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

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.