The Foundations of Quantum Theory

The Foundations of Quantum Theory discusses the correspondence between the classical and quantum theories through the Poisson bracket-commutator analogy. The book is organized into three parts encompassing 12 chapters that cover topics on one-and many-particle systems and relativistic quantum mechanics and field theory. The first part of the book discusses the developments that formed the basis for the old quantum theory and the use of classical mechanics to develop the theory of quantum mechanics. This part includes considerable chapters on the formal theory of quantum mechanics and the wave mechanics in one- and three-dimension, with an emphasis on Coulomb problem or the hydrogen atom. The second part deals with the interacting particles and noninteracting indistinguishable particles and the material covered is fundamental to almost all branches of physics. The third part presents the pertinent equations used to illustrate the relativistic quantum mechanics and quantum field theory. This book is of value to undergraduate physics students and to students who have background in mechanics, electricity and magnetism, and modern physics.

Preface

Part I One-Particle Systems

Chapter 1. Historical Aspects

I Black-Body Radiation

II Characteristic Modes within a Cavity

III The Rayleigh-Jeans (Classical) Theory

IV Planck's (Quantum) Theory

V The Photoelectric Effect

VI The Compton Effect

VII The Quantum Theory of Matter

VIII The de Broglie Hypothesis and the Davisson-Germer Experiment

IX The Bohr Theory of Hydrogen

X The Correspondence Principle

XI Summary

Suggested Reading

Problems

Chapter 2. Classical Mechanics

I The Newtonian Form of Mechanics (Nonrelativistic)

II Lagrange's Equations

III Hamilton's Equations

IV Poisson Brackets

V Relativistic Dynamics

Suggested Reading

Problems

Chapter 3. The Formalism of Quantum Mechanics

I Vectors in a Complex, N-Dimensional, Linear Space

II Linear Operators

III Eigenvalues and Eigenvectors

IV Eigenvalue-Eigenvector Algebra for Hermitian Operators

V The Commutator and the Eigenvalue Problem

VI The Projection Operator

VII The Postulates of Quantum Mechanics

VIII Quantum Dynamics

IX Stationary States

X The Dimensionality of "Quantum Space"

XI The Coordinate Representation

XII The Transition to Wave Mechanics

XIII The Schroedinger Wave Equation

XIV The Schroedinger Wave Equation and Probability Flow

Suggested Reading

Problems

Chapter 4. Wave Mechanics in One Dimension

I Classification of Stationary States in Wave Mechanics

II The Free Particle in One Dimension

III Scattering from One-Dimensional Barriers

IV The Rectangular Barrier

V Bound Stationary States in One Dimension

VI The Infinite Well

VII The Infinite Symmetric Well

VIII Parity

IX The Finite Symmetric Well

X The Harmonic Oscillator

XI Properties of Oscillator Eigenfunctions

XII Oscillations in Nonstationary States—Classical Correspondence

XIII The Oscillator Problem in Dirac Notation—The Ladder Method

Suggested Reading

Problems

Chapter 5. Wave Mechanics in Three Dimensions

I The Eigenvalue Problem in Three Dimensions

II The Free Particle (Cartesian Coordinates)

III The Particle in a Box

IV The Anisotropic Oscillator

V Curvilinear Coordinates

VI The Central Force Problem V=V(r)

VII Quantization of Angular Momentum

VIII The Free Particle (Spherical Coordinates)

IX The Isotropic Oscillator

X Bound States of an Attractive Coulomb Potential (V=−K/r)

XI The Hydrogen Atom

XII Parity and the Central Force Problem

XIII The Effect of a Uniform Magnetic Field on the Central Force Problem

XIV The Ladder Method

Suggested Reading

Problems

Chapter 6. Spin Angular Momentum

I Pauli's Theory of Electron Spin

II Transformation Properties of Spin Kets—The Total Angular Momentum

III Spin and the Central Force Problem

IV Spin Magnetism and the Spin-Orbit Interaction in Hydrogen

V External Magnetic Fields—The Paschen-Back Effect

Suggested Reading

Problems

Chapter 7. Methods of Approximation

I Perturbation Theory

II Nondegenerate-Bound-State-Stationary Perturbation Theory (Rayleigh-Schroedinger Method)

III An Application of the First-Order Theory

IV Second-Order Theory

V Perturbation of a Degenerate Level

VI An Application of the Perturbation Theory to a Degenerate Level—The Stark Effect in Hydrogen

VII The Hydrogen Atom with Spin-Orbit Interaction

VIII The Anomalous Zeeman Effect in Hydrogen

IX Time-Dependent Perturbation Theory

X Transitions Induced by a Constant Perturbation

XI First-Order Transitions—Fermi's Golden Rule

XII Higher-Order Corrections to the Golden Rule

XIII Transitions Induced by a Harmonic Perturbation

XIV Radiative Transitions in Hydrogen

XV Einstein's Approach to Spontaneous Emission—Detailed Balancing

XVI The Variational (Rayleigh-Ritz) Method

Suggested Reading

Problems

Chapter 8. The Theory of Scattering

I The Classical Theory of Scattering

II The Stationary (Steady-State) Quantum Theory of Scattering

III Rutherford Scattering (Quantum Case)

IV A Perturbation Treatment of Stationary Scattering—The Born Series

V The First Born Approximation

VI Higher Born Approximations

VII The Method of Partial Waves

VIII The Partial Phase Shift Approximation

IX s-Wave Scattering

X Dynamical Quantum Scattering and Transitions

XI Inelastic Scattering and Absorption

Suggested Reading

Problems

Part II Many-Particle Systems

Chapter 9. Noninteracting Particles

I Classical Mechanics

II The Transition to Quantum Mechanics

III The Coordinate Representation and Wave Mechanics

IV The Permutation Operator

V Distinguishable Ideal Systems

VI Indistinguishable Ideal Systems

VII Statistical Correlations in Ideal Bose and Fermi Systems

VIII The "Ideal" Helium Atom

IX Excited States in Helium

X The Quantum Ideal Gas

XI The N-Representation, the Density Operator, and Quantum Statistics

Suggested Reading

Problems

Chapter 10. Interacting Many-Particle Systems

I The Isolated Two-Body Problem

II Scattering from a Mobile Target

III The Helium Atom—A Perturbation Treatment

IV The Helium Atom—A Variational Approach

V The Statistical Model of Thomas and Fermi for Complex Atoms

VI The Self-Consistent Field Method and the Hartree-Fock-Slater Approximation

VII Properties of Atoms in the HFS Approximation

VIII Diatomic Molecules—The Adiabatic Approximation

IX The Hydrogen Molecule and the Covalent Bond (London-Heitler Theory)

X The Normal-Coordinate Transformation, the Linear Lattice, Phonons

Suggested Reading

Problems

Part III Relativistic Quantum Mechanics and Field Theory

Chapter 11. Relativistic Quantum Mechanics

I The Klein-Gordon Equation

II The Dirac Equation

III Free Dirac Particles

IV Negative Energy States

V A Dirac Particle in a Static Field

VI The Dirac Particle in a Coulomb Potential—Fine Structure in Hydrogen

Suggested Reading

Problems

Chapter 12. Quantum Field Theory

I Classical Theory of Fields

II The Hamiltonian Density

III Field Quantization

IV Classical Electrodynamics

V The Equivalence between Free Radiation and Oscillators

VI Quantization of the Free Radiation (Tranverse) Field

VII Quantum Electrodynamics—Radiative Transitions

VIII Broadening of Spectral Lines—The Energy-Time Uncertainty Relation

Suggested Reading

Problems

Appendix A. The Wentzel-Kramers-Brillouin (WKB or "Phase Integral") Approximation

Appendix B. The Heisenberg and Interaction Pictures

Index