Past exam of the quantum Condensed Matter Field Theory course of the University of Cambridge / 2012 / Paper 1 / 3 / a / Solution

For the untwisted quantum rotor, periodicity on the angle circle requires $\psi (\varphi +2\pi )=\psi (\varphi )$. The normalized energy eigenstates and their angular momentum areTaking the trace in this orthonormal basis therefore givesThe n=0 state is nondegenerate and each nonzero $|n|$ has two opposite-angular-momentum states. No flux or twisted angular boundary condition is present.