Fundamentals Of Quantum Mechanics
Author: V. A. Fock
Added by: mirtitles
Added Date: 20180918
Language: English
Subjects: quantum mechanics, Schrodinger equation, pauli electron, dirac electron, field theory, mir publishers, physics, wave equation, perturbation, atom, energy levels, zeeman effect, stark effect, matrix mechanics, electron
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Description
This unique course in quantum mechanics was written by an outstanding physics theoretician, who contributed much to the founding and development of the quantum theory. Many of the chapters bear the stamp of his scientific investigations.
The first Russian edition appeared in 1932 and for some years it was the only Soviet book on the quantum theory. It has retained its significance to this day, but has long been a bibliographical rarity.
For the second Russian edition the author has revised and greatly augmented the book, including his latest findings in quantum mechanics. He elaborates on the epistemological bases of the science, devoting several sections to specific problems that add to an understanding of the theory. Two of the new chapters consider Pauli's theory of the electron and the manyelectron problem in its application to the atomic theory.
Foreword 5 Preface to the Second Russian Edition 7
Preface to the First Russian Edition
8
PART I
BASIC CONCEPTS OF QUANTUM MECHANICS
Chapter I. The physical and epistemological bases of quantum mechanics 13
I. The need for new methods and concepts in describing atomic pheno mena 13 2. The classical description of phenomena 13
3. Range of application of the classical way of describing phenomena Heisenberg's and Bohr's uncertainty relations 15
4. Relativity with respect to the means of observation as the basis for
the quantum way of describing phenomena 17 5. Potential possibility in quantum mechanics 19
Chapter II. The mathematical apparatus of quantum mechanics 22
I. Quantum mechanics and the linearoperator problems 22 2. The operator concept and examples 23 3. Hermitian conjugate. Hermiticity 24 4. Operator and matrix multiplication 27 5. Eigenvalues and eigenfunctions 30 6. The Stieltjes integral and the operator corresponding to multiplica
tion into the independent variable 32 7. Orthogonality of eigenfunctions and normalization 34 8. Expansion in eigenfunctions. Completeness property of eigenfunc
tions 37 Chapter III. Quantum mechanical operators 41
I. Interpretation of the eigenvalues of an operator 41 2. Poisson brackets 42 3. Position and momentum operators 45 4. Eigenfunctions and eigenvalues of the momentum operator 48 5. Quantum description of systems 51 6. Commutativity of operators 52 7. Angular momentum 54 8. The energy operator 57 9. Canonical transformation 59
10. An example of canonical transformation 63 II. Canonical. transformation as an operator 64 12. Unitary invariants 66 13. Time evolution of systems. Time dependence of operators 69
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10
14. 15. 16.
Contents
Heisenberg's matrices 73 Semiclassical approximation 75 Relation between canonical transformation and the contact trans· formation of classical mechanics 80
Chapter IV. The probabilistic interpretation of quantum mechanics 85
1. Mathematical exprctation in the probability theory 85 2. Mathematical expectation in quantum mechanics 86 3. The probability formula 88

Time dependence of mathematical expectation 90

Correspondence between the theory of linear operators and the
quantum theory 92

The concept of statistical, ensemble in quantum mechanics 93
PART II
SCHRODINGER'S THEORY
Chapter I. The Schriidlnger equation. The harmonic oscillator 96
I. Equations of motion and the wave equation 96 2. Constants of the motion 98 3. The Schrodinger equation for the harmonic oscillator 99 4. The onedimensional harmonic oscillator 100 5. Hermite polynomials 103 6. Canonical transformation a; illustrated by the harmonicoscillator
problem 106 7. Heisenberg's uncertainty relations 110 8. The time dependence of matrices. A comparison with the classical
theory 112 9. An elementary criterion for the applicability of the formulas of
classical mechanics I15 Chapter II. Perturbation theory 119
I. Statement of the problem 119 2. Solution of the nonhomogeneous equation 120 3. Nondegenerate eigenvalues 123

Degenerate eigenvalues. Expansion in powers of the smallness para
meter 125

The eigenfunctions in the zerothorder approximation 126

The first and higher approximations 129

The case of adjacent eigenvalues 131

The anharmonic oscillator 133
Chapter III. Radiation, the theory of dispersion, and the law of decay 137
I. Classical formulas 137
2. Charge density and current density 139
3. Frequencies and intensities 143
4. Intensities in a continuous spectrum 146
5. Perturbation of an atom by a light wave 148
6. The dispersion formula 150
7. Penetration of a potential barrier by a particle 153
8. The law of decay of a quasistationary state 156
Contents 11 Chapter IV. An electron In a central field 160
1. General remarks 160
2. Conservation of angular momentum 161
3. Operators in spherical coordinates. Separation of variables 164
4. Solution of the differential equation for spherical harmonics 166
5. Some properties of spherical harmonics 170
6. Normalized spherical harmonics 173
7. The radial functions. A general survey 175
8. Description of the states of a valence electron. Quantum numbers 179
9. The selection rule 181
Chapter V. The Coulomb field 188
1. General remarks 188
2. The radial equation for the hydrogen atom. Atomic units 188
3. Solution of an auxiliary problem 190
4. Some properties of generalized Laguerre polynomials 193
5. Eigenvalues and eigenfunctions of the auxiliary problem 197
6. Energy levels and radial functions for the discrete hydrogen spectrum 198
7. Solution of the differential equation for the continuous spectrum in the form of a definite integral 201
8. Derivation of the asymptotic expression 204
9. Radial functions for the continuous hydrogen spectrum 207
10. Intensities in the hydrogen spectrum 211
11. The Stark effect. General remarks 215
12. The Schr5dinger equation in parabolic coordinates 216
13. Splitting of energy levels in an electric field 219
14. Scattering of a.particles. Statement of the problem 221
15. Solution of equations 223
16. The Rutherford·scattering law 225
17. The virial theorem in classical and in quantum mechanics 226
18. Some remarks concerning the superposition principle and the probabilistic interpretation of the wave function 229
PART III
PAULl'S THEORY OF THE ELECTRON
1. The electron angular momentum 232
2. The operators of total angular momentum in spherical coordinates 236
3. Spherical harmonics with spin 239
4. Some properties of spherical harmonics with spin 243
5. The Pauli wave equation 245
6. Operator P in spherical and cylindrical coordinates and its relation to .A 248
7. An electron in a magnetic field 254
PART IV
THE MANYELECTRON PROBLEM OF QUANTUM MECHANICS AND THE STRUCTURE OF ATOMS
1. Symmetry properties of the wave function 257
2. The Hamiltonian and its symmetry 262
3. The selfconsistent field method 263
4. The equation for the valence electron and the operator of quantum exchange 269
5. The selfconsistent field method in the theory of atoms 271
6. The symmetry of the Hamiltonian of a hydrogen·like atom 276
PART V
DIRAC'S THEORY OF THE ELECTRON
Chapter I. The Dirac equation 281
I. Quantum mechanics and the theory of relativity 281
2. Classical equations of motion 281
3. Derivation of the wave equation 283
4. The Dirac matrices 284
5. The Dirac equation for a free electron 288
6. Lorentz transformations 291
7. Form of matrix S for spatial rotations of axes and for Lorentz transformations 293
8. Current density 297
9. The Dirac equation in the case of a field. Equations of motion 298
10. Angular momentum and the spin vector in Dirac's theory 301
11. The kinetic energy of an electron 304
12. The second intrinsic degree of freedom of the electron 305
13. Secondorder equations 308
Chapter II. The use of the Dirac equation In physical problems 312
I. The free electron 312
2. An electron in a homogeneous magnetic field 316
3. Constants of the motion in the problem with spherical symmetry 320
4. Generalized spherical harmonics 322
5. The radial equation 325
6. Comparison with the SchrOdinger equation 327
7. General investigation of the radial equations 329
8. Quantum numbers 334
9. Heisenberg's matrices and the selection rule 336
10. Alternative derivation of the selection rule 340
11. The hydrogen atom. Radial functions 343 12. Finestructure levels of hydrogen 347
13. The Zeeman effect. Statement of the problem 350
14. Calculation of the perturbation matrix 352
15. Splitting of energy levels in a magnetic field 355
Chapter III. On the theory of positrons 359
I. Charge conjugation 359
2. Basic ideas of positron theory 360
3. Positrons as unfilled states 361
Index 362