Landau / Lifshitz Quantum Mechanics

Non-Relativistic Theory
3. Auflage 2013
ISBN: 978-1-4831-4912-7
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 688 Seiten

Quantum Mechanics, Third Edition: Non-relativistic Theory is devoted to non-relativistic quantum mechanics. The theory of the addition of angular momenta, collision theory, and the theory of symmetry are examined, together with spin, nuclear structure, motion in a magnetic field, and diatomic and polyatomic molecules. This book is comprised of 18 chapters and begins with an introduction to the basic concepts of quantum mechanics, with emphasis on the uncertainty principle, the principle of superposition, and operators, as well as the continuous spectrum and the wave function. The following chapters explore energy and momentum; Schrödinger's equation; angular momentum; and motion in a centrally symmetric field and in a magnetic field. Perturbation theory, spin, and the properties of quasi-classical systems are also considered. The remaining chapters deal with the identity of particles, atoms, and diatomic and polyatomic molecules. The final two chapters describe elastic and inelastic collisions. This monograph will be a valuable source of information for physicists.

Autoren/Hrsg.

Landau, L D

Lifshitz, E. M.

Inhaltsverzeichnis

1;Front Cover;1
2;Quantum Mechanics: Non-Relativistic Theory;4
3;Copyright Page;5
4;Table of Contents;6
5;From the Preface to the first English edition;12
6;Preface to the second English edition;13
7;Preface to the third Russian edition;14
8;Notation;15
9;CHAPTER 1. THE BASIC CONCEPTS OF QUANTUM MECHANICS;16
9.1;§1. The uncertainty principle;16
9.2;§2. The principle of superposition;21
9.3;§3. Operators;23
9.4;§4. Addition and multiplication of operators;28
9.5;§5. The continuous spectrum;30
9.6;§6. The passage to the limiting case of classical mechanics;34
9.7;§7. The wave function and measurements;36
10;CHAPTER 2. ENERGY AND MOMENTUM;40
10.1;§8. The Hamiltonian operator;40
10.2;§9. The differentiation of operators with respect to time;41
10.3;§10. Stationary states;42
10.4;§11. Matrices;45
10.5;§12. Transformation of matrices;50
10.6;§13. The Heisenberg representation of operators;52
10.7;§14. The density matrix;53
10.8;§15. Momentum;56
10.9;§16. Uncertainty relations;60
11;CHAPTER 3. SCHRÖDINGER'S EQUATION;65
11.1;§17. Schrödinger's equation;65
11.2;§18. The fundamental properties of Schrödinger's equation;68
11.3;§19. The current density;70
11.4;§20. The variational principle;73
11.5;§21. General properties of motion in one dimension;75
11.6;§22. The potential well;78
11.7;§23. The linear oscillator;82
11.8;§24. Motion in a homogeneous field;89
11.9;§25. The transmission coefficient;91
12;CHAPTER 4. ANGULAR MOMENTUM;97
12.1;§26. Angular momentum;97
12.2;§27. Eigenvalues of the angular momentum;101
12.3;§28. Eigenfunctions of the angular momentum;104
12.4;§29. Matrix elements of vectors;107
12.5;§30. Parity of a state;111
12.6;§31. Addition of angular momenta;114
13;CHAPTER 5. MOTION IN A CENTRALLY SYMMETRIC FIELD;117
13.1;§32. Motion in a centrally symmetric field;117
13.2;§33. Spherical waves;120
13.3;§34. Resolution of a plane wave;127
13.4;§35. Fall of a particle to the centre;129
13.5;§36. Motion in a Coulomb field (spherical polar coordinates);132
13.6;§37. Motion in a Coulomb field (parabolic coordinates);143
14;CHAPTER 6. PERTURBATION THEORY;148
14.1;§38. Perturbations independent of time;148
14.2;§39. The secular equation;153
14.3;§40. Perturbations depending on time;157
14.4;§41. Transitions under a perturbation acting for a finite time;161
14.5;§42. Transitions under the action of a periodic perturbation;166
14.6;§43. Transitions in the continuous spectrum;169
14.7;§44. The uncertainty relation for energy;172
14.8;§45. Potential energy as a perturbation;174
15;CHAPTER 7. THE QUASI-CLASSICAL CASE;179
15.1;§46. The wave function in the quasi-classical case;179
15.2;§47. Boundary conditions in the quasi-classical case;182
15.3;§48. Bohr and Sommerfeld's quantization rule;185
15.4;§49. Quasi-classical motion in a centrally symmetric field;190
15.5;§50. Penetration through a potential barrier;193
15.6;§51. Calculation of the quasi-classical matrix elements;200
15.7;§52. The transition probability in the quasi-classical case;204
15.8;§53. Transitions under the action of adiabatic perturbations;209
16;CHAPTER 8. SPIN;212
16.1;§54. Spin;212
16.2;§55. The spin operator;216
16.3;§56. Spinors;219
16.4;§57. The wave functions of particles with arbitrary spin;223
16.5;§58. The operator of finite rotations;228
16.6;§59. Partial polarization of particles;234
16.7;§60. Time reversal and Kramers' theorem;236
17;CHAPTER 9. IDENTITY OF PARTICLES;240
17.1;§61. The principle of indistinguishability of similar particles;240
17.2;§62. Exchange interaction;243
17.3;§63. Symmetry with respect to interchange;247
17.4;§64. Second quantization. The case of Bose statistics;254
17.5;§65. Second quantization. The case of Fermi statistics;260
18;CHAPTER 10. THE ATOM;264
18.1;§66. Atomic energy levels;264
18.2;§67. Electron states in the atom;265
18.3;§68. Hydrogen-like energy levels;269
18.4;§69. The self-consistent field;270
18.5;§70. The Thomas–Fermi equation;274
18.6;§71. Wave functions of the outer electrons near the nucleus;279
18.7;§72. Fine structure of atomic levels;280
18.8;§73. The Mendeleev periodic system;284
18.9;§74. X-ray terms;292
18.10;§75. Multipole moments;294
18.11;§76. An atom in an electric field;297
18.12;§77. A hydrogen atom in an electric field;302
19;CHAPTER 11. THE DIATOMIC MOLECULE;313
19.1;§78. Electron terms in the diatomic molecule;313
19.2;§79. The intersection of electron terms;315
19.3;§80. The relation between molecular and atomic terms;318
19.4;§81. Valency;322
19.5;§82. Vibrational and rotational structures of singlet terms in the diatomic molecule;329
19.6;§83. Multiplet terms. Case a;334
19.7;§84. Multiplet terms. Case b;338
19.8;§85. Multiplet terms. Cases c and d;342
19.9;§86. Symmetry of molecular terms;344
19.10;§87. Matrix elements for the diatomic molecule;347
19.11;§88. A-doubling;351
19.12;§89. The interaction of atoms at large distances;354
19.13;§90. Pre-dissociation;357
20;CHAPTER 12. THE THEORY OF SYMMETRY;369
20.1;§91. Symmetry transformations;369
20.2;§92. Transformation groups;372
20.3;§93. Point groups;375
20.4;§94. Representations of groups;383
20.5;§95. Irreducible representations of point groups;391
20.6;§96. Irreducible representations and the classification of terms;395
20.7;§97. Selection rules for matrix elements;398
20.8;§98. Continuous groups;402
20.9;§99. Two-valued representations of finite point groups;406
21;CHAPTER 13. POLYATOMIC MOLECULES;411
21.1;§100. The classification of molecular vibrations;411
21.2;§101. Vibrational energy levels;418
21.3;§102. Stability of symmetrical configurations of the molecule;420
21.4;§103. Quantization of the rotation of a top;425
21.5;§104. The interaction between the vibrations and the rotation of the molecule;434
21.6;§105. The classification of molecular terms;438
22;CHAPTER 14. ADDITION OF ANGULAR MOMENTA;446
22.1;§106. 3j-symbols;446
22.2;§107. Matrix elements of tensors;454
22.3;§108. 6j-symbols;457
22.4;§109. Matrix elements for addition of angular momenta;463
22.5;§110. Matrix elements for axially symmetric systems;465
23;CHAPTER 15. MOTION IN A MAGNETIC FIELD;468
23.1;§111. Schrödinger's equation in a magnetic field;468
23.2;§112. Motion in a uniform magnetic field;471
23.3;§113. An atom in a magnetic field;476
23.4;§114. Spin in a variable magnetic field;483
23.5;§115. The current density in a magnetic field;485
24;CHAPTER 16. NUCLEAR STRUCTURE;487
24.1;§116. Isotopie invariance;487
24.2;§117. Nuclear forces;491
24.3;§118. The shell model;495
24.4;§119. Non-spherical nuclei;504
24.5;§120. Isotopie shift;509
24.6;§121. Hyperfine structure of atomic levels;511
24.7;§122. Hyperfine structure of molecular levels;514
25;CHAPTER 17. ELASTIC COLLISIONS;517
25.1;§123. The general theory of scattering;517
25.2;§124. An investigation of the general formula;520
25.3;§125. The unitary condition for scattering;523
25.4;§126. Born's formula;527
25.5;§127. The quasi-classical case;533
25.6;§128. Analytical properties of the scattering amplitude;538
25.7;§129. The dispersion relation;544
25.8;§130. The scattering amplitude in the momentum representation;547
25.9;§131. Scattering at high energies;550
25.10;§132. The scattering of slow particles;557
25.11;§133. Resonance scattering at low energies;563
25.12;§134. Resonance at a quasi-discrete level;570
25.13;§135. Rutherford's formula;575
25.14;§136. The system of wave functions of the continuous spectrum;578
25.15;§137. Collisions of like particles;582
25.16;§138. Resonance scattering of charged particles;585
25.17;§139. Elastic collisions between fast electrons and atoms;590
25.18;§140. Scattering with spin-orbit interaction;594
25.19;§141. Regge poles;600
26;CHAPTER 18. INELASTIC COLLISIONS;606
26.1;§142. Elastic scattering in the presence of inelastic processes;606
26.2;§143. Inelastic scattering of slow particles;612
26.3;§144. The scattering matrix in the presence of reactions;614
26.4;§145. Breit and Wigner's formulae;618
26.5;§146. Interaction in the final state in reactions;626
26.6;§147. Behaviour of cross-sections near the reaction threshold;629
26.7;§148. Inelastic collisions between fast electrons and atoms;635
26.8;§149. The effective retardation;644
26.9;§150. Inelastic collisions between heavy particles and atoms;648
26.10;§151. Scattering of neutrons;651
26.11;§152. Inelastic scattering at high energies;655
27;MATHEMATICAL APPENDICES;662
27.1;§a. Hermite polynomials;662
27.2;§b. The Airy function;662
27.3;§c. Legendre polynomials;667
27.4;§d. The confluent hypergeometric function;670
27.5;§e. The hypergeometric function;674
27.6;§f. The calculation of integrals containing confluent hypergeometric functions;677
28;Index;682

Leseproben

CHAPTER I

THE BASIC CONCEPTS OF QUANTUM MECHANICS

Publisher Summary

An attempt to apply classical mechanics and electrodynamics to explain atomic phenomena leads to results that are in obvious conflict with experiment. This marked contradiction between theory and experiment indicates that the construction of a theory applicable to atomic phenomena, that is, phenomena occurring in particles of very small mass at very small distances demands a fundamental modification of the basic physical concepts and laws. As a starting-point for an investigation of these modifications, it is convenient to take the experimentally observed phenomenon known as electron diffraction. This chapter discusses the basic concepts of quantum mechanics. The mechanics which governs atomic phenomena—quantum mechanics or wave mechanics—is based on ideas of motion, which are fundamentally different from those of classical mechanics. In quantum mechanics, there is no such concept as the path of a particle. This forms the content of what is called the uncertainty principle, one of the fundamental principles of quantum mechanics. This principle in itself does not suffice as a basis to construct a new mechanics of particles. Quantum mechanics occupies a very unusual place among physical theories: it contains classical mechanics as a limiting case, yet at the same time, it requires this limiting case for its own formulation. A typical problem of quantum mechanics consists in predicting the result of a subsequent measurement from the known results of previous measurements. In comparison with classical mechanics, quantum mechanics restricts the range of values that can be taken by various physical quantities, for example, energy.

§1. The uncertainty principle

When we attempt to apply classical mechanics and electrodynamics to explain atomic phenomena, they lead to results which are in obvious conflict with experiment. This is very clearly seen from the contradiction obtained on applying ordinary electrodynamics to a model of an atom in which the electrons move round the nucleus in classical orbits. During such motion, as in any accelerated motion of charges, the electrons would have to emit electromagnetic waves continually. By this emission, the electrons would lose their energy, and this would eventually cause them to fall into the nucleus. Thus, according to classical electrodynamics, the atom would be unstable, which does not at all agree with reality.

This marked contradiction between theory and experiment indicates that the construction of a theory applicable to atomic phenomena—that is, phenomena occurring in particles of very small mass at very small distances—demands a fundamental modification of the basic physical concepts and laws.

As a starting-point for an investigation of these modifications, it is convenient to take the experimentally observed phenomenon known as .† It is found that, when a homogeneous beam of electrons passes through a crystal, the emergent beam exhibits a pattern of alternate maxima and minima of intensity, wholly similar to the diffraction pattern observed in the diffraction of electromagnetic waves. Thus, under certain conditions, the behaviour of material particles—in this case, the electrons—displays features belonging to wave processes.

How markedly this phenomenon contradicts the usual ideas of motion is best seen from the following imaginary experiment, an idealization of the experiment of electron diffraction by a crystal. Let us imagine a screen impermeable to electrons, in which two slits are cut. On observing the passage of a beam of electrons‡ through one of the slits, the other being covered, we obtain, on a continuous screen placed behind the slit, some pattern of intensity distribution; in the same way, by uncovering the second slit and covering the first, we obtain another pattern. On observing the passage of the beam through both slits, we should expect, on the basis of ordinary classical ideas, a pattern which is a simple superposition of the other two: each electron, moving in its path, passes through one of the slits and has no effect on the electrons passing through the other slit. The phenomenon of electron diffraction shows, however, that in reality we obtain a diffraction pattern which, owing to interference, does not at all correspond to the sum of the patterns given by each slit separately. It is clear that this result can in no way be reconciled with the idea that electrons move in paths.

Thus the mechanics which governs atomic phenomena or must be based on ideas of motion which are fundamentally different from those of classical mechanics. In quantum mechanics there is no such concept as the path of a particle. This forms the content of what is called the , one of the fundamental principles of quantum mechanics, discovered by W. Heisenberg in 1927.†

In that it rejects the ordinary ideas of classical mechanics, the uncertainty principle might be said to be negative in content. Of course, this principle in itself does not suffice as a basis on which to construct a new mechanics of particles. Such a theory must naturally be founded on some positive assertions, which we shall discuss below (§2). However, in order to formulate these assertions, we must first ascertain the statement of the problems which confront quantum mechanics. To do so, we first examine the special nature of the interrelation between quantum mechanics and classical mechanics. A more general theory can usually be formulated in a logically complete manner, independently of a less general theory which forms a limiting case of it. Thus, relativistic mechanics can be constructed on the basis of its own fundamental principles, without any reference to Newtonian mechanics. It is in principle impossible, however, to formulate the basic concepts of quantum mechanics without using classical mechanics. The fact that an electron‡ has no definite path means that it has also, in itself, no other dynamical characteristics.|| Hence it is clear that, for a system composed only of quantum objects, it would be entirely impossible to construct any logically independent mechanics. The possibility of a quantitative description of the motion of an electron requires the presence also of physical objects which obey classical mechanics to a sufficient degree of accuracy. If an electron interacts with such a “classical object”, the state of the latter is, generally speaking, altered. The nature and magnitude of this change depend on the state of the electron, and therefore may serve to characterize it quantitatively.

In this connection the “classical object” is usually called , and its interaction with the electron is spoken of as . However, it must be emphasized that we are here not discussing a process of measurement in which the physicist-observer takes part. By , in quantum mechanics, we understand any process of interaction between classical and quantum objects, occurring apart from and independently of any observer. The importance of the concept of measurement in quantum mechanics was elucidated by N. Bohr.

We have defined “apparatus” as a physical object which is governed, with sufficient accuracy, by classical mechanics. Such, for instance, is a body of large enough mass. However, it must not be supposed that apparatus is necessarily macroscopic. Under certain conditions, the part of apparatus may also be taken by an object which is microscopic, since the idea of “with sufficient accuracy” depends on the actual problem proposed. Thus, the motion of an electron in a Wilson chamber is observed by means of the cloudy track which it leaves, and the thickness of this is large compared with atomic dimensions; when the path is determined with such low accuracy, the electron is an entirely classical object.

Thus quantum mechanics occupies a very unusual place among physical theories: it contains classical mechanics as a limiting case, yet at the same time it requires this limiting case for its own formulation.

We may now formulate the problem of quantum mechanics. A typical problem consists in predicting the result of a subsequent measurement from the known results of previous measurements. Moreover, we shall see later that, in comparison with classical mechanics, quantum mechanics, generally speaking, restricts the range of values which can be taken by various physical quantities (for example, energy): that is, the values which can be obtained as a result of measuring the quantity concerned. The methods of quantum mechanics must enable us to determine these admissible values.

The measuring process has in quantum mechanics a very important property: it always affects the electron subjected to it, and it is in principle impossible to make its effect arbitrarily small, for a given accuracy of measurement. The more exact the measurement, the stronger the effect exerted by it, and only in measurements of very low accuracy can the effect on the measured object be small. This property of measurements is logically related to the fact that the dynamical characteristics of the electron appear only as a result of the measurement itself. It is clear that, if the effect of the measuring process on the object of it could be made arbitrarily small, this would mean that the...