THE METRIC OPERATORS FOR PSEUDO-HERMITIAN HAMILTONIAN

The Hamiltonian of a conventional quantum system is Hermitian, which ensures real spectra of the Hamiltonian and unitary evolution of the system. However, real spectra are just the necessary conditions for a Hamiltonian to be Hermitian. In this paper, we discuss the metric operators for pseudo-Hermi...

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Bibliographic Details
Published in:The ANZIAM journal 2023-07, Vol.65 (3), p.215-228
Main Authors: WANG, WEN-HUA, CHEN, ZHENG-LI, LI, WEI
Format: Article
Language:English
Subjects:
Eigenvalues
Hilbert space
Operators
Quantum physics
Quantum theory
Spectra
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Summary:The Hamiltonian of a conventional quantum system is Hermitian, which ensures real spectra of the Hamiltonian and unitary evolution of the system. However, real spectra are just the necessary conditions for a Hamiltonian to be Hermitian. In this paper, we discuss the metric operators for pseudo-Hermitian Hamiltonian which is similar to its adjoint. We first present some properties of the metric operators for pseudo-Hermitian Hamiltonians and obtain a sufficient and necessary condition for an invertible operator to be a metric operator for a given pseudo-Hermitian Hamiltonian. When the pseudo-Hermitian Hamiltonian has real spectra, we provide a new method such that any given metric operator can be transformed into the same positive-definite one and the new inner product with respect to the positive-definite metric operator is well defined. Finally, we illustrate the results obtained with an example.
ISSN:1446-1811
1446-8735
DOI:10.1017/S1446181123000184