Qualitative Methods in Quantum Theory by Migdal, A.B. -- Read -- Imperial Library of Trantor

Index

Cover Half Title Title Page Copyright Page Table of Contents Editor’s Foreword Translator’s Note Preface CHAPTER 1 DIMENSIONAL AND “MODEL” APPROXIMATIONS

1 Order-of-Magnitude Estimation of Mathematical Expressions

Estimation of Integrals The Method of Steepest Descents Properties of Integrals of Oscillating Functions. Estimates of the Higher Terms in Fourier Series Expansions Methods of Approximate Solution of Differential Equations

2 Atomic Physics

Estimate of the Velocities and Orbit Sizes of the Inner Atomic Electrons Stationary States Distribution of Electric Charge in the Atom The Rutherford Formula Inapplicability of Classical Mechanics for Large Impact Parameters Estimate of the Scattering Cross Section for Potentials which Fall Off Faster than the Coulomb Potential Resonance Effects in Scattering Interaction Between Atoms Ionization of Atoms Multiple Scattering

3 Interaction with Radiation

Zero-point Vibrations of the Electromagnetic Field The Photoeffect Lifetimes of Excited Atomic States Bremsstrahlung Pair Creation Creation of Soft Photons in the Scattering of Charged Particles (The “Infrared Catastrophe”) The Lamb Shift Asymptotic Character of Series in Quantum Electrodynamics

CHAPTER 2 VARIOUS TYPES OF PERTURBATION THEORY

1 Perturbation Theory in the Continuous Spectrum

Scattering of Charged Particles by the Atomic Nucleus

2 Perturbation of the Boundary Conditions

The Energy Levels of a Deformed Nucleus

3 Sudden Perturbations

Ionization of Atoms in β -decay Ionization of an Atom in Nuclear Reactions Transfer of Energy when a Photon is Emitted by a Nucleus in a Molecule (Mössbauer Effect)

4 Adiabatic Perturbations

Ionization of an Atom by the Passage of a Slow Heavy Particle Capture of an Atomic Electron by a Proton (Charge Exchange)

5 Fast and Slow Subsystems

Vibrational Energy Levels of a Molecule Excitation of Nuclear Dipole Levels by a Fast Particle Scattering of a Proton by a Hydrogen Atom (Charge Exchange)

6 Perturbation Theory for Adjacent Levels

A Particle in a Periodic Potential The Stark Effect in the Case of Adjacent Levels The Change of the Lifetime of the 2s 1/2 State of the Hydrogen Atom under an Applied Electric Field

CHAPTER 3 THE QUASICLASSICAL APPROXIMATION

1 The One-Dimensional Case

Asymptotic Series Matching of Quasiclassical Functions The Quantization Condition Accuracy of the Quasiclassical Approximation Normalization of Quasiclassical Functions The Correspondence Principle Mean Kinetic Energy Connection between the Quasiclassical Matrix Elements and the Fourier Components of Classical Motion Criterion for the Applicability of Perturbation Theory to the Calculation of Not Too Small Quantities Calculation of Matrix Elements in the Case of Fast Oscillating Functions Barrier Penetration Reflection Above a Barrier

2 The Three-Dimensional Case

Spherically Symmetric Field Modification of the Centrifugal Potential Energy Levels in the Coulomb Potential Quasiclassical Representation of Spherical Functions The Thomas-Fermi Distribution in the Atom Estimates of Nuclear Matrix Elements Noncentral Potential The Quasiclassical Scattering Problem Cross Section for Scattering of a Proton on a Hydrogen Atom

CHAPTER 4 THE ANALYTIC PROPERTIES OF PHYSICAL QUANTITIES

Dependence of the Moment of Inertia of a Nucleus on Deformation Dependence of the Frequency of Sound on the Wave Vector 1 Analytic Properties of the Dielectric Constant

Analytic Properties of the Dielectric Constant in a Simple Model

2 Analytic Properties of the Scattering Amplitude

Unitarity as a Consequence of the Superposition Principle and the Conservation of Probability The Dispersion Relation Resonance Scattering at Low Energies Nonresonant Scattering at Low Energies Scattering by a Potential Well Analytic Properties of the Wave Function Single-Particle Wave Functions of the Continuous Spectrum at Low Energy

3 The Use of Analyticity Properties in Physical Problems

Theory of Nuclear Reactions with the Formation of Slow Particles Interacting Particles in a Potential Well Theory of Direct Reactions Threshold Singularities of the Scattering Amplitude

CHAPTER 5 METHODS IN THE MANY-BODY PROBLEM

1 The Quasiparticle Method and Green’s Functions

The Transition Amplitude One-Particle Green’s Functions in a System of Noninteracting Particles (Quasiparticle Green’s Functions) The Green’s Function in a System of Interacting Particles Analytic Properties of the One-Particle Green’s Function Calculation of Observable Quantities The Fermion Momentum Distribution

2 The Graph Method

Graphical Representation of Processes The Interaction Between Quasiparticles The Local Quasiparticle Interaction

3 The Solution of Problems by the Green’s Function Method

Dyson’s Equation. The Basis of the Shell Model Instability of the Fermi Distribution in the Case of Attraction. The Occurrence of a Gap in the Energy Spectrum The Energy Spectrum of a Bose System. Superfluidity

4 A System in an External Field

Change of the Particle Distribution in a Field Spin Polarizability and Quasiparticle Magnetic Moments Sound Waves in a Fermi System (“Zero Sound”) Plasma Oscillations. Screening of a Charge in a Plasma Conservation Laws and Quasiparticle Charges for Different Fields

CHAPTER 6 QUALITATIVE METHODS IN QUANTUM FIELD THEORY

1 Construction of Relativistic Equations

Lorentz Invariance Maxwell’s Equations The Klein-Gordon-Fock Equation The Dirac Equation The Green’s Function of a Spinless Particle The Green’s Function of a Particle With Spin 1/2 The Photon Green’s Function

2 Divergences and Renormalizability

The Local Interaction Between Particles Feynman Graphs in a Scalar Theory Estimation of Divergences: The Idea of Renormalization The Logarithmic Approximation and Renormalizability

3 Quantum Electrodynamics at Small Distances

The Local Interaction in Quantum Electrodynamics Vacuum Polarization Radiative Corrections to Coulomb’s Law The Electromagnetic Interaction at Ultra-Small Distances

Index

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