= Magnetic flux spectrum of a quantum ring
{title2=$E_n=E_L(n-\alpha)^2$}
For circumference $L$ and <Hamiltonian operator> $(p-a)^2/(2m)$ with a <periodic boundary condition>, the <plane waves> $L^{-1/2}e^{2\pi inx/L}$ have energies $E_n=E_L(n-\alpha)^2$, where $E_L=2\pi^2\hbar^2/(mL^2)$ and $\alpha=aL/(2\pi\hbar)$. <Minimal coupling> gives $aL=q\Phi_C$, where $\Phi_C$ is the <magnetic flux> oriented consistently with increasing position around the ring. The ordered spectrum is periodic in $\alpha$ with period one, corresponding to <magnetic flux quantum> $h/|q|$.
On $|\alpha|\leq1/2$, its lowest two energies are $E_L\alpha^2$ and $E_L(1-|\alpha|)^2$. Their crossings at half-integer $\alpha$ can be opened by a potential that couples the relevant momentum states.
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