 Solution

ID: 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

ψ (ϕ + 2 π) = ψ (ϕ)

u_{n} (ϕ) = \frac{e ^{in ϕ}}{2 π}, ℓ_{n} = ℏ n, E_{n} = \frac{ℏ ^{2} n ^{2}}{2 I}, n \in Z .

Taking the trace in this orthonormal basis therefore gives

Z = n \in Z \sum exp [- \frac{β ℏ ^{2} n ^{2}}{2 I}] .

The n=0 state is nondegenerate and each nonzero

∣ n ∣

has two opposite-angular-momentum states. No flux or twisted angular boundary condition is present.

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