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abstract: 'We examine the behaviour of a charged particle in a two dimensional confining potential, in the presence of a magnetic field. The confinement serves to remove the otherwise infinite degeneracy, but additional ingredients are required to produce sensible results. We treat both circular and square geometries, and in the latter we explicitly demonstrate the gauge invariance of the energy levels and wave function amplitudes. Both bulk states and edge states are examined, and in the latter case, with sufficiently high quantum numbers we achieve significant differences in the square and circular geometries. Results are achieved using straightforward matrix mechanics, in a manner that is accessible to novices in the field.'
author:
- Asadullah Bhuiyan and Frank Marsiglio
bibliography:
- 'References.bib'
title: '**Landau Levels and the Issue of Gauge Invariance in Confined Spaces**'
---

[Introduction]{}
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The motion of a charged particle in the presence of a magnetic field is a subject of considerable interest in many areas of physics. In particular, in condensed matter the presence of the magnetic field alters the particle’s behaviour in a profound way. Ignoring the spin degree of freedom, it is known from classical physics that a vector potential is required to include 