# The Wigner function of a semiconfined harmonic oscillator model with a position-dependent effective mass

## TL;DR

This paper introduces an exact phase-space method to compute the Wigner function for a semiconfined quantum harmonic oscillator with a position-dependent mass, providing analytical solutions and exploring effects of external fields.

## Contribution

It presents a novel analytical approach to derive the Wigner function for a semiconfined oscillator with variable mass, addressing divergence issues in the quantum distribution calculation.

## Key findings

- Exact Wigner distribution functions expressed via Bessel and Laguerre functions.
- Analytical solutions for stationary states with and without external fields.
- Discussion of special cases and limiting behaviors.

## Abstract

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.