Generalized Laguerre Polynomials with Position-Dependent Effective Mass Visualized via Wigner's Distribution Functions

TL;DR

This paper analytically and numerically constructs Wigner distribution functions for solutions of the position-dependent mass Schrödinger equation involving generalized Laguerre polynomials, demonstrating the universality of Heisenberg's uncertainty principle.

Contribution

It introduces a method to derive Wigner functions for systems with position-dependent mass using generalized Laguerre polynomials and quantum transformations.

Findings

Wigner functions are successfully constructed for specific cases.

Expectation values confirm the Heisenberg uncertainty principle.

Analytical and numerical methods are combined for validation.

Abstract

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