= Free particle spectrum on a circle
{title2=$E_n=\hbar^2n^2/(2mR^2)$}
For a <particle on a circle> with arc length $x\sim x+2\pi R$ and <periodic boundary conditions>, the <Hamiltonian operator> $P^2/(2m)$ has <orthonormal basis> $\phi_n=e^{inx/R}/\sqrt{2\pi R}$, $n\in\mathbb Z$, and <energy eigenvalues> $E_n=\hbar^2n^2/(2mR^2)$. To see this, periodicity forces the <wave number> to be $n/R$, and applying the <momentum operator> $-i\hbar\partial_x$ gives $\hbar n/R$. The integral of $|\phi_n|^2$ over one turn is one. The zero <energy eigenvalue> is simple; each positive <energy eigenvalue> has <quantum degeneracy> two, from $n$ and $-n$.
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