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Author Moiseyev, Nimrod
Title Non-Hermitian Quantum Mechanics
Imprint Cambridge : Cambridge University Press, 2011
©2011
book jacket
Descript 1 online resource (410 pages)
text txt rdacontent
computer c rdamedia
online resource cr rdacarrier
Note Cover -- Half-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- 1 Different formulations of quantum mechanics -- 1.1 Hermitian operators: a brief review -- 1.2 Non-Hermitian potentials which support a continuous spectrum -- 1.3 Complex local potentials -- 1.4 Physical interpretation of complex expectation values -- 1.5 Concluding remarks -- 1.6 Solutions to the exercises -- 1.7 Further reading -- 2 Resonance phenomena in nature -- 2.1 Shape-type resonances -- 2.2 Feshbach-type resonances -- 2.3 Concluding remarks: on the ambiguity of the definitions of shape- and Feshbach-type resonances -- 2.4 Solutions to the exercises -- 2.5 Further reading -- 3 Resonances from Hermitian quantum-mechanical calculations -- 3.1 Resonances as metastable states -- 3.2 The poles of the S-matrix -- 3.3 Resonances from the spectra of density of states -- 3.4 Resonances from the asymptotes of continuum eigenfunctions -- 3.5 Resonances from the phase shifts -- 3.6 The scattering length -- 3.7 Resonances from stabilization calculations -- 3.8 Decay of resonance states -- 3.9 Real and complex poles of the scattering matrix from wavepacket propagation calculations -- 3.10 Concluding remarks -- 3.11 Solutions to the exercises -- 3.12 Further reading -- 4 Resonances from non-Hermitian quantum mechanical calculations -- 4.1 Resonances for a time-independent Hamiltonian -- 4.2 Transitions of bound states to anti-bound and resonance states -- 4.3 Bound, virtual and resonance states for a 1D potential -- 4.4 The mechanism of transition from a bound state to a resonance state -- 4.5 Concluding remarks on the physical and non-physical poles of the S-matrix -- 4.6 Resonances for a time-dependent Hamiltonian -- 4.7 Conservation of number of particles -- 4.8 Solutions to the exercises -- 4.9 Further reading -- 5 Square integrable resonance wavefunctions
5.1 The Zel'dovich transformation -- 5.2 The complex scaling transformation -- 5.3 The exterior scaling transformation -- 5.4 The smooth exterior scaling transformation -- 5.5 Dilation of the Hamiltonian matrix elements into the complex plane -- 5.6 Square integrability of field induced resonances -- 5.7 Partial widths from the tails of the wavefunctions -- 5.8 Concluding remarks -- 5.9 Solutions to the exercises -- 5.10 Further reading -- 6 Bi-orthogonal product (c-product) -- 6.1 The c-product -- 6.2 Completeness of the spectrum -- 6.3 Advantages of calculating survival probabilities by c-product -- 6.4 The c-product for non-Hermitian time-periodic Hamiltonians -- 6.5 The F-product for time propagated wavepackets -- 6.6 The F-product and the conservation of the number of particles -- 6.7 Concluding remarks -- 6.8 Solutions to the exercises -- 6.9 Further reading -- 7 The properties of the non-Hermitian Hamiltonian -- 7.1 The turn-over rule -- 7.2 The complex analog of the variational principle -- 7.2.1 Linear c-variational calculations -- 7.2.2 Non-linear variational approaches -- 7.3 The complex analogs of the virial and hypervirial theorem -- 7.4 The complex analog of the Hellmann-Feynman theorem -- 7.5 Cusps and theta-trajectories -- 7.6 Upper and lower bounds of the resonance positions and widths -- 7.6.1 The Hermitian representation of the solutions of the non-Hermitian time-independent Schrodinger equation -- 7.7 Perturbation theory for non-Hermitian Hamiltonians -- 7.8 Concluding remarks -- 7.9 Solutions to the exercises -- 7.10 Further reading -- 8 Non-Hermitian scattering theory -- 8.1 Full collision processes for time-independent systems -- 8.1.1 The complex coordinate scattering theory and the Kohn variational principle -- 8.1.2 Resonance scattering: partial widths -- 8.2 Half collision processes for time-independent systems
8.2.1 Wave packet propagation on complex potential energy surfaces -- 8.3 Time-independent scattering theory for time-dependent systems -- 8.3.1 Resonance photo-induced ionization decay rates by the non-Hermitian formalism of quantum mechanics -- 8.3.2 The complex-scaled resonance wave functions of atoms/molecules in intense time-periodic laser fields -- 8.3.3 The non-Hermitian adiabatic theorem for time-dependent open systems -- 8.3.4 Non-Hermitian quantum mechanical theory of high order harmonic generation -- 8.3.5 Non-Hermitian theory of above threshold ionization (ATI) spectra -- 8.4 Solutions to the exercises -- 8.5 Further reading -- 9 The self-orthogonality phenomenon -- 9.1 The phenomenon of self-orthogonality -- 9.1.1 The self-orthogonal eigenvectors of non-Hermitian matrices -- 9.2 On self-orthogonality and the closure relations -- 9.2.1 On self-orthogonality for complex potentials -- 9.3 Calculations of the radius of convergence of perturbational expansion of the eigenvalues in V0 -- 9.4 The effect of self-orthogonality on c-expectation values -- 9.5 Zero resonance contribution to the cross section -- 9.6 Geometric phases (Berry phases) -- 9.7 Concluding remarks -- 9.8 Solutions to the exercises -- 9.9 Further reading -- 10 The point where QM branches into two formalisms -- 10.1 Feshbach resonances -- 10.2 The point where QM branches into two formalisms -- 10.3 Concluding remarks -- 10.4 Solutions to the exercises -- 10.5 Further reading -- Index
The first book to present NHQM, useful to advanced graduate students and researchers in physics, chemistry and engineering
Description based on publisher supplied metadata and other sources
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2020. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries
Link Print version: Moiseyev, Nimrod Non-Hermitian Quantum Mechanics Cambridge : Cambridge University Press,c2011 9780521889728
Subject Quantum theory -- Mathematics.;Hermitian structures.;Resonance.;Hermitian symmetric spaces
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