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arXiv:quant-ph/0601005 · Communications in Theoretical Physics

Localization of s -Wave and Quantum Effective Potential of a Quasi-free Particle with Position-Dependent Mass

Ju Guo-Xing, Xiang Yang, Zhongzhou Ren

Published Nov 1, 2006• 8 citations
arXivHTMLPDF

Abstract

The properties of the s -wave for a quasi-free particle with position-dependent mass (PDM) have been discussed in details. Differed from the system with constant mass in which the localization of the s -wave for the free quantum particle around the origin only occurs in two dimensions, the quasi-free particle with PDM can experience attractive forces in D dimensions except D = 1 when its mass function satisfies some conditions. The effective mass of a particle varying with its position can induce effective interaction, which may be attractive in some cases. The analytical expressions of the eigenfunctions and the corresponding probability densities for the s -waves of the two- and three-dimensional systems with a special PDM are given, and the existences of localization around the origin for these systems are shown.

DOI
10.1088/0253-6102/46/5/010

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The properties of the s -wave for a quasi-free particle with position-dependent mass (PDM) have been discussed in details. Differed from the system with constant mass in which the localization of the s -wave for the free quantum particle around the origin only occurs in two dimensions, the quasi-free particle with PDM can experience attractive forces in D dimensions except D = 1 when its mass function satisfies some conditions. The effective mass of a particle varying with its position can induce effective interaction, which may be attractive in some cases. The analytical expressions of the eigenfunctions and the corresponding probability densities for the s -waves of the two- and three-dimensional systems with a special PDM are given, and the existences of localization around the origin for these systems are shown.
为一个没有伪的粒子的 s 波浪的性质 withposition 依赖的质量(PDM ) 在细节被讨论了。与为在起源附近的免费的量粒子的 s 波浪的本地化仅仅在发生在二种尺寸的经常的质量不同于系统，除了 D = ，有 PDM 的没有伪的粒子能在 Ddimensions 经历吸引人的力量 1 当它的集体功能满足一些条件时。与它的位置变化的一个粒子的有效的团能导致有效相互作用，它可能在一些情况中是吸引人的。为有特殊 PDM 的二维、三维的系统的 s 波浪的特徵函数和相应概率密度的分析表达式被给，并且在为这些系统的起源附近的本地化的存在被显示出。
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.
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.
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.
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.
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.