 Solution

Single-valued rotor wavefunctions are periodic in $\varphi$. The normalized eigenfunctions are $⟨\varphi |n⟩=e^{in\varphi }/\sqrt{2\pi }$ with integer $n$, angular momentum $\hslash n$ and energy $E_n={\hslash }^2{n}^2/(2I)$. The quantum rotor consequently hasThe nonzero levels have the $n\leftrightarrow -n$ degeneracy; the ground state is $n=0$. No twisted boundary condition or applied flux has been introduced.