We now come to Fundamental of Quantum Mechanics by *V. A. Fock.*

Vladimir Aleksandrovich Fock was one of the group of brilliant physics theoreticians whose work built the magnificent edifice of the quantum theory.

From the vast subject of the quantum theory the author has chosen material limited in two respects. First, the book considers none but the main principles and simplest applications of quantum mechanics, It concerns itself exclusively with the one-body problem. It does not deal with the many-body problem or the Pauli exclusion principle, basic to that problem. Second, the author has sought to confine himself to that part of the theory that is’ considered proved, that is, quantum mechanics proper. He has not examined quantum electrodynamics since this theory has yet to be fully elaborated.

The author’s main purpose is to introduce the reader to a new set of ideas differing greatly from the classical theory. He has endeavoured to avoid using images from the classical theory as being inapplicable to quantum physics. Rather, he has attempted to familiarize the reader with the basic concepts underlying a quantum description of the states of atomic systems.

The second edition of this book, unlike the first, devotes a separate chapter to the nonrelativistic theory of the electron spin (Pauli’s theory of the electron) and contains a chapter on the many-electron problem of quantum mechanics. In addition, some of the author’s findings have been incorporated as separate sections. Otherwise, the subject matter of the book (both the mathematical theory and its physical interpretation) remains the same, except for certain new formulations of an epistemological character (the concepts of relativity with respect to the means of observation and of potential possibility), which has necessitated changing the expression “the statistical interpretation of quantum mechanics” to “the probabilistic interpretation”. The new formulations are more precise than the previous ones.

The title of the book speaks for itself. The word “fundamentals” can be understood as “basic principles” or as “introductory facts”.

*About the author:*

Vladimir Aleksandrovich Fock was one of the group of brilliant physics theoreticians whose work built the magnificent edifice of the quantum theory. A contemporary of Niels Bohr, Lev landau, Werner Heisenberg, and Paul Dirac, he contributed much to practically all fields of theoretical and mathematical physics. His books The Theory of Space, Time and Gravitation, and Electromagnetic Diffraction and Propagation Problems have been translated into English (Pergamon Press). In 1936 Vladimir Fock merited the Mendeleev prize for his work in quantum theory of atoms, in 1946 the State prize for his work in the propagation of radio waves, and in 1960 the Lenin prize for his work in quantum field theory. In 1932 Vladimir Fock became a Corresponding Member of the USSR Academy of Sciences and in 1939 a Full Member.

The book was translated from the Russian by Eugene Yankovsky and was published by Mir in 1978, 1982 and 1986. The present scan is from the 1986 print.

PDF | Cover | OCR | Bookmarked | 600 dpi

The Internet Archive Link

*Edit: Removed, old, dead links. Added a link to a fresh scan and a new cover. 19 September 2018.*

**Contents**

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 phenomena 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

1. 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 multiplication into the independent variable 32

7. Orthogonality of eigenfunctions and normalization 34

8. Expansion in eigenfunctions. Completeness property of eigenfunctions 37

Chapter III. Quantum mechanical operators 41

1. 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

11. Canonical. transformation as an operator 64

12. Unitary invariants 66

13. Time evolution of systems. Time dependence of operators 69

14. Heisenberg’s matrices 73

15. Semiclassical approximation 75

16. Relation between canonical transformation and the contact transformation of classical mechanics 80

Chapter IV. The probabilistic interpretation of quantum mechanics 85

1. Mathematical expectation in the probability theory 85

2. Mathematical expectation in quantum mechanics 86

3. The probability formula 88

4. Time dependence of mathematical expectation 90

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

6. The concept of statistical, ensemble in quantum mechanics 93

PART II SCHRODINGER’S THEORY

Chapter I. The Schrodinger 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

1. Statement of the problem 119

2. Solution of the nonhomogeneous equation 120

3. Nondegenerate eigenvalues 123

4. Degenerate eigenvalues. Expansion in powers of the smallness parameter 125

5. The eigenfunctions in the zeroth-order approximation 126

6. The first and higher approximations 129

7. The case of adjacent eigenvalues 131

8. The anharmonic oscillator 133

Chapter III. Radiation, the theory of dispersion, and the law of decay 137

1. 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

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 Schrodinger 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

1. 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

1. 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. 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

1. Charge conjugation 359

2. Basic ideas of positron theory 360

3. Positrons as unfilled states 361

Index 362

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