= Solution
For the <magnetic flux spectrum of a quantum ring>, the <periodic boundary condition> permits the normalized <plane waves> $L^{-1/2}e^{2\pi inx/L}$, with $n\in\mathbb Z$. Their <energy eigenvalues> are
$$
E_n(a)=\frac1{2m}\left(\frac{2\pi\hbar n}{L}-a\right)^2
=E_L(n-\alpha)^2,\qquad
E_L=\frac{2\pi^2\hbar^2}{mL^2},\quad
\alpha=\frac{aL}{2\pi\hbar}.
$$
Shifting $\alpha$ by an integer relabels $n$, so the ordered <energy levels> have period one in $\alpha$, or $2\pi\hbar/L$ in $a$. For $-1/2\leq\alpha\leq1/2$, the lowest two are
$$
\boxed{\epsilon_0=E_L\alpha^2,\qquad
\epsilon_1=E_L(1-|\alpha|)^2,}
$$
extended periodically. The lowest branches cross at every half-integer $\alpha$, with energy $E_L/4$. At integer $\alpha$, the first excited <energy level> has multiplicity two. The corresponding <magnetic flux> period is the electron <magnetic flux quantum> $h/e$.
\Image[/past-exam-of-the-advanced-quantum-physics-course-of-the-university-of-cambridge/2022-q4-ring-spectrum.png]
{title=Two lowest ring energies from free motion to a deep cosine potential}
{description=Original calculations in a <plane wave> basis. From left to right: the free spectrum, a weak cosine potential with $\lambda/E_L=0.18$, and a deep potential with $\lambda/E_L=8$. The middle panel shows an <avoided crossing> at half-integer $\alpha=aL/(2\pi\hbar)$; the deep-potential branches are almost horizontal and well separated. The dashed levels in the final panel are the <parabolic approximation>, whose leading anharmonic errors are visible at this finite depth.}
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