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arXiv:0907.5069 · Europhysics Letters (EPL)

Potential algebra approach to position-dependent mass Schrödinger equations

T. K. Jana, P. Roy

Published Aug 1, 2009• 11 citations
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Abstract

It is shown that for a class of position-dependent mass Schrödinger equations the shape invariance condition is equivalent to a potential symmetry algebra. Explicit realization of such algebras has been obtained for some shape-invariant potentials.
DOI
10.1209/0295-5075/87/30003

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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.