= Solution
{c}
For a <free particle>, the <Hamiltonian operator> is $H=P^2/(2m)$, where $P=-i\hbar\,d/dx$ is the <momentum operator>. The <time-independent Schrödinger equation> is therefore
$$
\boxed{-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2}=E\psi,\qquad \psi(x+2\pi)=\psi(x).}
$$
The full <stationary state> is $\Psi(x,t)=e^{-iEt/\hbar}\psi(x)$. The <periodic boundary condition> identifies $x$ and $x+2\pi$, giving a <particle on a circle> with <wavefunction normalization> over one period.
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