= Solution
A <stationary state> has $\Psi(t,x)=e^{-iEt/\hbar}\psi(x)$. Substitution into the <Schrödinger equation> gives the <energy eigenvalue> equation
$$
-\frac{\hbar^2}{2m}\psi''=E\psi.
$$
The trial <wavefunction> $Ce^{ikx}$ satisfies it with $E=\hbar^2k^2/(2m)$. The <periodic boundary condition> requires $e^{2\pi ikR}=1$, hence $k=n/R$, $n\in\mathbb Z$. The <wavefunction normalization> is $1=2\pi R|C|^2$, so a convenient common <global phase> gives the <orthonormal basis>
$$
\boxed{\phi_n(x)=\frac{e^{inx/R}}{\sqrt{2\pi R}},\qquad E_n=\frac{\hbar^2n^2}{2mR^2},\qquad n\in\mathbb Z.}
$$
Indeed, $\int_0^{2\pi R}\phi_n^*\phi_q\,dx=\delta_{nq}$. This is the <free particle spectrum on a circle>. The <ground state> $\phi_0$ has <energy> zero and is nondegenerate. Each positive <energy level> $\hbar^2n^2/(2mR^2)$, $n=1,2,\ldots$, has <energy eigenspace> $\operatorname{span}\{\phi_n,\phi_{-n}\}$ and <quantum degeneracy> two. Real sine and cosine combinations give another basis of the same <energy eigenspace>.
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