On the Three-Dimensional Pauli Equation in Noncommutative Phase-Space

TL;DR

This paper derives the three-dimensional noncommutative Pauli equation for spin-1/2 particles in electromagnetic fields, analyzes the effects of noncommutativity on current and magnetization, and computes related thermodynamic properties.

Contribution

It introduces a novel derivation of the noncommutative Pauli equation in three dimensions and explores its implications on current density and magnetic properties.

Findings

Noncommutativity affects the order of the magnetic field in the equations.

The noncommutative effects influence the magnetization and magnetic susceptibility.

The classical semi-classical partition function was successfully derived for noncommutative systems.

Abstract

In this paper, we obtained the three-dimensional Pauli equation for a spin-1/2 particle in the presence of an electromagnetic field in noncommutative phase-space, as well the corresponding deformed continuity equation, where the cases of a constant and non-constant magnetic field are considered. Due to the absence of the current magnetization term in the deformed continuity equation as expected, we had to extract it from the noncommutative Pauli equation itself without modifying the continuity equation. It is shown that the non-constant magnetic field lifts the order of the noncommutativity parameter in both the Pauli equation and the corresponding continuity equation. However, we successfully examined the effect of the noncommutativity on the current density and the magnetization current. By using a classical treatment, we derived the semi-classical noncommutative partition function of the three-dimensional Pauli system of the one-particle and N-particle systems. Then, we employed it for calculating the corresponding Helmholtz free energy followed by the magnetization and the magnetic susceptibility of electrons in both commutative and noncommutative phase-spaces. Knowing that with both the three-dimensional Bopp-Shift transformation and the Moyal-Weyl product, we introduced the phase-space noncommutativity in the problems in question.