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Journal of Applied Mathematics and Physics

A New Class of Exactly Solvable Models within the Schrödinger Equation with Position Dependent Mass

A. Dhahbi, YASSINE CHARGUI, Adel Trablesi

Published Jan 1, 2019• 11 citations
Publisher

Abstract

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.

DOI
10.4236/jamp.2019.75068

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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 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.
A new method for generating exactly solvable Schrödinger equations with a position-dependent mass is proposed. It is based on a relation with some deformed Schrödinger equations, which can be dealt with by using a supersymmetric quantum mechanical approach combined with a deformed shape-invariance condition. The solvability of the latter is shown to impose the form of both the deformed superpotential and the position-dependent mass. The conditions for the existence of bound states are determined. A lot of examples are provided and the corresponding bound-state spectrum and wavefunctions are reviewed.
Quantum dots, liquid crystals, compositionally graded crystals, and other condensed matter systems all use position dependent effective mass (PDEM) Hamiltonians to describe the dynamics of electrons. Because the PDEM quantum Hamiltonians are not Hermitian, we employ the effective mass kinetic energy operator in Von Ross?s two-parameter form, which is Hermitian by default and contains additional reasonable forms as special instances. It is shown that Hamiltonians of the form H(s)=(1-s)H_-+sH_+,0?s?1 where H_? are supersymmetric partner Hamiltonians corresponding to position dependent mass Schr?dinger equations are exactly solvable for a number of deformed shape invariant potentials.
We consider the application of the supersymmetric quantum-mechanical formalism to the Schr\"odinger equation describing a particle characterized by a position-dependent effective mass m(x).m(x).m(x). We show that any one-dimensional quantum system with effective mass has a supersymmetric partner system characterized by the same position dependence of the mass, but with a new potential function. The form of this supersymmetric partner potential V2(x){V}_{2}(x)V2​(x) depends on both the form of the original potential V1(x){V}_{1}(x)V1​(x) and the form of the mass x dependence. We also generalize the concept of shape invariance to the nonconstant mass scenario. As illustrative examples we provide, for a given form m(x)m(x)m(x) of the effective mass, shape-invariant potentials exhibiting (a) harmonic-oscillator-like spectra and (b) Morse-like spectra. In both cases the energy eigenvalues and eigenfunctions can be obtained in algebraic fashion.
During recent years, exact solutions of position-dependent mass Schrödinger equations have inspired intense research activities, based on the use of point canonical transformations, Lie algebraic methods or supersymmetric quantum mechanical techniques. Here we highlight the interest of another approach to such problems, relying on quadratic algebras. We illustrate this point by constructing spectrum generating algebras for a class of d -dimensional radial harmonic oscillators with d ≥ 2 (including the one-dimensional oscillator on the line via some minor changes) and a specific mass choice. This provides us with a counterpart of the well-known su(1, 1) Lie algebraic approach to the constant-mass oscillators.
In this work, the quantum operator approach is applied to both, the position-dependent mass Schrödinger equation (PDMSE) and the Schrodinger equation with constant mass (CMSE). This fact enable us to find the factorization operators that relates both Hamiltonians by means of a kinetic energy operator that comes from the proposal of Morrow and Brownstein. With this approach is possible to find the exactly-solvable PDMSE, for any value of the parameters α and γ in the von Roos's Hamiltonian. For that, our proposal can be considered as a unified treatment of the PDMSE because it contains as particular cases, the kinetic energy operators of various authors such as BenDaniel-Duke, Gora-Williams, Zhu-Kroemer and Li-Kuhn among others. To show the usefulness of our result, we show the solvable PDMSE that comes from the harmonic oscillator potential model for the CMSE. The proposal is general and can easily be extended to other potential models and mass distributions which will be given in the extended paper.