# Fundamentals Of Quantum Mechanics | V. A. Fock

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Fundamentals Of Quantum Mechanics
Original Title Fundamentals Of Quantum Mechanics
Author V. A. Fock
Publication date

Topics 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
Language English
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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 find
ings in quantum mechanics.
He elaborates on the
epistemological bases of the
science, devoting several sections to specific problems that
the theory. Two of the new
chapters consider Pauli’s
theory of the electron and the
many-electron 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 linear-operator 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

9

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

1. Time dependence of mathematical expectation 90

2. Correspondence between the theory of linear operators and the

quantum theory 92

3. 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 one-dimensional harmonic oscillator 100
5. Hermite polynomials 103
6. Canonical transformation a, illustrated by the harmonic-oscillator

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

1. Degenerate eigenvalues. Expansion in powers of the smallness para-

meter 125

2. The eigenfunctions in the zeroth-order approximation 126

3. The first and higher approximations 129

4. The case of adjacent eigenvalues 131

5. 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 quasi-stationary 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 MANY-ELECTRON 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 self-consistent field method 263

4. The equation for the valence electron and the operator of quantum
exchange 269

5. The self-consistent 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. Second-order 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

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. Fine-structure 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

Fundamentals Of Quantum Mechanics | V. A. Fock
Fundamentals Of Quantum Mechanics Original Title Fundamentals Of Quantum Mechanics Author
Fundamentals Of Quantum Mechanics | V. A. Fock
Fundamentals Of Quantum Mechanics Original Title Fundamentals Of Quantum Mechanics Author
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