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”.

This book was translated from the Russian by* Eugene Yankovsky.* The book was published by first Mir Publishers in 1978 with a reprint

in 1982.

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Table of Contents

Foreword

Preface to the Second Russian Edition

Preface to the First Russian Edition

**PART I**

** BASIC CONCEPTS OF QUANTUM MECHANICS**

Chapter I. The physical and epistemological bases of quantum mechanics 13

1. 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 11

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 21

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

15. Semiclassical approximation

16. Relation between canonical transformation and the contact

transformation of classical mechanics

**Chapter IV. The probabilistic Interpretation of quantum mechanics**

1. Mathemalical expectation in the probability theory

2. Mathematical expectation In quantum mechanics

3. The probability formula

4. Time dependence of mathematical expectation

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

6. The concept of statistical ensemble In quantum mechanics

**PART II**

** SCHRODINGER’S THEORY**

**Chapter I. The Schrodinger equation. The harmonic oscillator**

1. Equations of motion and the wave equation

2. Constants of the motion

3. The Schrodinger equation for the harmonic oscillator

4. The one-dimensional harmonic oscillator

5. Hermite polynomials

6. Canonical transformation a Iillustrated by the harmonic-oscillator

problem

7. Heisenberg’s uncertainty relations

8. The time dependence of matrices. A comparison with the classical theory

9. An elementary criterion for the applicability of the formuIas of

classical mechanics

** Chapter II. Perturbation theory**

I. Statement of the problem

2. Solution of the nonhomogeneous equation

3. Nondegenerate eigenvalues

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

5. The eigenfunctions in the zeroth-order approximation

6. The first and higher approxirnatlcns

7. The case of adjacent eigenvalues

8. The anharmonic oscillator

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

I. Classical formulas

2. Charge density and current density

3. Frequencies and intensities

4. Intensities in a continuous spectrum

5. Perturbation of an atom by a tight wave

6. The dispersion formula

7. Penetration of a potential barrier by a particle

8. The law of decay of a quasi-stationary state

**Chapter IV. An electron In a central Ileld**

1. General remarks

2. Conservation of angular momentum

3. Operators in spherical coordinates. Separation of variables

4. Solution of the differential equation for spherical harmonics

5. Some properties of spherical harmonics

6. Normalized spherical harmonics

7. The radial functions. A general survey

8. Description of the states of a valence electron. Quantum numbers

9. The selection rule

**Chapter V. The Coulomb field**

1. General remarks

2. The radial equation for the hydrogen atom. Atomic units

3. Solution of an auxiliary problem

4. Some properties of generalized Laguerre polynomials

5. Eigenvalues and eigenfunctions of the auxiliary problem

6. Energy levels and radial functtons for the discrete hydrogen spectrum

7. Solution of the differential equation for the continuous spectrum

in the form of a definite integral

8. Derivation of the asymptotic expression

9. Radial functions for the continuous hydrogen spectrum

10. Intensities in the hydrogen spectrum

11. The Stark effect. General remarks

12. The SchrOdinger equation in parabolic coordinates

13. Splitting of energy levels in an electric field

14. Scattering of \aplha -particles. Statement of the problem

I5. Solution of equations

16. The Rutherford scattering law

17. The virial theorem in classical and in quantum mechanics

18. Some remarks concerning the superposition principle and the

probabilstic interpretation of the wave function

**PART III**

** PAULI’S THEORY OF THE ELECTRON**

1. The electron angular momentum

2. The operators of total angular momentum in spherical coordinates

3. Spherical harmonics with spin

4. Some properties of spherical harmonica with spin

5. The Pauli wave equation

6. Operator P in spherical and cylindrical coordinates and its

relation R

7. An electron In a magnetic field

**PART IV**

** THE MANY-ELECTRON PROBLEM OF QUANTUM MECHANICS**

** AND THE STRUCTURE OF ATOMS**

1. Symmetry properties of the wave function

2. The Hamiltonian and Its symmetry

3. The self -conslstent Held method

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

5. The self-consistent field method in the theory of atoms

6. The symmetry of the Hamiltonian of a hydrogen-like atom

**PART V**

** DIRAC’S THEORY OF THE ELECTRON**

**Chapter I. The Dirac equation 2811**

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 lor a Free electron 288

6. Lorentz translormations

7. Form of matrix S for spatial rotations of axes and for Lorentz

transformations 293

8. Current density

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

**Chapter II. The use of the Dirac equation In physical problems**

1. The Free electron

2. An electron in a homogeneous magnetic field

3. Constants of the motion in the problem with spherical

4. Generalized spherical harmonics

5. The radial equation

6. Comparison with the Schrodinger equation

7. General investigation of the radial equations

8. Quantum numbers

9. Heisenberg’s matrices and the selection rule

10. Alternative derivation of the selection rule

11. The hydrogen atom. Radial funclions

12. Fine-structure levels of hydrogen

13. The Zeeman ellect. Statement of the problem

14. Calculatlon of the perturbation matrix

15. Splitting of energy levels in a magnetic field

** Chapter III. On the theory of positrons**

1. Charge conjugation

2. Basic ideas 01 positron theory

3. Positrons as unfilled slates

Index

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