Modern Many Particle Physics

Lipparini981-238-345-X, 981-238-346-8

Table of contents :
Contents……Page 8
Preface……Page 6
PART 1……Page 12
1.1 Introduction……Page 13
1.2 Bosons……Page 14
1.3 Fermions……Page 15
1.4 Matrix Elements of One-Body Operators……Page 17
1.5 Matrix Elements of Two-Body Operators……Page 19
1.6 Density Matrices……Page 21
1.7 The Ideal Bose Gas Confined in a Harmonic Potential……Page 23
1.8 The Fermi Gas……Page 25
1.8.1 Excited States……Page 29
1.8.2 Polarized Fermi Gas……Page 33
1.9 Finite Temperature and Quasiparticles……Page 34
2.1 Introduction……Page 43
2.2 The Hartree-Fock Method for Fermions……Page 44
2.2.1 Examples of Physical Systems treated by the Hartree-Fock method……Page 48
2.2.2 Examples of Infinite Systems treated by the Hartree-Fock method……Page 58
2.3 The Hartree-Fock Method for Bosons……Page 63
2.4 The Gross-Pitaevskii Equations……Page 64
2.5 Hartree-Fock in Second Quantization Language……Page 65
2.6 Hartree-Fock at Finite Temperature……Page 69
2.7 Hartree-Fock-Bogoliubov and BCS……Page 73
3.2 The Lippman-Schwinger Equation……Page 81
3.3 The Bethe-Goldstone Equation……Page 83
3.4 The One-Dimensional Fermion System……Page 84
3.5 Numerical Results of BHF Calculation in Different Systems……Page 87
3.6.1 Decomposition in Partial Waves……Page 90
3.6.2 The Separable Approximation……Page 94
3.6.3 The g Matrix Expansion……Page 98
3.6.4 Numerical Results and Discussion……Page 101
3.6.5 Appendix……Page 104
4.2 The Density Functional Formalism……Page 109
4.3.1 The Thomas-Fermi Theory for the Atom……Page 111
4.3.2 The Gross-Pitaevskii Theory for the Ground State of a Diluted Gas of Bosons……Page 113
4.4 The Kohn-Sham Equations……Page 114
4.5 The Local Density Approximation for the Exchange-Correlation Energy……Page 116
4.6 The Local Spin Density Approximation (LSDA)……Page 117
4.7 Inclusion of Current Terms in the DFT (CDFT)……Page 120
4.8 Ensemble Density Functional Theory……Page 123
4.9 DFT for Strongly Correlated Systems: Nuclei and Helium……Page 124
4.10 DFT for Mixed Systems……Page 133
4.11 Symmetries and Mean Field Theories……Page 138
5.2 The Independent-Particle Model……Page 143
5.2.1 The Wo >> Wc Case……Page 144
5.2.2 The Wc >> Wo Case……Page 145
5.2.3 The MDD (Maximum Density Droplet) State……Page 147
5.3 Fractional Regime……Page 149
5.4 Hall Effect……Page 151
5.5 Elliptical Quantum Dots……Page 152
5.5.1 Analogies with the Bose-Einstein Condensate in a Rotating Trap……Page 157
5.6 Spin-Orbit Coupling and Spintronics……Page 158
5.7 The DFT for Quantum Dots in a Magnetic Field……Page 163
5.8 The Aharanov-Bohm Effect and Quantum Rings……Page 177
6.2 Standard Quadrature Formulae……Page 185
6.3 Random Variable Distributions and Central Limit Theorem……Page 187
6.4 Calculation of Integrals by the Monte Carlo Method……Page 188
6.5 Markov Chains……Page 189
6.6 The Metropolis Algorithm [M(RT)2]……Page 192
6.7 Variational Monte Carlo for Liquid 4He……Page 194
6.8 Monte Carlo Methods and Quantum Mechanics……Page 196
6.9 Propagation of a State in Imaginary Time……Page 197
6.10 Schrodinger Equation in Imaginary Time……Page 198
6.11 Importance Sampling……Page 200
6.12 Fermion Systems and the Sign Problem……Page 204
PART 2……Page 207
7.1 Introduction……Page 208
7.2 General Formalism……Page 212
7.3 Linear Response Function and Sum Rules……Page 215
7.4 Finite Temperature……Page 219
7.5 The Density Response……Page 221
7.6 The Current Response to an Electromagnetic Field……Page 225
7.7 The Density Response for Non-Interacting Homogeneous Systems……Page 230
8.1 The Linear Response Function in Landau Theory……Page 238
8.2 Time-Dependent Hartree (TDH) for Homogeneous Systems: The RPA……Page 251
8.3 TDH for the Density Matrix and the Landau Equations……Page 256
8.4 The RPA for Electron Gas in Different Dimensions: The Plasmon……Page 258
8.5 The RPA for Bosons……Page 264
8.6 The Time-Dependent Gross-Pitaevskii Theory……Page 268
8.7 Time Dependent Hartree-Fock (TDHF) and the Matrix RPAE……Page 274
8.8.1 RPA with Separable Interactions……Page 280
8.8.2 RPAE for Metal Clusters……Page 283
8.9 Adiabatic Time Dependent LSDA (TDLSDA)……Page 289
8.9.1 The TDLSDA Longitudinal Response Function……Page 291
8.9.2 The TDLSDA Transverse Response Function……Page 296
8.10 RPA and TDLSDA Commutators and Symmetry Restoration……Page 299
8.10.1 The Kohn and Larmor Theorems……Page 303
8.10.2 Magneto-Conductivity and Quantum Hall Effect……Page 305
8.11 The Linear Response Based on the Green Functions RPAE……Page 308
8.12 Screened Response Function and Dielectric Constant……Page 311
8.13 Examples of Application of the TDLSDA Theory……Page 312
8.13.1 Quantum Wells under very High External Magnetic Field……Page 313
8.13.2 Quantum Dots under Magnetic Field……Page 324
9.2 Interaction Energy and Correlation Energy……Page 350
9.3 The RPA Correlation Energy……Page 353
9.3.1 The RPA Correlation Energy for the Cold and Dilute Gases of Bosons and Fermions……Page 354
9.4 Theories Beyond the RPA……Page 357
9.5 STLS Theory……Page 358
9.6 Comparison of Different Theories for Electron Gas in 2D……Page 361
9.7 Quasiparticle Properties……Page 362
9.8 Nonlocal Effects……Page 365
9.9 Mean Energy of Many-Particle Excitations……Page 373
9.10 The Polarization Potential Model……Page 375
9.11 The Gross-Kohn Model……Page 378
9.12 The Method of Lorentz Transforms……Page 381
10.1 The Hydrodynamic Model for Bosons……Page 388
10.1.1 Backflow……Page 390
10.1.2 Compression and Surface Modes of Spherical Drops……Page 391
10.1.3 Compression and Surface Modes of a Bose Gas in a Magnetic Trap……Page 394
10.1.4 Moment of Inertia and the Scissors Mode of a Bose Gas in a Magnetic Trap……Page 396
10.1.5 Vortices in the Bose Gas in a Magnetic Trap……Page 401
10.2 The Fluidodynamic and Hydrodynamic Model for Fermions……Page 404
10.2.1 Dipolar Modes in Metal Clusters……Page 412
10.2.2 The Scalar Quadrupole Mode in Confined Systems……Page 414
10.2.3 The Scissors Mode in Fermi Systems……Page 415
10.2.4 The Moment of Inertia of Quantum Dots……Page 418
10.2.5 The Vibrating Potential Model……Page 421
10.3.1 Surface Vibrations of Charged Metal Clusters……Page 426
10.3.2 Edge Vibrations of Quantum Dots……Page 428
Index……Page 438