Algebraic solutions of shape-invariant position-dependent effective mass systems

TL;DR

This paper develops an algebraic method using supersymmetric quantum mechanics and shape invariance to solve quantum systems with position-dependent effective mass, providing explicit eigenstates and energies for non-linear oscillators.

Contribution

It introduces a general algebraic framework for solving position-dependent mass systems, including explicit solutions and properties of associated polynomials, extending previous methods.

Findings

Explicit eigenenergies and eigenfunctions for position-dependent mass oscillators

Properties of generalized Hermite polynomials related to these systems

Recovery of harmonic oscillator results in the linear limit

Abstract

Keeping in view the ordering ambiguity that arises due to the presence of position-dependent effective mass in the kinetic energy term of the Hamiltonian, a general scheme for obtaining algebraic solutions of quantum mechanical systems with position-dependent effective mass is discussed. We quantize the Hamiltonian of the pertaining system by using symmetric ordering of the operators concerning momentum and the spatially varying mass, initially proposed by von Roos and Levy-Leblond. The algebraic method, used to obtain the solutions, is based on the concepts of supersymmetric quantum mechanics and shape invariance. In order to exemplify the general formalism a class of non-linear oscillators has been considered. This class includes the particular example of a one-dimensional oscillator with different position-dependent effective mass profiles. Explicit expressions for the eigenenergies and eigenfunctions in terms of generalized Hermite polynomials are presented. Moreover, properties of these modified Hermite polynomials, like existence of generating function and recurrence relations among the polynomials have also been studied. Furthermore, it has been shown that in the harmonic limit, all the results for the linear harmonic oscillator are recovered.