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arXiv:1611.05394 · Journal of Mathematical Physics

Isospectral Hamiltonian for position-dependent mass for an arbitrary quantum system and coherent states

Sid-Ahmed Yahiaoui, M. Bentaïba

Published Jun 1, 2017• 8 citations
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Abstract

By means of the unitary transformation, a new way for discussing the ordering prescription of the Schrödinger equation with a position-dependent mass (PDM) for isospectral Hamiltonian operators is presented. We show that the ambiguity parameter choices in the kinetic part of the Hamiltonian can be explained through an exact SUSY QM symmetry as well as a consequence of an accidental symmetry under the Z2 action. By making use of the unitary transformation, we construct coherent states for a family of PDM isospectral Hamiltonians from a suitable choice of ladder operators. We show that these states preserve the usual structure of Klauder-Perelomov’s states and thus saturate and minimize the position-momentum uncertainty relation (PMUR) under some special restrictions. We show that PMUR properties can be used to determine the sign of the superpotential.

DOI
10.1063/1.4984981

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By means of the unitary transformation, a new way for discussing the ordering prescription of the Schrödinger equation with a position-dependent mass (PDM) for isospectral Hamiltonian operators is presented. We show that the ambiguity parameter choices in the kinetic part of the Hamiltonian can be explained through an exact SUSY QM symmetry as well as a consequence of an accidental symmetry under the Z2 action. By making use of the unitary transformation, we construct coherent states for a family of PDM isospectral Hamiltonians from a suitable choice of ladder operators. We show that these states preserve the usual structure of Klauder-Perelomov’s states and thus saturate and minimize the position-momentum uncertainty relation (PMUR) under some special restrictions. We show that PMUR properties can be used to determine the sign of the superpotential.
In this paper, we propose an algorithm to construct coherent states for an exactly solvable position dependent mass Schrödinger equation. We use point canonical transformation method and obtain ground state eigenfunction of the position dependent mass Schrödinger equation. We fix the ladder operators in the deformed form and obtain explicit expression of the deformed superpotential in terms of mass distribution and its derivative. We also prove that these deformed operators lead to minimum uncertainty relations. Further, we illustrate our algorithm with two examples, in which the coherent states given for the second example are new.
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'.
The interest in physical systems with position-dependent mass (PDM) is always topical. In this work we present a new way to construct kinetic energy operators (KEO’s) for particles endowed with a position varying mass within the Schrödinger equation. Taking recent developments of non-Hermitian quantum mechanics into account, we start with a non-Hermitian Hamiltonian and we recall that it is quasi-Hermitian and can be always associated to a Hermitian partner. The latter is then used along with the formalism of supersymmetric quantum mechanics (SUSY QM) to generate exactly solvable models on the basis of shape invariance condition. Some known and new concrete examples are considered as illustrations of the proposed scheme.
The study of physical systems endowed with a position-dependent mass (PDM) remains a fundamental issue of quantum mechanics. In this paper we use a new approach, recently developed by us for building the quantum kinetic energy operator (KEO) within the Schrodinger equation, in order to construct a new class of exactly solvable models with a position varying mass, presenting a harmonic-oscillator-like spectrum. To do so we utilize the formalism of supersymmetric quantum mechanics (SUSY QM) along with the shape invariance condition. Recent outcomes of non-Hermitian quantum mechanics are also taken into account.
The von Roos’ position dependent mass Schrödinger Equation (PDMSE) is given in terms of an effective potential, which is constructed by the potential under study and additional terms involving the position dependent mass and the ambiguity ordering parameters. At this regard, some of the most important approaches to the ambiguity parameters are those of BenDaniel-Duke, Gora-Williams, Zhu-Kroemer, Li-Kuhn and so on. In this work we show that for the cases of Zhu-Kroemer and BenDaniel-Duke it is possible to express any solvable potential through a couple of quantum superpotentials. For this, we stablish the factorization of the PDMSE Hamiltonian and the use of a point canonical transformation; these allow to identify a first superpotential W(x). Next, the term in the effective potential with ambiguity parameters is expressed through a function Z(x) and later the cases when Z(x) is a quantum superpotential are discussed. This way, the proposed superpotentials determine the solvable potentials in the PDMSE, the energy spectra and mass functions. We give some examples to explain the approach which can be straightforwardly applied to the solution of the PDMSE for other different potential models and mass distributions.
We study the generalized harmonic oscillator that has both the position-dependent mass and the potential depending on the form of mass function in a more general framework. The explicit expressions of the eigenvalue and eigenfunction for such a system are given, they have the same forms as those for the usual harmonic oscillator with constant mass. The coherent state and its properties for the system with PDM are also discussed. We give the corresponding effective potentials for several mass functions, the systems with such potentials are isospectral to the usual harmonic oscillator.