Free particle spectrum on a circle ($E_n={\hslash }^2{n}^2/(2m{R}^2)$)

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 ${\varphi }_n=e^{inx/R}/\sqrt{2\pi R}$, $n\in \mathbb{Z}$, and energy eigenvalues $E_n={\hslash }^2{n}^2/(2m{R}^2)$. To see this, periodicity forces the wave number to be $n/R$, and applying the momentum operator $-i\hslash {\partial }_x$ gives $\hslash n/R$. The integral of $|{\varphi }_n{|}^2$ over one turn is one. The zero energy eigenvalue is simple; each positive energy eigenvalue has quantum degeneracy two, from $n$ and $-n$.