Quantum behavior of a charged particle in an axial magnetic field

TL;DR

This paper explores the quantum behavior of a charged particle in magnetic fields, analyzing uniform and non-uniform cases, and deriving energy levels that generalize Landau levels for non-uniform fields.

Contribution

It introduces methods for analyzing quantum systems in non-uniform magnetic fields and derives energy levels in a Helmholtz coil, extending Landau level concepts.

Findings

Clarified use of angular momentum and momentum eigenstates in uniform fields

Developed perturbative and non-perturbative methods for non-uniform fields

Derived quantized energy levels in a Helmholtz coil

Abstract

We study some aspects of the quantum theory of a charged particle moving in a time-independent, uni-directional 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 non-uniform but weakly varying, we discuss both perturbative and non-perturbative 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.