Non-Hermitian Quantum Mechanics

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 13
1 Different formulations of quantum mechanics......Page 17
1.1 Hermitian operators: a brief review......Page 19
1.2 Non-Hermitian potentials which support a continuous spectrum......Page 20
1.3 Complex local potentials......Page 26
1.4 Physical interpretation of complex expectation values......Page 27
1.5 Concluding remarks......Page 28
1.6 Solutions to the exercises......Page 29
1.7 Further reading......Page 35
2.1 Shape-type resonances......Page 37
2.2 Feshbach-type resonances......Page 41
2.3 Concluding remarks: on the ambiguity of the definitions of shape- and Feshbach-type resonances......Page 47
2.4 Solutions to the exercises......Page 49
2.5 Further reading......Page 55
3.1 Resonances as metastable states......Page 57
3.2 The poles of the S-matrix......Page 61
3.3 Resonances from the spectra of density of states......Page 62
3.4 Resonances from the asymptotes of continuum eigenfunctions......Page 66
3.5 Resonances from the phase shifts......Page 70
3.6 The scattering length......Page 73
3.7 Resonances from stabilization calculations......Page 76
3.8 Decay of resonance states......Page 80
3.9 Real and complex poles of the scattering matrix from wavepacket propagation calculations......Page 86
3.10 Concluding remarks......Page 87
3.11 Solutions to the exercises......Page 88
3.12 Further reading......Page 98
4 Resonances from non-Hermitian quantum mechanical calculations......Page 100
4.1 Resonances for a time-independent Hamiltonian......Page 102
4.2 Transitions of bound states to anti-bound and resonance states......Page 107
4.3 Bound, virtual and resonance states for a 1D potential......Page 111
4.4 The mechanism of transition from a bound state to a resonance state......Page 113
4.5 Concluding remarks on the physical and non-physical poles of the S-matrix......Page 117
4.6 Resonances for a time-dependent Hamiltonian......Page 118
4.7 Conservation of number of particles......Page 120
4.8 Solutions to the exercises......Page 122
4.9 Further reading......Page 131
5 Square integrable resonance wavefunctions......Page 132
5.1 The Zel'dovich transformation......Page 134
5.2 The complex scaling transformation......Page 136
5.3 The exterior scaling transformation......Page 143
5.4 The smooth exterior scaling transformation......Page 145
5.5 Dilation of the Hamiltonian matrix elements into the complex plane......Page 149
5.6 Square integrability of field induced resonances......Page 152
5.7 Partial widths from the tails of the wavefunctions......Page 158
5.8 Concluding remarks......Page 163
5.9 Solutions to the exercises......Page 165
5.10 Further reading......Page 185
6.1 The c-product......Page 190
6.2 Completeness of the spectrum......Page 199
6.3 Advantages of calculating survival probabilities by c-product......Page 202
6.4 The c-product for non-Hermitian time-periodic Hamiltonians......Page 204
6.5 The F-product for time propagated wavepackets......Page 206
6.6 The F-product and the conservation of the number of particles......Page 211
6.7 Concluding remarks......Page 212
6.8 Solutions to the exercises......Page 213
6.9 Further reading......Page 226
7.1 The turn-over rule......Page 227
7.2 The complex analog of the variational principle......Page 229
7.2.1 Linear c-variational calculations......Page 230
7.2.2 Non-linear variational approaches......Page 231
7.3 The complex analogs of the virial and hypervirial theorem......Page 241
7.4 The complex analog of the Hellmann–Feynman theorem......Page 242
7.5 Cusps and theta-trajectories......Page 243
7.6.1 The Hermitian representation of the solutions of the non-Hermitian time-independent Schrodinger equation......Page 246
7.7 Perturbation theory for non-Hermitian Hamiltonians......Page 251
7.8 Concluding remarks......Page 253
7.9 Solutions to the exercises......Page 254
7.10 Further reading......Page 263
8 Non-Hermitian scattering theory......Page 266
8.1 Full collision processes for time-independent systems......Page 270
8.1.1 The complex coordinate scattering theory and the Kohn variational principle......Page 274
8.1.2 Resonance scattering: partial widths......Page 276
8.2 Half collision processes for time-independent systems......Page 282
8.2.1 Wave packet propagation on complex potential energy surfaces......Page 290
8.3 Time-independent scattering theory for time-dependent systems......Page 291
8.3.1 Resonance photo-induced ionization decay rates by the non-Hermitian formalism of quantum mechanics......Page 294
8.3.2 The complex-scaled resonance wave functions of atoms/molecules in intense time-periodic laser fields......Page 298
8.3.3 The non-Hermitian adiabatic theorem for time-dependent open systems......Page 302
8.3.4 Non-Hermitian quantum mechanical theory of high order harmonic generation......Page 310
8.3.5 Non-Hermitian theory of above threshold ionization (ATI) spectra......Page 320
8.4 Solutions to the exercises......Page 325
8.5 Further reading......Page 334
9 The self-orthogonality phenomenon......Page 339
9.1 The phenomenon of self-orthogonality......Page 340
9.1.1 The self-orthogonal eigenvectors of non-Hermitian matrices......Page 342
9.2 On self-orthogonality and the closure relations......Page 350
9.2.1 On self-orthogonality for complex potentials......Page 353
9.3 Calculations of the radius of convergence of perturbational expansion of the eigenvalues in V0......Page 358
9.4 The effect of self-orthogonality on c-expectation values......Page 359
9.5 Zero resonance contribution to the cross section......Page 366
9.6 Geometric phases (Berry phases)......Page 367
9.7 Concluding remarks......Page 374
9.8 Solutions to the exercises......Page 375
9.9 Further reading......Page 389
10.1 Feshbach resonances......Page 391
10.2 The point where QM branches into two formalisms......Page 395
10.4 Solutions to the exercises......Page 403
10.5 Further reading......Page 407
Index......Page 409