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International Journal of Quantum Chemistry

Nonperturbative solutions for one‐dimensional Schrödinger equation with position‐dependent mass

Mohammad Solimannejad, S. K. Moayedi, Majid Tavakoli

Published Nov 4, 2005• 4 citations
Publisher

Abstract

Abstract In this article, we discuss the relationship between exact solvability of the position‐dependent mass Schrödinger equation with analytical solutions of an ordinary Schrödinger equation in quantum mechanics. Furthermore, it is demonstrated that nonperturbative solutions of ordinary Schrödinger equation for a sextic potential leads to a large class of exact solutions for the position‐dependent mass Schrödinger equation. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

DOI
10.1002/qua.20861

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Abstract In this article, we discuss the relationship between exact solvability of the position‐dependent mass Schrödinger equation with analytical solutions of an ordinary Schrödinger equation in quantum mechanics. Furthermore, it is demonstrated that nonperturbative solutions of ordinary Schrödinger equation for a sextic potential leads to a large class of exact solutions for the position‐dependent mass Schrödinger equation. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006
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
Abstract The aim of the present study is to obtain exact solutions for the one‐dimensional position‐dependent effective mass Schrödinger equation. After the change of wave function on position‐dependent effective mass Schrödinger equation and comparison between the resulted equation with the PT‐symmetric invariant anharmonic potential, three distinct classes of exact solutions are obtained for the Schrödinger equation. As an illustrative example for a Gaussian form of m(x), energy spectrum, potential energy function and wave function are obtained explicitly. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
Abstract Exactly solvable Schrödinger equation (SE) with a position‐dependent mass distribution allowing Morse‐like eigenvalues is presented. For this, the position‐dependent mass Schrödinger equation is transformed into a standard SE, with constant mass, by means of the point canonical transformation scheme. In that method, the choice of potential for the position‐dependent mass Schrödinger equation allows us to obtain the transformation that should be used to find the exactly solvable SE. As a useful application of the proposal, the equivalent of the Witten superpotential is chosen to be constant to find the position‐dependent mass distribution and the exactly solvable potential V(m(x)) allowing Morse‐type energy spectra. This V(m(x)) is shown to have a Coulomb potential structure and can be useful in the study of the electronic properties of materials in which the carrier effective mass depends on the position. Moreover, the worked example, the approach is general and can be applied in the search of new potentials suitable on the study of quantum chemical systems. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010
An algebraic method of constructing potentials for which the Schrödinger equation with position dependent mass can be solved exactly is presented. A general form of the generators of su (1,1) algebra has been employed with a unified approach to the problem. Our systematic approach reproduces a number of earlier results and also leads to some novelties. We show that the solutions of the Schrödinger equation with position dependent mass are free from the choice of parameters for position dependent mass. Two classes of potentials are constructed that include almost all exactly solvable potentials.
Using the coordinate transformation method, we solve the one-dimensional Schrödinger equation with position-dependent mass. The explicit expressions for the potentials, energy eigenvalues, and eigenfunctions of the systems are given. The eigenfunctions can be expressed in terms of the Jacobi, Hermite, and generalized Laguerre polynomials. All potentials for these solvable systems have an extra term V m , which is produced from the dependence of mass on the position, compared with those for the systems of constant mass. The properties of V m for several mass functions are discussed.