For a quantum mechanical particle moving freely on a circle of length $2 \pi$, the wavefunction $\psi(t, x)$ satisfies the Schrödinger equation

$$i \hbar \frac{\partial \psi}{\partial t}=-\frac{\hbar^{2}}{2 m} \frac{\partial^{2} \psi}{\partial x^{2}}$$

on the interval $0 \leqslant x \leqslant 2 \pi$, and also the periodicity conditions $\psi(t, 2 \pi)=\psi(t, 0)$, and $\frac{\partial \psi}{\partial x}(t, 2 \pi)=\frac{\partial \psi}{\partial x}(t, 0)$. Find the allowed energy levels of the particle, and their degeneracies.

The current is defined as

$$j=\frac{i \hbar}{2 m}\left(\psi{\frac{\partial \psi^{*}}{\partial x}}^{*}-\psi^{*} \frac{\partial \psi}{\partial x}\right)$$

where $\psi$ is a normalized state. Write down the general normalized state of the particle when it has energy $2 \hbar^{2} / m$, and show that in any such state the current $j$ is independent of $x$ and $t$. Find a state with this energy for which the current has its maximum positive value, and find a state with this energy for which the current vanishes.