Green 's Functions in Quantum Physics (Third Edition) By Eleftherios N. Economou
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Green 's Functions in Quantum Physics (Third Edition)
written by
Eleftherios N. Economou .
The new edition of a standard reference will be of interest to advanced students wishing to become familiar with
the method of Green's functions for obtaining simple and general solutions to basic problems in quantum physics.
The main part is devoted to the simplest kind of Green's functions, namely the solutions of linear differential
equations with a -function source. It is shown that these familiar Green's functions are a powerful tool for
obtaining relatively simple and general solutions of basic problems such as scattering and boundlevel
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. This third edition is 50%
longer than the previou and offers end-of-chapter problems and solutions (40% are solved) and additional
appendices to helpit is to serve as an effective self-tutorial and self-sufficient reference. Throughout, it
demonstrates the powerful and unifying formalism of Green's functions across many applications, including
transport properties, carbon nanotubes, and photonics and photonic crystals.
Green 's Functions in Quantum Physics (Third Edition)
written by
Eleftherios N. Economou
cover the following topics.
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 ?
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 Functions
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 41
3.1 General Relations
3.2 The Free-Particle (H0 = p2/2m) 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 Function
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
6.1 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 Significance 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
Appendix
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
Print 2 + k2
Print (R) = 345
C.1.1 Cartesian Coordinates x, y, z
C.1.2 Cylindrical Coordinates z, f,
C.1.3 Spherical coordinates r, ?, f
C.2 Vector Derivatives 347
C.2.1 Spherical Coordinates r, ?, f .
C.2.2 Cylindrical Coordinates z, , f
C.3 Schr¨odinger 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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