We consider the application of the supersymmetric quantum-mechanical formalism to the Schrödinger equation describing a particle characterized by a position-dependent effective mass $m\left(x\right).$ 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 $V_2\mathrm{}\left(x\right)\mathrm{}$ depends on both the form of the original potential $V_1\mathrm{}\left(x\right)\mathrm{}$ 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 $m\left(x\right)\mathrm{}$ 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.

• Received 22 June 1999

DOI:https://doi.org/10.1103/PhysRevA.60.4318