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Non-Hermitian Hamiltonians and Quantum Transport in Multi-Terminal Conductors
Open AccessArticle

Time-Dependent Pseudo-Hermitian Hamiltonians and a Hidden Geometric Aspect of Quantum Mechanics

Departments of Mathematics and Physics, Koç University, Sarıyer, 34450 Istanbul, Turkey
Entropy 2020, 22(4), 471; https://doi.org/10.3390/e22040471
Received: 18 March 2020 / Revised: 11 April 2020 / Accepted: 16 April 2020 / Published: 20 April 2020
(This article belongs to the Special Issue Quantum Dynamics with Non-Hermitian Hamiltonians)
A non-Hermitian operator H defined in a Hilbert space with inner product · | · may serve as the Hamiltonian for a unitary quantum system if it is η -pseudo-Hermitian for a metric operator (positive-definite automorphism) η . The latter defines the inner product · | η · of the physical Hilbert space H η of the system. For situations where some of the eigenstates of H depend on time, η becomes time-dependent. Therefore, the system has a non-stationary Hilbert space. Such quantum systems, which are also encountered in the study of quantum mechanics in cosmological backgrounds, suffer from a conflict between the unitarity of time evolution and the unobservability of the Hamiltonian. Their proper treatment requires a geometric framework which clarifies the notion of the energy observable and leads to a geometric extension of quantum mechanics (GEQM). We provide a general introduction to the subject, review some of the recent developments, offer a straightforward description of the Heisenberg-picture formulation of the dynamics for quantum systems having a time-dependent Hilbert space, and outline the Heisenberg-picture formulation of dynamics in GEQM. View Full-Text
Keywords: pseudo-Hermitian operator; time-dependent Hilbert space; energy observable; Heisenberg picture pseudo-Hermitian operator; time-dependent Hilbert space; energy observable; Heisenberg picture
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Mostafazadeh, A. Time-Dependent Pseudo-Hermitian Hamiltonians and a Hidden Geometric Aspect of Quantum Mechanics. Entropy 2020, 22, 471.

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