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Journal of Mathematical Physics

An exactly soluble Schrödinger equation with smooth position-dependent mass

Liès Dekar, L. Chétouani, Théophile F. Hammann

Published May 1, 1998• 167 citations
Publisher

Abstract

The one-dimensional generalized Schrödinger equation for a system with smooth potential and mass step is resolved exactly. The wave function depends on the Heun’s function, which is a solution of a second-order Fuchsian equation with four singularities. The behavior of the transmission coefficient as a function of energy is compared to that of the case of an abrupt potential and mass step. Two limiting cases are also studied: when the width of the mass step is vanishing, and when the smooth potential and mass step tend to an abrupt potential and mass step.

DOI
10.1063/1.532407

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The one-dimensional generalized Schrödinger equation for a system with smooth potential and mass step is resolved exactly. The wave function depends on the Heun’s function, which is a solution of a second-order Fuchsian equation with four singularities. The behavior of the transmission coefficient as a function of energy is compared to that of the case of an abrupt potential and mass step. Two limiting cases are also studied: when the width of the mass step is vanishing, and when the smooth potential and mass step tend to an abrupt potential and mass step.
The Schrödinger equation with a position-dependent mass (SEPDM) is employed in many areas of quantum physics. Exact solutions for the SEPDM lie at the center of interest of the professional public because it helps us to understand the behavior of quantum particles in the cases in which their mass varies spatially. For this purpose, we used the mass function represented by a quartic polynomial and a quadratic potential function, which extends the current class of exact solutions of the SEPDM. The exact analytical solution of the problem is expressed as a linear combination of local Heun functions. Heun’s equation contains many parameters, resulting in its general nature. We studied how limit changes in some of these parameters will affect the solution of the SEPDM. The obtained solutions are particularly suitable for the transfer matrix method and solutions of scattering problems; this is demonstrated by the calculation of bound states.
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Using the coordinate transformation method, we solve the one-dimensional Schrödinger equation with position-dependent mass(PDM). 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, Hemite and generalized Laguerre polynomials. All potentials for these solvable systems have an extra term VmV_mVm​ which produced from the dependence of mass on the coordinate, compared with that for the systems of constant mass. The properties of VmV_mVm​ for several mass functions are discussed.
The Schrödinger equation under the application of a position-dependent mass (PDM) with an exponential form is presented. Several physical models are carried out by choosing different external potential fields including the free field or a confined hard-all potential, the linear potential plus an attractive centrifugal-like term and harmonic oscillator. The eigenfunction of the first case is given by a Bessel function. The calculations of the second case are found to have the Airy function and we are able to get a general form of energy levels based on the zeros of Airy function of the first kind. The last case is found that the eigenfunctions are given by the popular associated Laguerre function. To provide a better physical insight into the solutions, some figures are plotted graphically.
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