Wigner Functions for the Landau Problem in Noncommutative Spaces

TL;DR

This paper derives Wigner functions for the Landau problem in noncommutative spaces, revealing conditions under which these functions are unaffected by noncommutativity parameters, thus extending quantum phase space analysis.

Contribution

It introduces a generalized *-genvalue equation for noncommutative phase space and finds conditions where Wigner functions remain independent of noncommutativity.

Findings

Wigner functions are obtained using solutions of the Schrödinger equation.

Under specific noncommutativity relations, Wigner functions are independent of noncommutativity parameters.

The approach extends phase space analysis to noncommutative geometries.

Abstract

An electron moving on plane in a uniform magnetic field orthogonal to plane is known as the Landau problem. Wigner functions for the Landau problem when the plane is noncommutative are found employing solutions of the Schroedinger equation as well as solving the ordinary *-genvalue equation in terms of an effective Hamiltonian. Then, we let momenta and coordinates of the phase space be noncommutative and introduce a generalized *-genvalue equation. We solve this equation to find the related Wigner functions and show that under an appropriate choice of noncommutativity relations they are independent of noncommutativity parameter.