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Annals of Physics

A hybrid approach for quantizing complicated motion of a charged particle in time-varying magnetic field

Salah Menouar, Jeong Ryeol Choi

Published Dec 6, 2014• 13 citations
Publisher
DOI
10.1016/j.aop.2014.11.014

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We study some aspects of the quantum theory of a charged particle moving in a time-independent, unidirectional magnetic field. When the field is uniform, we make a few clarifying remarks on the use of angular momentum eigenstates and momentum eigenstates with the diamagnetism of a free electron gas as an example. When the field is nonuniform but weakly varying, we discuss both perturbative and nonperturbative methods for studying a quantum mechanical system. As an application, we derive the quantized energy levels of a charged particle in a Helmholtz coil, which go over to the usual Landau levels in the limit of a uniform field.
For a spin-1/2 particle moving in a background magnetic field in noncommutative phase space, the 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 of increasing 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.