Dirac equation with a magnetic field in 3D non-commutative phase space

TL;DR

This paper solves the Dirac equation for a spin-1/2 particle in a magnetic field within a 3D non-commutative phase space, revealing how motion along the field affects the effective magnetic field and particle dynamics.

Contribution

It provides a novel solution to the Dirac equation in 3D non-commutative phase space with off-plane motion, analyzing the effects on magnetic field and particle trajectories.

Findings

Motion along the magnetic field increases the effective magnetic field.

Classical limit shows conserved momentum along the effective magnetic field.

Velocity operator definitions influence the complexity of particle motion.

Abstract

For a spin-1/2 particle moving in a background magnetic field in noncommutative phase space, Dirac equation is solved when the particle is allowed to move off the plane that the magnetic field is perpendicular to. It is shown that the motion of the charged particle along the magnetic field has the effect to increase the magnetic field. In the classical limit, matrix elements of the velocity operator related to the probability give a clear physical picture: Along an effective magnetic field the mechanical momentum is conserved and the motion perpendicular to the effective magnetic field follows a round orbit. If using the velocity operator defined by the coordinate operators, the motion becomes complicated.