Non-Hermitian quantum mechanics

Non-Hermitian quantum mechanics (NHQM) is an important alternative to the standard (Hermitian) formalism of quantum mechanics, enabling the solution of otherwise difficult problems. The first book to present this theory, it is useful to advanced graduate students and researchers in physics, chemistry and engineering. NHQM provides powerful numerical and analytical tools for the study of resonance phenomena - perhaps one of the most striking events in nature. It is especially useful for problems whose solutions cause extreme difficulties within the structure of a conventional Hermitian framework. NHQM has applications in a variety of fields, including optics, where the refractive index is complex; quantum field theory, where the parity-time (PT) symmetry properties of the Hamiltonian are investigated; and atomic and molecular physics and electrical engineering, where complex potentials are introduced to simplify numerical calculations.

Author Notes

Nimrod Moiseyev is Bertha Hartz Axel Professor of Chemistry and Physics and a member of the joint Minerva Centre for Nonlinear Physics of Complex Systems at the Technion-Israel Institute of Technology. He has won numerous awards for his work, including the Humboldt Prize, the Marie Curie Prize, and the Landau Prize for major contribution to the development of the non-Hermitian theory of quantum mechanics.

Preface	p. xi
1 Different formulations of quantum mechanics	p. 1
1.1 Hermitian operators: a brief review	p. 3
1.2 Non-Hermitian potentials which support a continuous spectrum	p. 4
1.3 Complex local potentials	p. 10
1.4 Physical interpretation of complex expectation values	p. 11
1.5 Concluding remarks	p. 12
1.6 Solutions to the exercises	p. 13
1.7 Further reading	p. 19
2 Resonance phenomena in nature	p. 21
2.1 Shape-type resonances	p. 21
2.2 Feshbach-type resonances	p. 25
2.3 Concluding remarks	p. 31
2.4 Solutions to the exercises	p. 33
2.5 Further reading	p. 39
3 Resonances from Hermitian quantum-mechanical calculations	p. 41
3.1 Resonances as metastable states	p. 41
3.2 The poles of the S-matrix	p. 45
3.3 Resonances from the spectra of density of states	p. 46
3.4 Resonances from the asymptotes of continuum eigenfunctions	p. 50
3.5 Resonances from the phase shifts	p. 54
3.6 The scattering length	p. 57
3.7 Resonances from stabilization calculations	p. 60
3.8 Decay of resonance states	p. 64
3.9 Real and complex poles of the scattering matrix from wavepacket propagation calculations	p. 70
3.10 Concluding remarks	p. 71
3.11 Solutions to the exercises	p. 72
3.12 Further reading	p. 82
4 Resonances from non-Hermitian quantum mechanical calculations	p. 84
4.1 Resonances for a time-independent Hamiltonian	p. 86
4.2 Transitions of bound states to anti-bound and resonance states	p. 91
4.3 Bound, virtual and resonance states for a 1D potential	p. 95
4.4 The mechanism of transition from a bound state to a resonance state	p. 97
4.5 Concluding remarks on the physical and non-physical poles of the S-matrix	p. 101
4.6 Resonances for a time-dependent Hamiltonian	p. 102
4.7 Conservation of number of particles	p. 104
4.8 Solutions to the exercises	p. 106
4.9 Further reading	p. 115
5 Square integrable resonance wavefunctions	p. 116
5.1 The Zel'dovich transformation	p. 118
5.2 The complex scaling transformation	p. 120
5.3 The exterior scaling transformation	p. 127
5.4 The smooth exterior scaling transformation	p. 129
5.5 Dilation of the Hamiltonian matrix elements into the complex plane	p. 133
5.6 Square integrability of field induced resonances	p. 136
5.7 Partial widths from the tails of the wavefunctions	p. 142
5.8 Concluding remarks	p. 147
5.9 Solutions to the exercises	p. 149
5.10 Further reading	p. 169
6 Bi-orthogonal product (c-product)	p. 174
6.1 The c-product	p. 174
6.2 Completeness of the spectrum	p. 183
6.3 Advantages of calculating survival probabilities by c-product	p. 186
6.4 The c-product for non-Hermitian time-periodic Hamiltonians	p. 188
6.5 The F-product for time propagated wavepackets	p. 190
6.6 The F-product and the conservation of the number of particles	p. 195
6.7 Concluding remarks	p. 196
6.8 Solutions to the exercises	p. 197
6.9 Further reading	p. 210
7 The properties of the non-Hermitian Hamiltonian	p. 211
7.1 The turn-over rule	p. 211
7.2 The complex analog of the variational principle	p. 213
7.3 The complex analogs of the virial and hypervirial theorem	p. 225
7 4 The complex analog of the Hellmann-Feynman theorem	p. 226
7.5 Cusps and 0-trajectories	p. 227
7.6 Upper and lower bounds of the resonance positions and widths	p. 230
7.7 Perturbation theory for non-Hermitian Hamiltonians	p. 235
7.8 Concluding remarks	p. 237
7.9 Solutions to the exercise	p. 238
7.10 Further reading	p. 247
8 Non-Hermitian scattering theory	p. 250
8.1 Full collision processes for time-independent Systems	p. 254
8.2 Half collision processes for time-independent Systems	p. 266
8.3 Time-independent scattering theory for time-dependent Systems	p. 275
8.4 Solutions to the exercises	p. 309
8.5 Further reading	p. 318
9 The self-orthogonality phenomenon	p. 323
9.1 The phenomenon of self-orthogonality	p. 324
9.2 On self-orthogonality and the closure relations	p. 334
9.3 Calculations of the radius of convergence of perturbational expansion of the eigenvalues in V 0	p. 342
9.4 The effect of self-orthogonality on c-expectation values	p. 343
9.5 Zero resonance contribution to the cross section	p. 350
9.6 Geometric phases (Berry phases)	p. 351
9.7 Concluding remarks	p. 358
9.8 Solutions to the exercises	p. 359
9.9 Further reading	p. 373
10 The point where QM branches into two formalisms	p. 375
10.1 Feshbach resonances	p. 375
10.2 The point where QM branches into two formalisms	p. 379
10.3 Concluding remarks	p. 387
10.4 Solutions to the exercises	p. 387
10.5 Further reading	p. 391
Index	p. 393

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