results to be used later, we present a discussion of results for circular confinement. We will tend to use dimensionless units wherever possible. However, it is important to convert our results to physical values. As we have already seen, the flux quantum plays an important role in these problems; its value in SI units is $h/e \approx 4140$ T (nm)$^2$. We will use a dimensionless measure of flux, $G = BA/(h/e)$, where $A$ is the area of the sample through which a (constant) magnetic field penetrates. For circular geometry, we use $\gamma \equiv B\pi a^2/(h/e)$, where $a$ is the radius. So, for example, applying a $10$ T magnetic field through a sample with radius 100 nm gives $\gamma \approx 76$.

[The Infinite Circular Well]{}
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Let us first briefly review the problem of a particle confined to an infinite circular well. The infinite circular well potential $V(r)$ is simply given by: $$V(r) = \left\{
        \begin{array}{ll}
            0 & \quad r \leq a \\
            \infty & \quad r > a
        \end{array}
    \right. 
    \label{circ_pot}$$ where $a$ is the radius of the well. With this potential, the Hamiltonian in polar coordinates becomes: $$H=-\frac{\hbar^2}{2m}(\partial_r^2+\frac{1}{r}\partial_r)+\frac{L_z^2}{2mr^2}.
    \label{ham_polar}$$ Since $[H,L_z]=0$, we again write the eigenstates $\Psi(r,\phi) $ of 