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arXiv:quant-ph/0511182 · Journal of Physics A Mathematical and General

Pseudo-Hermitian versus Hermitian position-dependent-mass Hamiltonians in a perturbative framework

Bijan Kumar Bagchi, C. Quesne, Rajkumar Roychoudhury

Published Jan 24, 2006• 22 citations
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Abstract

We formulate a systematic algorithm for constructing a whole class of Hermitian position-dependent-mass Hamiltonians which, to lowest order of perturbation theory, allow a description in terms of -symmetric Hamiltonians. The method is applied to the Hermitian analogue of the -symmetric cubic anharmonic oscillator. A new example is provided by a Hamiltonian (approximately) equivalent to a -symmetric extension of the one-parameter trigonometric Pöschl–Teller potential.

DOI
10.1088/0305-4470/39/6/l01

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We formulate a systematic algorithm for constructing a whole class of Hermitian position-dependent-mass Hamiltonians which, to lowest order of perturbation theory, allow a description in terms of -symmetric Hamiltonians. The method is applied to the Hermitian analogue of the -symmetric cubic anharmonic oscillator. A new example is provided by a Hamiltonian (approximately) equivalent to a -symmetric extension of the one-parameter trigonometric Pöschl–Teller potential.
We perform a perturbative calculation of the physical observables, in particular, pseudo-Hermitian position and momentum operators, the equivalent Hermitian Hamiltonian operator and the classical Hamiltonian for the -symmetric cubic anharmonic oscillator, . Ignoring terms of order 4 and higher, we show that this system describes an ordinary quartic anharmonic oscillator with a position-dependent mass, and real and positive coupling constants. This observation elucidates the classical origin of the reality and positivity of the energy spectrum. We also discuss the quantum–classical correspondence for this -symmetric system, compute the associated conserved probability density and comment on the issue of factor ordering in the pseudo-Hermitian canonical quantization of the underlying classical system.
In this paper, we build on our earlier proposal for the construction of a positive inner product for pseudo-Hermitian Hamiltonians and present examples to clarify the procedure. We focus on two detailed calculations where the method is used, namely, a simple (generalized 2×2 matrix) pseudo-Hermitian Hamiltonian, which can be diagonalized, and a second system where the Hamiltonian cannot be diagonalized, but can be described as a perturbation of the harmonic oscillator. When the quantum mechanical system cannot be diagonalized exactly, our construction can be carried out perturbatively and we develop the general formalism for such a perturbative calculation systematically (for real eigenvalues).
In the context of the factorization method, we investigate the pseudo-Hermitian coherent states and their Hermitian counterpart coherent states under the generalized quantum condition in the framework of a position-dependent mass. By considering a specific modification in the superpotential, suitable annihilation and creation operators are constructed in order to reproduce the Hermitian counterpart Hamiltonian in the factorized form. We show that by means of these ladder operators, we can construct a wide range of exactly solvable potentials as well as their accompanying coherent states. Alternatively, we explore the relationship between the pseudo-Hermitian Hamiltonian and its Hermitian counterparts, obtained from a similarity transformation, to construct the associated pseudo-Hermitian coherent states. These latter preserve the structure of Perelomov's states and minimize the generalized position–momentum uncertainty principle. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.