Shape-invariant quantum Hamiltonian with position-dependent effective mass through second order supersymmetry

TL;DR

This paper develops a second order supersymmetric method to analyze quantum systems with position-dependent mass, deriving new shape-invariant potentials that allow exact solutions with spectra similar to harmonic oscillators.

Contribution

It introduces a novel relation between potential and mass functions enabling the construction of exactly solvable models with shape invariance in position-dependent mass systems.

Findings

Derived a new shape-invariance condition for position-dependent mass systems.

Obtained algebraic solutions for spectra of specific potentials.

Identified classes of potentials with harmonic-oscillator-like spectra.

Abstract

Second order supersymmetric approach is taken to the system describing motion of a quantum particle in a potential endowed with position-dependent effective mass. It is shown that the intertwining relations between second order partner Hamiltonians may be exploited to obtain a simple shape-invariant condition. Indeed a novel relation between potential and mass functions is derived, which leads to a class of exactly solvable model. As an illustration of our procedure, two examples are given for which one obtains whole spectra algebraically. Both shape-invariant potentials exhibit harmonic-oscillator-like or singular-oscillator-like spectra depending on the values of the shape-invariant parameter.