The main part of this book is devoted to the simplest kind of Green’s functions, namely the solutions of that differential equations with a delta function source. It is shown that these familiar Green’s functions are a powerful tool for obtaining relatively simple and general solutions of basic quantum problems such as scattering and bound-level information. The bound-level treatment gives a clear physical understanding of “difficult”questions such as superconductivity, the Kondo effect, and, to a lesser degree, disorder-induced localization. The more advanced subject of many-body Green’s functions is presented in the last part of the book.
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目录
Part I Green's Functions in Mathematical Physics 1 Time-Independent Green's Functions 1.1 Formalism 1.2 Examples 1.2.1 Three-Dimensional Case (d=3) 1.2.2 Two-Dimensional Case (d=2) 1.2.3 One-Dimensional Case (d=1) 1.2.4 Finite Domain Ω 1.3 Summary 1.3.1 Definition 1.3.2 Basic Properties 1.3.3 Methods of Calculation 1.3.4 Use Further Reading Problems 2 Time-Dependent Green's Function 2.1 First-Order Case 2.1.1 Examples 2.2 Second-Order Case 2.2.1 Examples 2.3 Summary 2.3.1 Definition 2.3.2 Basic Properties 2.3.3 Definition 2.3.4 Basic Properties 2.3.5 Use Further Reading Problems Part II Green's Functions in One-Body Quantum Problems 3 Physical Significance of G. Application to the Free-Particle Case 3.1 General Relations 3.2 The Free-Particle (H0=p22m) Case 3.2.1 3-d Case 3.2.2 2-d Case 3.2.3 1-d Case 3.3 The Free-Particle Klein-Gordon Case 3.4 Summary Further Reading Problems 4 Green's Functions and Perturbation Theory 4.1 Formalism 4.1.1 Time-Independent Case 4.1.2 Time-Dependent Case 4.2 Applications 4.2.1 Scattering Theory (E>0) 4.2.2 Bound State in Shallow Potential Wells (E<0) 4.2.3 The KKR Method for Electronic Calculations in Solids 4.3 Summary Further Reading Problems 5 Green's Functions for Tight-Binding Hamiltonians 5.1 Introductory Remarks 5.2 The Tight-Binding Hamiltonian (TBH) 5.3 Green's Functions 5.3.1 One-Dimensional Lattice 5.3.2 Square Lattice 5.3.3 Simple Cubic Lattice 5.3.4 Green's Functions for Bethe Lattices (Cayley Trees) 5.4 Summary Further Reading Problems 6 Single Impurity Scattering 6 Single Impurity Scattering 6.1 Formalism 6.2 Explicit Results for a Single Band 6.2.1 Three-Dimensional Case 6.2.2 Two-Dimensional Case 6.2.3 One-Dimensional Case 6.3 Applications 6.3.1 Levels in the Gap 6.3.2 The Cooper Pair and Superconductivity 6.3.3 The Kondo Problem 6.3.4 Lattice Vibrations in Crystals Containing “Isotope” Impurities 6.4 Summary Further Reading Problems 7 Two or More Impurities; Disordered Systems 7.1 Two Impurities 7.2 Infinite Number of Impurities 7.2.1 Virtual Crystal Approximation (VCA) 7.2.2 Average t-Matrix Approximation (ATA) 7.2.3 Coherent Potential Approximation (CPA) 7.2.4 The CPA for Classical Waves 7.2.5 Direct Extensions of the CPA 7.2.6 Cluster Generalizations of the CPA 7.3 Summary Further Reading Problems 8 Electrical Conductivity and Green's Functions 8.1 Electrical Conductivity and Related Quantities 8.2 Various Methods of Calculation 8.2.1 Phenomenological Approach 8.2.2 Boltzmann's Equation 8.2.3 A General，Independent-Particle Formula for Conductivity 8.2.4 General Linear Response Theory 8.3 Conductivity in Terms of Green's Functions 8.3.1 Conductivity Without Vertex Corrections 8.3.2 CPA for Vertex Corrections 8.3.3 Vertex Corrections Beyond the CPA 8.3.4 Post-CPA Corrections to Conductivity 8.4 Summary Further Reading Problems 9 Localization，Transport，and Green's Functions 9.1 An Overview 9.2 Disorder，Diffusion，and Interference 9.3 Localization 9.3.1 Three-Dimensional Systems 9.3.2 Two-Dimensional Systems 9.3.3 One-Dimensional and Quasi-One-Dimensional Systems 9.4 Conductance and Transmission 9.5 Scaling Approach 9.6 Other Calculational Techniques 9.6.1 Quasi-One-Dimensional Systems and Scaling 9.6.2 Level Spacing Statistics 9.7 Localization and Green's Functions 9.7.1 Green's Function and Localization in One Dimension 9.7.2 Renormalized Perturbation Expansion (RPE) and Localization 9.7.3 Green's Functions and Transmissions in Quasi-One-Dimensional Systems 9.8 Applications 9.9 Summary Further Reading Problems Part III Green's Functions in Many-Body Systems 10 Definitions 10.1 Single-Particle Green's Functions in Terms of Field Operators 10.2 Green's Functions for Interacting Particles 10.3 Green's Functions for Noninteracting Particles 10.4 Summary Further Reading Problems 11 Properties and Use of the Green's Functions 11.1 Analytical Properties of gs and gs 11.2 Physical Snificance and Use of gs and gs 11.3 Quasiparticles 11.4 Summary 11.4.1 Properties 11.4.2 Use Further Reading Problems 12 Calculational Methods for g 12.1 Equation of Motion Method 12.2 Diagrammatic Method for Fermions at T=0 12.3 Diagrammatic Method for T≠0 12.4 Partial Summations. Dyson's Equation 12.5 Other Methods of Calculation 12.6 Summary Further Reading Problems 13 Applications 13.1 Normal Fermi Systems. Landau Theory 13.2 High-Density Electron Gas 13.3 Dilute Fermi Gas 13.4 Superconductivity 13.4.1 Diagrammatic Approach 13.4.2 Equation of Motion Approach 13.5 The Hubbard Model 13.6 Summary Further Reading Problems A Dirac's delta Function B Dirac's bra and ket Notation C Solutions of Laplace and Helmholtz Equations in Various Coordinate Systems C.1 Helmholtz Equation (Δ2+k2)ψ(r)=0 C.1.1 Cartesian Coordinates x，y，z C.1.2 Cylindrical Coordinates z，Φ，p C.1.3 Spherical coordinates r，Θ，Φ C.2 Vector Derivatives C.2.1 Spherical Coordinates r，Θ，Φ C.2.2 Cylindrical Coordinates z，p，Φ C.3 Schrodinger Equation in Centrally Symmetric 3-and 2-Dimensional Potential V D Analytic Behavior of G(z) Near a Band Edge E Wannier Functions F Renormalized Perturbation Expansion (RPE) G Boltzmann's Equation H Transfer Matrix，S-Matrix，etc. I Second Quantization Solutions of Selected Problems References Index
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