the degeneracy associated with the eigenstates. The dilemma is solved by taking an appropriate linear combination of the degenerate eigenstates (which is still an eigenstate), and these can be chosen to result in a state that differs from one in the other gauge by a simple phase factor, as illustrated in Refs. \[\] and \[\]. We will encounter this dilemma once more in a situation where the degeneracy has (nominally) been lifted, and outline a resolution by means different than that used in Refs. \[\] and \[\].

![Contour plots of the free space probability density vs. $x$ and $y$, in length units of $\ell_B$, for (a) the Symmetric gauge centred about the origin, with $\ell=5$ and (b) the Landau gauge, with $k_y\ell_B=5$. The probability densities in each gauge have strikingly different spatial structure. It is clear that these eigenstates are not related by just a simple phase factor.[]{data-label="fig: fs symm vs landau"}](free-space-landau-symm.pdf)

[Circular Confinement: Symmetric Gauge]{} {#circ}
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We will revisit the issue of degeneracy in different gauges in Section \[square\] when we discuss an electron confined to a square dot. First, however, partly to review results due to Lent,[@lent91] and partly to introduce some technical aspects and more general 