A New Approach for a Position Dependent Mass Problem

Sara Cruz y Cruz

Federica Agostini

José Juan Peña, Gerardo Ovando, J. Morales +2 more

Abstract In this work, an alternative approach for finding exact solutions of Schrödinger equations with a position‐dependent mass is presented. Essentially, the method consists in transforming the position‐dependent mass Schrödinger equation into a standard constant mass Schrödinger equation. The constant mass equation is formulated in terms of a new variable and includes, as effective potential, a term related to the position‐dependent mass and another term connected with the original potential. For exactly solvable potentials, our proposal leads to a Riccati equation for an equivalent Witten superpotential, which means that this kind of problems can be studied within the quantum supersymmetric framework with the aim to find the explicit solutions and the energy spectra. To show the usefulness of the proposed approach, we consider the particular case of exactly solvable Schrödinger equations for different position‐dependent mass situations in a null potential. However, the method can directly be applied to other effective potentials as well as to find isospectral potentials for specific position‐dependent mass distributions. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

B. BAGCHI, P. GORAIN, C. QUESNE +1 more

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.

Gerardo Ovando, José Juan Peña, J. Morales

The exactly-solvable position-dependent mass Schrodinger equation (PDMSE) for the Thomas-Fermi potential is presented. The PDMSE is transformed into a standard Schrodinger equation (SSE) with constant mass by means of a point canonical transformation scheme. By proposing an exponential type potential of the SSE it is possible to determine a PDMSE with the Thomas-Fermi potential. The resulting PDMSE is carried to the Sturm- Liouville form and the corresponding theory is developed for the particular resulting problem in order to obtain closed solutions in terms of Bessel functions of the first kind. Beyond the case considered in this work, the approach is general and can be useful in the study of electronic properties of non-uniform materials in which the carrier effective mass depends on the position as well as in the search of new solvable potentials suitable for quantum systems.

Óscar Rosas-Ortiz