Contents
Part I Many-Electron Systems and the Electron Propagator
1 Systems of Identical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.1 Many-Electron Wave Functions . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2 Matrix Elements for Many-Electron States . . . . . . . . . . . . . . . .
9
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Second Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1 Definition of Creation and Destruction Operators . . . . . . . . . . . 19
2.2 Anticommutation Relations for Creation and Destruction
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Operators in Second Quantization . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Combining Second Quantization and Slater–Condon Rules . . . . 26
2.5 Change of the One-Particle Representation . . . . . . . . . . . . . . . . 27
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 One-Particle Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1 Definition and Relation to Physical Quantities . . . . . . . . . . . . . 31
3.2 Ground-State Expectation Values . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Ground-State Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4 Free One-Particle Green’s Function . . . . . . . . . . . . . . . . . . . . . 40
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Part II Formalism of Diagrammatic Perturbation Theory
4 Perturbation Theory for the Electron Propagator . . . . . . . . . . . . . . 45
4.1 Time-Development Operator in the Interaction Picture . . . . . . . 46
4.2 The Gell-Mann and Low Theorem . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Expectation Values of Heisenberg Operators . . . . . . . . . . . . . . 53
4.4 Comparison with Rayleigh–Schrödinger Perturbation Theory . . . 55
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
ix
Part I Many-Electron Systems and the Electron Propagator
1 Systems of Identical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.1 Many-Electron Wave Functions . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2 Matrix Elements for Many-Electron States . . . . . . . . . . . . . . . .
9
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Second Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1 Definition of Creation and Destruction Operators . . . . . . . . . . . 19
2.2 Anticommutation Relations for Creation and Destruction
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Operators in Second Quantization . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Combining Second Quantization and Slater–Condon Rules . . . . 26
2.5 Change of the One-Particle Representation . . . . . . . . . . . . . . . . 27
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 One-Particle Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1 Definition and Relation to Physical Quantities . . . . . . . . . . . . . 31
3.2 Ground-State Expectation Values . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Ground-State Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4 Free One-Particle Green’s Function . . . . . . . . . . . . . . . . . . . . . 40
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Part II Formalism of Diagrammatic Perturbation Theory
4 Perturbation Theory for the Electron Propagator . . . . . . . . . . . . . . 45
4.1 Time-Development Operator in the Interaction Picture . . . . . . . 46
4.2 The Gell-Mann and Low Theorem . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Expectation Values of Heisenberg Operators . . . . . . . . . . . . . . 53
4.4 Comparison with Rayleigh–Schrödinger Perturbation Theory . . . 55
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
ix