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arXiv:quant-ph/0601004

The coordinate transformation and the exact solutions of the Schrödinger equation with position-dependent effective mass

Ju Guo-Xing, Cai Chang-Ying, Xiang, Yang, Zhongzhou Ren

Published Dec 31, 2005• 1 citation
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Abstract

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 $V_m$ which produced from the dependence of mass on the coordinate, compared with that for the systems of constant mass. The properties of $V_m$ for several mass functions are discussed.

DOI
10.48550/arxiv.quant-ph/0601004

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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.
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.
The Schrodinger equation with position-dependent mass (PDM) becomes one of interesting subjects in the study of quantum systems, because of its wide applications in many physical problems. Meanwhile, harmonic oscillator becomes important model in most physical problems. In this paper, analytical solutions of the PDM Schrodinger equation, i.e. energy eigenvalues En and eigenfunctions , of a one dimensional harmonic oscillator system using transformation method has been studied. The results showed that there was no difference between the energy eigenvalues, En of the harmonic oscillator systems with and without PDM, whereas it is not for the eigenfunctions,  of these two harmonic oscillator systems. This difference was due to the presence of dimensionless mass function  and the changes of x becomes  variable for the eigenfunctions of the harmonic oscillator system with PDM.  Moreover, the differences in the eigenfunctions will affect the dynamical properties of the quantum systems. It can be concluded that the harmonic oscillator system with PDM is the generalization of the harmonic oscillator system with constant mass.
An approximate method is proposed to solve the position-dependent mass (PDM) Schrödinger equation. The procedure suggested here leads to the solution of the PDM Schrödinger equation without transforming the potential function to the mass space or vice versa. The method based on the asymptotic Taylor expansion of the function produces an approximate analytical expression for eigenfunction and numerical results for eigenvalues of the PDM Schrödinger equation. The results show that the PDM and constant mass Schrödinger equations are not isospectral. The calculations are carried out with the aid of a computer system of symbolic or numerical calculation by constructing a simple algorithm.
Considering the position-dependent effective mass in the study of quantum mechanical systems, a wide range of solvable potentials has been obtained. These potentials are obtained by applying canonical transformations to the Schrödinger equation. In this method, the internal functions introduced by Levai for solvable potentials with constant mass have been used, and the eigenfunctions and eigenvalues have been fully obtained. The eigenfunction of these solvable potentials can be obtained based on specific known functions (Jacobian, generalized Laguerre, and Hermit polynomials). Some of these potentials are Scarf-II, Pöschl-Teller, Rosen-Mörse-II and Eckart, whose applications have been described in physical systems. Finally, all the results are placed in a table and the figures of the desired functions is drawn for specific values
We study the generalized harmonic oscillator that has both the position-dependent mass and the potential depending on the form of mass function in a more general framework. The explicit expressions of the eigenvalue and eigenfunction for such a system are given, they have the same forms as those for the usual harmonic oscillator with constant mass. The coherent state and its properties for the system with PDM are also discussed. We give the corresponding effective potentials for several mass functions, the systems with such potentials are isospectral to the usual harmonic oscillator.