A systematic procedure to study one-dimensional Schrödinger equation with a position-dependent effective mass (PDEM) in the kinetic energy operator is explored. The conventional free-particle problem reveals a new and interesting situation in that, in the presence of a mass background, formation of bound states is signalled. We also discuss coordinate-transformed, constant-mass Schrödinger equation, its matching with the PDEM form and the consequent decoupling of the ambiguity parameters. This provides a unified approach to many exact results known in the literature, as well as to a lot of new ones.Running head: Effective-Mass Schrödinger Equation
By using the point canonical transformation approach in a manner distinct from previous ones, we generate some new exactly solvable or quasi-exactly solvable potentials for the one-dimensional Schrödinger equation with a position-dependent effective mass. In the latter case, SUSYQM techniques provide us with some additional new potentials.
A one-dimensional Schrödinger equation with position-dependent effective mass in the kinetic energy operator is studied in the framework of an so(2, 1) algebra. New mass-deformed versions of Scarf II, Morse and generalized Pöschl-Teller potentials are obtained. Consistency with an intertwining condition is pointed out.
We provide some explicit examples wherein the Schrödinger equation for the Morse potential remains exactly solvable in a position-dependent mass background. Furthermore, we show how in such a context, the map from the full line (−∞, ∞) to the half line (0, ∞) may convert an exactly solvable Morse potential into an exactly solvable Coulomb one. This generalizes a well-known property of constant-mass problems.
Running head: Morse and Coulomb Potentials
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