Stochastic Variational Approach to QM Few-Body Problems

Lecture Notes in Physics: Monographs 54 ......Page 2
Stochastic Variational Approach to Quantum-Mechanical Few-Body Problems ......Page 4
Table of Contents ......Page 6
1. Introduction ......Page 10
2.1 Hamiltonian ......Page 16
2.2 RelatiVe coordinates ......Page 18
2.3 Symmetrization ......Page 24
2.4 Permutation of the Jacobi coordinates ......Page 25
2.1 An N-particle Hamiltonian in the heavy-particle center coordinate set ......Page 27
2.2 Canonical Jacobi coordinates ......Page 28
3.1 Variational principles ......Page 30
3.2 The variance of local energy ......Page 39
3.3 The virial theorem ......Page 42
4.1 Basis optimization ......Page 48
4.2 A practical example ......Page 52
4.2.1 Geometric progression ......Page 53
4.2.3 Random basis ......Page 56
4.2.5 Trial and error search ......Page 59
4.2.6 Refining ......Page 62
4.2.7 Comparison of different optimizing strategies ......Page 63
4.3 Optimization for excited states ......Page 65
4.1 Minimization of energy versus variance ......Page 70
5.1 Quantum Monte Carlo method: The imaginary-time evolution of a system ......Page 74
5.2 Hyperspherical harmonics expansion method ......Page 76
5.3 Faddeev method ......Page 79
5.4 The generator coordinate method ......Page 81
6.1 Correlated Gaussians and correlated Gaussian-type geminals ......Page 84
6.2 Orbital functions with arbitrary angular momentum ......Page 91
6.3 Generating function ......Page 96
6.4 The spin fanction ......Page 103
6.1 Nodeless harmonic-oscillator functions as a basis ......Page 105
6.3 Angular momentum recoupling ......Page 115
6.4 Separation of the center-of-mass motion from correlated Gaussians ......Page 121
6.6 Four electrons in an arbitrary spin arrangement ......Page 124
6.7 Six electrons with S = 0 ......Page 125
Exercises ......Page 127
7.1 Matrix elements of the generating function ......Page 132
7.2 Correlated Gaussians ......Page 134
7.3 Correlated Gaussians in two-dimensional systems ......Page 138
7.4 Correlated Gaussian-type geminals ......Page 140
7.5 Nonlocal potentials ......Page 143
7.6 Semirelativistic kinetic energy ......Page 146
7. 1 Sherman-MorTison fonnula ......Page 152
Exercises ......Page 154
8.1 Coulombic systems ......Page 158
8.2 Coulombic three-body systems ......Page 159
8.3 Four or more particles ......Page 163
8.4 Small molecules ......Page 174
8.1 The cusp condition for the Coulomb potential ......Page 176
8.2 The chemical bond. The H^+_2 ion ......Page 178
8.3 Stability of hydrogen-like molecules ......Page 181
8.4 Application of global vectors to muonic molecules ......Page 183
9. Baryon spectroscopy ......Page 186
9.2 One-gluon exchange model ......Page 187
9.3 Meson-exchange model ......Page 190
10. Few-body problems in solid state physics ......Page 196
10.1 Excitonic complexes ......Page 197
10.2 Quantum dots ......Page 200
10.3 Quantum dots in magnetic field ......Page 205
10.4 Quantum dots in the generator coordinate method ......Page 211
10.1 Two-dimensional electron motion in a magnetic field ......Page 213
11.1 Introductory remark on nucleon-nucleon potentials ......Page 222
11.2 Few-nucleon systems with central forces ......Page 225
11.1 Correlations in few-nucleon systems ......Page 239
11-3 Quark Pauli effect in s-shell A hypernuclei ......Page 248
11.4 The 12C nucleus as a system of three alpha-particles ......Page 251
11.2 Convergence of partial-wave expansions ......Page 242
A.1.1 Overlap of the basis functions ......Page 256
A.1.2 Kinetic energy ......Page 258
A.1.3 Two-body interactions ......Page 259
A.1.4 Density multipole operators ......Page 265
A.2 Correlated Gaussians with different coordinate sets ......Page 266
A.3 Correlated Gaussian-type geminals ......Page 271
A.4 Spin matrix elements ......Page 272
A.5 Three-body problem with central, tensor and spin-orbit forces ......Page 274
A.I Matrix elements o central potentials ......Page 289
A.2 Matrix elements of density multipoles ......Page 292
A.3 Overlap matrix elements of the correlated Gaussians for a threeparticle system ......Page 294
Exercises ......Page 297
References ......Page 308
Index ......Page 316