SOLVABLE POTENTIALS WITH POSITION-DEPENDENT EFFECTIVE MASS AND CONSTANT MASS SCHRÖDINGER EQUATION

Vol. 41 (2010) ACTA PHYSICA POLONICA B No 1 H. Panahit , Z. Bakhshi Department of Physics, University of Guilan Rasht 51335-1914, Iran (Received November 12, 2008; revised version received June 3, 2009; final version received October 16, 2009) Abstract Using the point canonical transformation method, we show that a large class of solvable potentials with Position-Dependent Effective Mass (PDEM) can be obtained by using the internal functions which are introduced by Levai for solvable potentials with constant mass. We also obtain the ex- plicit expressions for some of these solvable potentials and show that their eigenfunctions can be obtained in terms of the known special functions such as Jacobi, generalized Laguerre and Hermit polynomials. PACS numbers: 03.65.–w, 03.65.Fd, 03.65.Ge, 11.30.Pb 1. Introduction In recent years, quantum mechanical systems with a Position-Dependent Effective Mass (PDEM) have attracted a lot of attention due to their appli- cations in condensed matter physics, nuclear physics, semiconductor theory and other related fields [1–8]. In theoretical researches, many different meth- ods have been used in the study of systems with constant mass such as the Lie algebraic techniques [9, 10], point canonical transformation [11–13], factor- ization method [14] and supersymmetric quantum mechanics together with shape invariance techniques [11, 15–17]. During the last few years, some of these developments have been generalized to the systems with PDEM and a number of interesting results has been produced [18–29]. For systems with constant mass, Levai used the point canonical transformation approach and calculated eigen spectrum of a large class of exactly solvable potentials, by transforming the Schrödinger equation into the second order differential equation which has solutions of the special functions [12]. By using of the † Corresponding author: t-panahi@guilan.ac.ir (11)