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arXiv:quant-ph/0702205

Non-Hermitian Hamiltonians with Real Spectra in Quantum Mechanics

C. Yüce

Published Feb 22, 2007• 1 citation
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Abstract

A condition to have a real spectrum for a non-Hermitian Hamiltonian is given. As special cases, it is shown that the condition is reduced to Hermiticity and PT symmetric conditions.
DOI
10.48550/arxiv.quant-ph/0702205

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We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudo-supersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler-DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.
It is know that PT-symmetric models have real spectra provided the symmetry is not spontaneously broken. Even pseudo-hermitian models have real spectra, which enlarge the the class of non-hermitian models possessing real spectra. We however consider non-hermitian models in higher dimensions which are not necessarily explicit PT-symmetric nor pseudo-hermitian. We show that the models may generate real spectra depending upon the coupling constants of the Hamiltonian. Our models thus further generalize the class of non-hermitian systems, which possess real spectra.
One of the postulates of quantum mechanics is that the Hamiltonian is Hermitian, as this guarantees that the eigenvalues are real. Recently there has been an interest in asking if H†=HH^\dagger = HH†=H is a necessary condition, and has lead to the development of PT-symmetric quantum mechanics. This note shows that any finite physically acceptable non-Hermitian Hamiltonian is equivalent to doing ordinary quantum mechanics in a non-orthogonal basis. In particular, this means that there is no experimental distinction between PT-symmetric quantum mechanics and ordinary quantum mechanics for finite systems. In particular, the claim that PT-symmetric quantum mechanics allows for faster evolution than Hermitian quantum mechanics is shown to be a problem of physical interpretation.