= Solution
{c}
For a <momentum eigenstate> with <eigenvalue> $p$, the equation $-i\hbar\psi'=p\psi$ has solution $\psi(x)=Ce^{ipx/\hbar}$. The <periodic boundary condition> imposes $e^{2\pi ip/\hbar}=1$, so $p/\hbar\in\mathbb Z$. The <wavefunction normalization> gives $1=2\pi|C|^2$. Choosing a convenient <global phase> yields
$$
\boxed{\psi_n(x)=\frac{e^{inx}}{\sqrt{2\pi}},\qquad p_n=n\hbar,\qquad n\in\mathbb Z.}
$$
These <momentum eigenstates> form an <orthonormal basis>, the <Fourier basis> for the <particle on a circle>:
$$
\langle\psi_n,\psi_k\rangle=\frac1{2\pi}\int_0^{2\pi}e^{i(k-n)x}\,dx=\delta_{nk}.
$$
Here $\delta_{nk}$ is the <Kronecker delta>.
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