Quantum Theory of Many-Particle Systems
MWF 1:00 - 2:30 pm, Compton 245
Wim Dickhoff
Fall 2003
Quantum Theory of Many-Particle Systems
Physics 540, Fall 2003 Tentative Schedule August 19, 2003
Instructor: Wim Dickhoff
Textbook: Mattuck - recommended: cheapest physics book in the bookstore and funny book; not required since its treatment doesn't overlap very well with the presentation of the course. There is a textbook being written by myself and Dimitri Van Neck from Belgium. This material will be made available during the semester.
The book outline below corresponds roughly to the course plan for the Semester.
Book Outline
1. Identical particles
1.1 Some simple considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Bosons and fermions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Antisymmetric and symmetric two-particle states. . . . . . . . . . . . . . . . . . 4
1.4 Some experimental consequences related to identical particles. . . . . . . . 9
1.5 Antisymmetric and symmetric many-particle states . . . . . . . . . . . . . . . 11
1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15
2. Second quantization
2.1 Fermion addition and removal operators . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Boson addition and removal operators . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 One-body operators in Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Two-body operators in Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . .24
2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28
3. Independent-particle model for fermions in finite systems
3.1 General results and the independent-particle model . . . . . . . . . . . . . . 31
3.2 Electrons in atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Nucleons in nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3.1 Empirical Mass Formula and Nuclear Matter . . . . . . . . . . . . . . . . 44
3.4 Second quantization and isospin . . . . . . . . . . . . . . . . . . . . . . . . . . . .46
3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4. Noninteracting Fermi gas
4.1 The Fermi gas at zero temperature . . . . . . . . . . . . . . . . . . . . . . . . . . .51
4.2 Electron gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54
4.3 Nuclear matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55
4.4 Neutron matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.5 Liquid 3He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55
4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5. Two-particle states and interactions
5.1 Symmetry considerations for two-particle states . . . . . . . . . . . . . . . . 57
5.2 Two particles outside closed shells . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3 General discussion of two-body interactions . . . . . . . . . . . . . . . . . . . .61
5.4 Examples of relevant two-body interactions . . . . . . . . . . . . . . . . . . . .64
5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6. Bosons and fermions at finite temperature
6.1 Some statistical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69
6.2 Bosons at finite T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71
6.2.1 Bose-Einstein condensation in infinite systems . . . . . . . . . . . . . . . . . 71
6.2.2 Bose-Einstein condensation in traps . . . . . . . . . . . . . . . . . . . . . . . . 74
6.2.3 Trapped bosons at finite temperature: thermodynamic limit . . . . . . 77
6.3 Fermions at finite T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.3.1 Noninteracting fermion systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.3.2 Fermion atoms in traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7. Propagators in one-particle quantum mechanics
7.1 Time evolution and propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.2 Expansion of the propagator and diagram rules . . . . . . . . . . . . . . . . . 88
7.2.1 Diagram Rules for the Single-Particle Propagator . . . . . . . . . . . . . . 89
7.3 Solution in the case of discrete states . . . . . . . . . . . . . . . . . . . . . . . . . .92
7.4 Scattering theory using propagators . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.4.1 Partial Waves and Phase Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102
8. Single-particle propagator in many-particle systems
8.1 Fermion single-particle propagator . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8.2 Lehmann representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105
8.3 Spectral functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106
8.4 Expectation values of operators in the correlated ground state . . . . . 108
8.5 Propagator for noninteracting systems . . . . . . . . . . . . . . . . . . . . . . . .110
8.6 Direct knock-out reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8.7 Comparison with (e, 2e) data for atoms . . . . . . . . . . . . . . . . . . . . . . 116
8.8 Comparison with (e, e'p) data for nuclei . . . . . . . . . . . . . . . . . . . . . . 116
8.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .116
9. Perturbation expansion of the single-particle propagator
9.1 Time evolution in the interaction picture . . . . . . . . . . . . . . . . . . . . . .117
9.2 Expansion in terms of noninteracting propagators . . . . . . . . . . . . . . 118
9.3 Wick's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .120
9.4 Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
9.5 Diagram rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
9.5.1 Time-dependent version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
9.5.2 Energy formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
9.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
10. Dyson Equation
10.1 Analysis perturbation expansion, self-energy, and Dyson's equation .153
10.2 Equation of motion method for propagators . . . . . . . . . . . . . . . . . . 159
10.3 Two-particle propagator, vertex function, and self-energy . . . . . . . . 161
10.4 Dyson equation and the vertex function . . . . . . . . . . . . . . . . . . . . . . .166
10.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
11. Mean-field approximation at T = 0
11.1 Derivation of the Hartree-Fock equations . . . . . . . . . . . . . . . . . . . . . 171
11.2 Variational content of the HF equations . . . . . . . . . . . . . . . . . . . . . . .174
11.3 Hartree-Fock in infinite systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
11.3.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
11.3.2 The electron gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
11.3.3 Nuclear matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .177
11.4 Hartree-Fock method in atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
11.5 Comparison with experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . .177
11.6 Hartree-Fock for molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
11.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
12. Interacting boson systems at T = 0
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .179
12.2 Boson propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
12.2.1 Single-particle propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
12.2.2 Equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
12.2.3 Non-interacting propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .181
12.3 Hartree-Bose approximation through variational ansatz . . . . . . . . . . . .182
12.4 Standard perturbation expansion for bosons . . . . . . . . . . . . . . . . . . . . . 183
12.4.1 Bogoliubov method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .183
12.4.2 Number conserving approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
12.5 Gross-Pitaevski equation for dilute systems . . . . . . . . . . . . . . . . . . . . . .184
12.6 Computer exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .184
13. Beyond the mean-field approximation
13.1 Second-order self-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .185
13.2 Schematic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
13.3 Application to nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
13.4 Application to atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
13.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .185
14. Excited states
14.1 Polarization propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .188
14.2 Random Phase Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .192
14.3 Schematic model in nite systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .196
14.4 Lindhard function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .196
14.5 Plasmons in the electron gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .196
14.6 Excitations of a normal Fermi liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . .196
14.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .196
15. Excited states in A +/- 2 systems
15.1 Two-particle problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
15.2 Ladder diagrams and short-range correlations . . . . . . . . . . . . . . . . . . . . 197
15.3 Pair excitations in nuclear matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
15.4 Cooper problem and pairing instability . . . . . . . . . . . . . . . . . . . . . . . . . .197
15.5 Particle-particle hole-hole RPA in nite systems . . . . . . . . . . . . . . . . . . 197
15.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .197
16. Dynamical treatment of the self-energy
16.1 Spectral function of the electron gas . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
16.2 Spectral function of nuclear matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
16.3 Long-range correlations in finite systems . . . . . . . . . . . . . . . . . . . . . . . . 199
16.4 Baym-Kadanoff procedure for excited states . . . . . . . . . . . . . . . . . . . . .199
16.5 Computer problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .199
17. Pairing phenomena
17.1 Anomalous propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
17.2 BCS gap equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
17.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
17.3.1 Superconductivity in metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
17.3.2 Superfluid 3He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
17.3.3 Superfluidity in neutron stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
17.4 Computer problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 201
18. Perturbation theory at finite temperature
18.1 Matsubara formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .203
18.2 Diagram rules at finite T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
18.3 Hartree-Fock at finite T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .203
18.4 BCS at finite T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .203
18.5 Spectral functions at finite T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
18.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .203
Appendix A Pictures and Gell-Mann and Low Theorem
A.1 Schrodinger Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .205
A.2 Interaction Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
A.3 Heisenberg Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
A.4 The adiabatic switching one of the full interaction . . . . . . . . . . . . . . . . .212
A.5 Gell-Mann and Low Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .214
Bibliography
Index
1. FORMAT OF COURSE:
i. Three lectures per week on Monday, Wednesday, and Friday from 1-1:30 pm in Compton 245ii. Appropriate review / discussion time during class
iii. Regular assignments covering class material
iv. Some computer assignments
v. No exams
vi. A paper discussing the material of a relevant set of articles from the literature related to the material of the course is to be turned in before the last class. This paper must be written in revtex format (used in Physical Review journals) and should contain a proper set of references. Using the documentstyle [prc,aps] the paper should be at least 7 but not more than 10 pages long and may include equations but not detailed derivations (if necessary they can be included in an appendix). Half a page containing a proposal for the topic of the paper is due during the last class before Fall break.
vii. A 25/30-minute presentation on the material of the paper is required. Attendance at all talks by other students is also required. This talk should include a motivation, a discussion of the method of solution and experimental data (where appropriate), a discussion of the results, and a summary plus conclusions of the presented material. The use of overhead transparencies is recommended.
viii. Classroom participation and discussion is useful.
ix. Reading assignments are also useful.
2. GRADING POLICY:
You can download the course schedule and information from the regular website Acrobat Reader.
- Tentative Course Schedule
- Information sheet
- Ch.1 Identical particles
- Ch.2 Second quantization
- Ch.3 Part 1
- Ch.3 Part 2
- Ch.4 without figures
- Ch.5 without figures
- Ch.6
- Ch.7
- Ch.8
- Ch.9 Part 1
- Ch.9 Part 2
- Ch.10
- Ch.11 Part 1
- Ch.11 Part 2
- Ch.11 Part 3
- Ch.13 Part 1
- Ch.14 Part 1
- Ch.14 Part 2
- Problem Set 1
- Problem Set 2
- Problem Set 3
- Problem Set 4
- Problem Set 5
- Computer assignment 1
- Computer assignment 2