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Journal of Mathematical Physics

Generalized Laguerre polynomials with position-dependent effective mass visualized via Wigner’s distribution functions

Othmane Cherroud, Sid-Ahmed Yahiaoui, M. Bentaïba

Published Jun 1, 2017• 17 citations
Publisher

Abstract

We construct, analytically and numerically, the Wigner distribution functions for the exact solutions of the position-dependent effective mass Schrödinger equation for two cases belonging to the generalized Laguerre polynomials. Using a suitable quantum canonical transformation, expectation values of position and momentum operators are obtained analytically in order to verify the universality of Heisenberg’s uncertainty principle.

DOI
10.1063/1.4984310

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We construct, analytically and numerically, the Wigner distribution functions for the exact solutions of the position-dependent effective mass Schrödinger equation for two cases belonging to the generalized Laguerre polynomials. Using a suitable quantum canonical transformation, expectation values of position and momentum operators are obtained analytically in order to verify the universality of Heisenberg’s uncertainty principle.
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.
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.
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 propose a phase-space representation concept in terms of the Wigner function for a quantum harmonic oscillator model that exhibits the semiconfinement effect through its mass varying with the position. The new method is used to compute the Wigner distribution function exactly for such a semiconfinement quantum system. This method suppresses the divergence of the integrand in the definition of the quantum distribution function and leads to the computation of its analytical expressions for the stationary states of the semiconfined oscillator model. For this quantum system, both the presence and absence of the applied external homogenous field are studied. Obtained exact expressions of the Wigner distribution function are expressed through the Bessel function of the first kind and Laguerre polynomials. Furthermore, some of the special cases and limits are discussed in detail.