Past exam of the quantum theory course of the University of Oxford / 2023 / 1 / b / i / Solution

Using ${\sin }^{2}u=\frac{1}{2}-\frac{1}{4}{e}^{2iu}-\frac{1}{4}{e}^{-2iu}$, the energy eigenstate decomposition isOnly the energy eigenvalues zero and $E_{\star }=2{\hslash }^2/(m{R}^2)$ occur. The last two terms are distinct momentum eigenstates in the same energy eigenspace.