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Physica Scripta

Wigner function of the position-dependent effective Schrödinger equation

Zidong Chen, Gang Chen

Published Mar 9, 2006• 10 citations
Publisher

Abstract

In this paper, we calculate the Wigner distribution function to reveal the quantum–classical connection for the Schrödinger equation with position-dependent effective mass. On mapping this wave equation into an isospectral one with constant mass, the Wigner distribution function, which can be used to calculate matrix elements and expectation values of operators, can be obtained analytically for a given mass distribution and potential function.

DOI
10.1088/0031-8949/73/4/007

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AIP conference proceedings•Jan 1, 2010•0 citations
In this paper, we calculate the Wigner distribution function to reveal the quantum–classical connection for the Schrödinger equation with position-dependent effective mass. On mapping this wave equation into an isospectral one with constant mass, the Wigner distribution function, which can be used to calculate matrix elements and expectation values of operators, can be obtained analytically for a given mass distribution and potential function.
In this paper, two novel semiclassical methods including the standard and supersymmetric WKB quantization conditions are suggested to discuss the Schrödinger equation with position-dependent effective mass. From a proper coordinate transformation, the formalism of the Schrödinger equation with position-dependent effective mass is mapped into isospectral one with constant mass and therefore for a given mass distribution and physical potential function the bound state energy spectrum can be determined easily by above method associated with a simple integral formula. It is shown that our method can give the analytical results for some exactly-solvable quantum systems.
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.
One of the most important scientific contributions of Professor Marcos Moshinsky has been his study on the harmonic oscillator in quantum theory vis à vis the standard Schrödinger equation with constant mass [1].However, a simple description of the motion of a particle interacting with an external environment such as happen in compositionally graded alloys consist of replacing the mass by the so-called effective mass that is in general variable and dependent on position.Therefore, honoring in memoriam Marcos Moshinsky, in this work we consider the position-dependent mass Schrodinger equations (PDMSE) for the harmonic oscillator potential model as former potential as well as with equi-spaced spectrum solutions, i.e. harmonic oscillator isospectral partners.To that purpose, the point canonical transformation method to convert a general second order differential equation (DE), of Sturm-Liouville type, into a Schrödinger-like standard equation is applied to the PDMSE.In that case, the former potential associated to the PDMSE and the potential involved in the Schrödinger-like standard equation are related through a Riccati-type relationship that includes the equivalent of the Witten superpotential to determine the exactly solvable positions-dependent mass distribution (PDMD) m(x).Even though the proposed approach is exemplified with the harmonic oscillator potential, the procedure is general and can be straightforwardly applied to other DEs.