Fundametals Of Theoretical Physics Vol 2 | I. V. Savelyev

Fundamentals of Theoretical Physics Volume 2 Quantum Mechanics by I. V. Savelyev

The book was translated from the Russian by G. Leib. The book was first published in 1982, revised from the 1977 Russian edition by Mir Publishers.

The present book is the second volume of a guide to theoretical physics. As in the first Volume I have adhered to the rule of omitting what is explained in sufficient detail in modern general courses of physics. In particular, the experimental fundamentals of quantum physics are not discussed.

With a view to the fact that the mastering of the mathematical apparatus of quantum mechanics involves great difficulties, I have done everything in my power to make calculations as simple and as clear as possible. For this purpose, special care was taken in choosing the notation.

The book is provided with mathematical appendices. Sometimes I refer to the mathematical appendices of Volume 1. The book has been conceived first of all as a training aid for students of non-theoretical specialities of higher educational establishments. Acquaintance with it will facilitate a more detailed studying of the subject with the aid of fundamental guides.

Preface 5

Chapter I. Foundations of Quantum Mechanics 9

1. Introduction 9
2. State 10
3. The Superposition Principle 12
4. The Physical Meaning of the Psi-Function 14
5. The Schrodinger Equation16
6. Probability Flux Density 20

Chapter II. Mathematical Tools of Quantum Mechanics 23

7. Fundamental Postulates 23
8. Linear Operators 27
9. Matrix Representation uf Operators 31
10. The Algebra of Operators 38
11. The Uncertainty Relation 45
12. The Continuous Spectrum 48
13. Dirac Notation 51
14. Transformation of Functions and Operators from One Representation to Another 55

Chapter III. Eigenvalues and Eigenfunctions of Physical Quantities 63

15. Operators of Physical Quantities 63
16. Rules for Commutation of Operators of Physical Quantities 67
17. Eigenfunctions of the Coordinate and Momentum Operators 71
18. Momentum and Energy Representation 74
19. Eigenvalues and Eigenfunctions of tho Angular Momentum Operator 78
20. Parity 81

Chapter IV. Time Dependence of Physical Quantities 83

21. The Time Derivative of an Operator 83
22. Time Dependence of Matrix Elements  86

Chapter V. Motion of a Particle in Force Fields 89

23. A Particle in a Central Force Field . 89
24. An Electron in a Coulomb Field. The Hydrogen Atom 94
25. The Harmonic Oscillator 106
26. Solution of the Harmonic Oscillator Problem in the Matrix Form 109
27. Annihilation and Creation Operators 116

Chapter VI. Perturbation Theory 123

28. Introduction 123
29. Time-Independent Perturbations 123
30. Case of Two Close Levels 132
31. DegenerateCase 136
32. Examples of Application of tho Stationary Perturbation Theory 141
33. Time-Dependent Perturbations  148
34. Perturbations Varying Harmonically with Time 156
35. Transitions in a Continuous Spectrum 163
36. Potential Energy as a Perturbation 164

Chapter VII. The Quasiclassical Approximation 169

37. The Classical Limit 169
38. Boundary Conditions at a Turning Point 174
39. Bohr-Sommerfeld Quantization Rule 184
40. Penetration of a Potential Barrier 188

Chapter VIII. Semiempirical Theory of Particles with Spin 192

41. Psi-Function of a Particle with Spin 192
42. Spin Operators 194
43. Eigenvalues and Eigenfunctions of Spin Operators 202
44.Spinors 205

Chapter IX. Systems Consisting of Identical Particles   214

45. Principle of Indistinguishability of Identical Particles 214
46. Psi-Functions for Systems of Particles. The Pauli Principle 216
47. Summation of Angular Momenta 222
48. Psi-Function of System of Two Particles Having a Spin of 1/2 225
49. Exchange Interaction 229
50. SecondQuantization 233
51. Second Quantization Applied to Bosons 235
52. Second Quantization Applied to Fermions 250

Chapter X. Atoms and Molecules  258

53. Methods of Calculating Atomic Systems 258
54. The Helium Atom 259
55. The Variation Method 263
56. The Method or the Self-Consistent Field 268
57. The Thomas-Fermi Method 275
58. The Zeeman Effect 278
59. The Theory of Molecules in the Adiabatic Approximation 281
60. TheHydrogen Molecule 285

Chapter XI. Radiation Theory 291

61. Quantization of an Electromagnetic Field 291
62. Interaction of an Electromagnetic Field with a Charged Particle 301
63. One-Photon Processes 305
64. Dipole Radiation 308
65. Selection Rules 312

Chapter XII. Scattering Theory 315

66. Scattering Cross Section 315
67. Scattering Amplitude  317
68. Born Approximation  319
69. Method of Partial Waves 321
70. Inelastic Scattering 328

Appendices 331

I. Angular Momentum Operators in Spherical Coordinates 331
II. Spherical Functions 332
III. Chebyshev-Hermite Polynomials 340
IV. Some Information from the Theory of Functions of a Complex Variable 345
V. Airy Function 354
VI. Method of Green’s Functions 355
VII. Solution of the Fundamental Equation of the Scattering Theory by the Method of Green’s Functions 358
VIII. The Dirac Delta Function 361
Index 364