of the form: $$H=\frac{\boldsymbol{\pi}^2}{2m}=\frac{(\mathbf{p}-q\mathbf{A})^2}{2m}
    \label{ham_original}$$ where $q$ is the charge and $m$ is the mass of the particle of interest, and $\boldsymbol{\pi}=\mathbf{p}-q\mathbf{A}$ is the *kinetic* momentum operator. Hereafter we adopt $q = -e$ for the electron, where $e>0$ is the magnitude of the charge of the electron. It is important to note that in the presence of a magnetic field, $\boldsymbol{\pi}$ represents the true momentum of the particle rather than $\mathbf{p}$. In this problem, we consider an electron free to move in a 2-dimensional system in a uniform magnetic field, $\mathbf{B}=B\boldsymbol{\hat{z}}$. For such a magnetic field, there are three common gauge choices, as already mentioned in the introduction. The two Landau gauges are handled very similarly mathematically, and so we will focus only on $\mathbf{A}_1$. However, $\mathbf{A}_1$ and $\mathbf{A}_S$ are handled very differently mathematically and we now discuss each in turn. A number of resources are available that provide considerable detail for this problem. These include textbooks like Refs. \[\], \[\], \[\] and \[\] and online lecture notes by Tong[@qheTong] and Murayama.[@Murayama]

[The Symmetric Gauge]{}
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Substituting $\mathbf{A}_S$ into the Hamiltonian (\[ham\_original\]) leads to $$H=\frac{\mathbf{p}^2}{2m}-i\frac{\hbar\omega_c}{2}(x\partial_y-y\partial_x)+\frac{1}{8}m\omega_c^2(x^2+y^2),
\label{ham_symm1}$$ where $\partial_x \equiv \partial/\partial x$, and similarly for $y$ and later for radial coordinates 