Keeping in view the ordering ambiguity that arises due to the presence ofposition-dependent effective mass in the kinetic energy term of theHamiltonian, a general scheme for obtaining algebraic solutions of quantummechanical systems with position-dependent effective mass is discussed. Wequantize the Hamiltonian of the pertaining system by using symmetric orderingof the operators concerning momentum and the spatially varying mass, initiallyproposed by von Roos and Levy-Leblond. The algebraic method, used to obtain thesolutions, is based on the concepts of supersymmetric quantum mechanics andshape invariance. In order to exemplify the general formalism a class ofnon-linear oscillators has been considered. This class includes the particularexample of a one-dimensional oscillator with different position-dependenteffective mass profiles. Explicit expressions for the eigenenergies andeigenfunctions in terms of generalized Hermite polynomials are presented.Moreover, properties of these modified Hermite polynomials, like existence ofgenerating function and recurrence relations among the polynomials have alsobeen studied. Furthermore, it has been shown that in the harmonic limit, allthe results for the linear harmonic oscillator are recovered.

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

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