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Acta Physica Sinica

Analytical solutions of the three-dimensional Schr?dinger equation with an exponentially changing effective mass

Cai Chang-Ying, Zhongzhou Ren, Ju Guo-Xing, (1)南京大学物理系，南京 210008; (2)南京大学物理系，南京 210008;井冈山学院物理系，吉安 343009; (3)南京大学物理系，南京 210008;兰州重离子加速器国家实验室原子核理论中心，兰州 730000

Published Jan 1, 2005• 2 citations
Publisher

Abstract

For an effective mass distribution that is an exponential function of the radial coordinate, the analytical solutions of the three-dimensional Schrdinger equ ation are obtained by using the coordinate transformation method for the referen ce problems with Coulomb potential, Kratzer potential and spherically square pot ential well of infinite depth, respectively. The explicit analytical expressions for the energy eigenvalues and the corresponding eigenfunctions of the three sy stems are presented.

DOI
10.7498/aps.54.2528

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For an effective mass distribution that is an exponential function of the radial coordinate, the analytical solutions of the three-dimensional Schrdinger equ ation are obtained by using the coordinate transformation method for the referen ce problems with Coulomb potential, Kratzer potential and spherically square pot ential well of infinite depth, respectively. The explicit analytical expressions for the energy eigenvalues and the corresponding eigenfunctions of the three sy stems are presented.
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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 method of solving the radial Schrdinger equation is studied by under the tight coupling condition of several positivepower and inversepower potential functions. The precise analytic solutions and the conditions that determine the existence of analytic solution are searched when the potential of the radial Schrdinger equation is V（r）=α1r8＋α2r3+α3r2+β3r-1＋β2r-3＋β1r-4. According to the single valued, bounded and continuous stipulations of wave function in a quantum system, firstly, the asymptotic solution is solved when the radial coordinate r→∞ and r→0; secondly, the asymptotic solutions are combined with the series solutions in the neighborhood of irregular singularities; and then the power series coefficients are compared. A series of analytic solutions of the stationary state wave function and the corresponding energy level structure are deduced by tight coupling between the coefficients of potential functions for the radial Schrdinger equation. And the solutions are discussed and the conclusions are made.