopic fields). Under these conditions Planck's constant is not significant, and may be regarded as negligibly small. Diffraction phenomena also are unimportant. In the small-scale micro universe classical mechanics is replaced by quantum mechan­ics. Thus the object of study in quantum mechanics is the motion of microparticles.

Quantum mechanics is a statistical theory. For example, it can be used to predict the mean distribution over a photographic plate of electrons reflected from a crystal, but only a probability can be derived regarding the point of incidence of each individual electron, in the form of a statement that it will appear in a given place with a given probability.

A similar situation occurs in statistical mechanics, but there is a profound differ­ence between quantum mechanics and classical statistical mechanics. The latter is based on Newtonian mechanics, which in principle allows the history of each particle to be traced. Modern quantum mechanics, by contrast, is not based on any theory of individual microprocesses. It deals with the individual properties of micro­particles and individual microprocesses by working with statistical ensembles. These are defined by properties taken over from classical macroscopic physics, such as momentum, energy and co-ordinate. When, therefore, the reproducibility of a micro­phenomenon is discussed in quantum mechanics (e.g. the repetition of a given experi­ment), this refers to the reproducing of the macroscopic conditions for the microscopic

phenomenon, i.e. the establishment of the same statistical ensemble. Thus quantum mechanics considers statistical ensembles of micro particles in their

relation to macroscopic measuring apparatus with which the 'state of the particles' can be determined, i.e. the statistical ensemble can be specified.

Within the scope defined by the foregoing formulation, quantum mechanics is a great advance in the development of twentieth-century atomic physics - which has, indeed, outstepped the bounds of physics and entered the realm of the industrial arts.