spin degree of freedom. Spin will cause a breaking of degeneracy in the presence of a magnetic field, owing to the direct coupling with field through the so-called Zeeman term. We begin in the next section with a textbook review of the free particle, from which the idea of Landau levels first arise. In particular we will explore both the Landau and the Symmetric gauges, and seemingly derive very different results for the wave functions. We further note that confinement within a parabolic trap can also be treated analytically, following Rontani.[@rontani99] Next we will review and expand upon Lent’s treatment of the circular quantum dot,[@lent91] and finally, we will show numerical results for the square quantum dot in both aforementioned gauges. This will hopefully make clear the role of degeneracy, and we explicitly illustrate the equivalence of the results in the two gauge formulations by removing the degeneracy. We will present the results of calculations for simple properties only, such as probability density and current density.

[Brief Review of Landau Quantization in Free Space]{}
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In the presence of a magnetic field $\mathbf{B}$, the canonical momentum $\mathbf{p}$ is shifted by the magnetic vector potential $\mathbf{A}$ to give us a Hamiltonian 