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Physica Scripta

Wigner distribution for a class of isospectral position-dependent mass systems

ALVARO DE SOUZA DUTRA, Juliano Antonio de Oliveira

Published Aug 28, 2008• 12 citations
Publisher

Abstract

In this paper, we show how to construct a general class of non-relativistic quantum systems with position-dependent masses, which are isospectral to a given system with constant mass. Then, we use this approach in order to construct the Wigner functions in three examples which are isospectral to the harmonic oscillator. Finally, we discuss possible general behavior of these systems.

DOI
10.1088/0031-8949/78/03/035009

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In this paper, we show how to construct a general class of non-relativistic quantum systems with position-dependent masses, which are isospectral to a given system with constant mass. Then, we use this approach in order to construct the Wigner functions in three examples which are isospectral to the harmonic oscillator. Finally, we discuss possible general behavior of these systems.
In this work we construct a general class of exactly solvable non-relativistic bi-dimensional quantum systems with position-dependent masses (PDM). These systems are isospectral to a given system with constant mass. The case of a charged particle with a PDM interacting with an external magnetic field is included in the present investigation. We apply the approach in order to construct the SU(2) coherent states in some examples which are isospectral to the two-dimensional anisotropic harmonic oscillator, and discuss the impact of the introduction of special non-homogeneous external magnetic fields.
In this work we construct a general class of exactly solvable nonrelativistic bi-dimensional quantum systems with position-dependent masses (PDM). These systems are isospectral to a given system with constant mass. The case of a charged particle with a PDM interacting with an external magnetic field is included in the present investigation. We apply the approach in order to construct the SU(2) coherent states in some examples which are isospectral to the two-dimensional anisotropic harmonic oscillator, and discuss the impact of the introduction of special non-homogeneous external magnetic fields.
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.
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.