as $$\Psi(r,\phi)=e^{i \ell \phi}\psi(r),
    \label{separable}$$ where $\hbar \ell$ is the eigenvalue of the $L_z$ operator. Using periodic boundary conditions, $\Psi(r,\phi+2\pi)=\Psi(r,\phi)$, requires $\ell$ to be an integer, and the eigenvalues of $L_z$ are thus quantized, with $\ell$ the ‘azimuthal’ or ‘angular momentum’ quantum number. This results in the radial ordinary differential equation, $$\left[-\frac{\hbar^2}{2m}\left(\frac{d^2}{dr^2}+\frac{1}{r}\frac{d}{dr}\right)+\frac{\hbar^2 \ell^2}{2mr^2}+\frac{1}{2}m\left(\frac{\omega_c}{2}\right)^2r^2 +\frac{\hbar \omega_c\ell}{2} \right]\psi(r) = E\psi(r).
      \label{ham_rad}$$ But Eq. (\[ham\_rad\]) is clearly just the 2D radial quantum harmonic oscillator Hamiltonian with frequency ${\omega_c}/{2}$ and with an additional energy shift by the constant term ${\hbar \omega_c\ell}/{2}$; the normalized 2D radial eigenstates are given by:[@jain2007composite] $$\Psi_{n_r,\ell}(r,\phi)=\frac{1}{\ell_B}\sqrt{\frac{n_r!}{\left(n_r+|\ell|\right)!}}\left(\frac{r}{\sqrt{2}\ell_B}\right)^{|\ell|}e^{-\frac{r^2}{4\ell_B^2}}L^{|\ell|}_{n_r}\left[\frac{r^2}{2\ell_B^2}\right]\frac{e^{i\ell\phi}}{\sqrt{2\pi}}
    \label{psi_rad}$$ where $L^{|\ell|}_{n_r}[x]$ are the associated Laguerre polynomials and $\ell_B \equiv \sqrt{{\hbar}/{eB}}$ is the magnetic length. The magnetic length plays an important role in what follows, and represents the characteristic length for the wave function of a charged particle in a magnetic field. For $B \approx 1$ Tesla, $\ell_B \approx 26$ nm. The quantum number $n_r$ is the ‘radial quantum number’ and simply counts the number of nodes that the radial part of the wave function has ($n_r$ is a non-negative integer). The corresponding probability density takes the appearance of concentric rings that are radially localized increasingly outwards from $r=0$ with increasing $|\ell|$. The 