Many-Body Theory of Solids: An Introduction

by John C. Inkson

Overview

here exists a gap in the present literature on quantum mechanics T and its application to solids. It has been difficult to find an intro­ ductory textbook which could take a student from the elementary quan­ tum mechanical ideas of the single-particle Schrodinger equations, through the formalism and new physical concepts of many-body theory, to the level where the student would be equipped to read the scientific literature and specialized books on specific topics. The present book, which I believe fills this gap, grew out of two courses which I have given for a number of years at the University of Cambridge: "Advanced Quan­ tum Mechanics," covering the quantization of fields, representations, and creation and annihilation operators, and "Many Body Theory," on the application of quantum field theory to solids. The first course is a final-year undergraduate physics course while the second is a joint first­ and fourth-year undergraduate math­ year postgraduate physics course ematics course. In an American context this would closely correspond to a graduate course at the masters level. In writing this book I have tried to stress the physical aspects of the mathematics preferring where possible to introduce a technique by using a simple illustrative example rather than develop a purely formal treatment. In order to do this I have assumed a certain familiarity with solid­ state physics on the level of a normal undergraduate course, but the book should also be useful to those without such a background.

Product Details

ISBN-13:	9781475702286
Publisher:	Springer US
Publication date:	11/26/2012
Edition description:	Softcover reprint of the original 1st ed. 1984
Pages:	334
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

1. The Interacting System.- 1.1. The Basic Problem.- 1.2. The Jellium Solid.- 1.3. Hartree Theory—The Sommerfeld Model.- 1.4. Hartree-Fock.- 1.5. Exchange and Correlation Holes.- 1.6. Correlation Effects and the Thomas-Fermi Model.- Problems.- 2. Green’s Functions of the Single-Particle Schrödinger Equation.- 2.1. Green’s Functions of the Schrödinger Equation.- 2.2. Green’s Functions and Perturbation Theory.- 2.3. Time-Dependent Green’s Functions.- 2.4. Green’s Function Diagrams.- 2.5. Green’s Functions or Wave Functions?.- Problems.- 3. Quantization of Waves (Second Quantization).- 3.1. Waves and Particles.- 3.2. The Linear Chain of Atoms.- 3.3. The General Quantization of a Wave System.- 3.4. Quantization of the Electromagnetic Field.- 3.5. Elementary Excitations and “Particles”.- 3.6. Perturbations and the Elementary Excitations.- 3.7. Summary.- Problems.- 4. Representations of Quantum Mechanics.- 4.1. Schrödinger Representation.- 4.2. Heisenberg Representation.- 4.3. Interaction Representation.- 4.4. Occupation Number Representation.- 4.5. Interaction between Waves and Particles.- 4.6. Field Operators.- Problems.- 5. Interacting Systems and Quasiparticles.- 5.1. Single-Particle States.- 5.2. Absorbing Media.- 5.3. Exact and Approximate Eigenstates.- 5.4. Landau Quasiparticles.- Problems.- 6. Many-Body Green’s Functions.- 6.1. Definition of the Many-Body Green’s Function.- 6.2. Relationship to Single-Particle Green’s Function.- 6.3. Energy Structure and the Green’s Function.- 6.4. The Lehman Representation and Quasiparticles.- 6.5. Expectation Values.- 6.6. Equation of Motion for the Green’s Function.- 6.7. Hartree and Hartree-Fock Approximations.- 6.8. The Self-Energy.- Problems.- 7. The Self-Energy and Perturbation Series.- 7.1.Functional Derivatives and the Calculation of G and—.- 7.2. Iterative Solution for the Green’s Function and Self-Energy.- 7.3. Screening and the Perturbation Series.- 7.4. The Screened Interaction and Selective Summations.- 7.5. The Uniform System.- Problems.- 8. Diagrammatic Interpretation of the Green’s Function Series.- 8.1. Diagrammatic Interpretation of the Perturbation Series.- 8.2. Diagrammatic Expansion.- 8.3. Infinite Series and Irreducible Diagrams.- 8.4. The Hartree Potential.- 8.5. The Uniform System.- 8.6. Rules for Evaluating Diagrams.- 8.7. Selective Summations.- 8.8. Practical Aspects of Diagrammatics.- Problems.- 9. The Normal System.- 9.1. The Jellium Solid Response Function.- 9.2. The Self-Energy (Physical Considerations).- 9.3. Evaluation of the Self-Energy and Quasiparticle Properties.- 9.4. Landau Quasiparticles.- 9.5. Insulating Systems.- 9.6. Surfaces.- Problems.- 10. Thermal Effects on the Green’s Function.- 10.1. The Density Matrix.- 10.2. Statistical Mechanics.- 10.3. The “Thermal” Heisenberg Representation.- 10.4. Evaluation of the Perturbation Expansion.- 10.5. Periodicity of G and the Extension to Energy Dependency.- 10.6. Real-Time Thermal Green’s Functions.- Problems.- 11. Boson Particles.- 11.1. Collective Excitations in Solids.- 11.2. Electron-Phonon System.- 11.3. Plasmons and the Total Interaction.- 11.4. Boson Systems with a Condensate.- Problems.- 12. Special Methods.- 12.1. The Density Functional Method (Nearly Uniform Electron Gases).- 12.2. Highly Localized Systems (Anderson-Hubbard Models).- 12.3. Canonical Transformations.- 12.4. Mean-Field Theory.- Problems.- 13. Superconductivity.- 13.1. Cooper Pairs.- 13.2. Canonical Transformations.- 13.3. Propagator Approach.- Problems.- Appendix: List of Symbols.