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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/iii/paper-46/1/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 46 / 1 / a / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

For the quantum rotor, the transition kernel is the path integral

K (θ_{f}, θ_{i}; T) = \int_{θ (0) = θ_{i}}^{θ (T) = θ_{f} (mod 2 π)} D θ e^{i S [θ] /ℏ}, S [θ] = \frac{Λ}{2} \int_{0}^{T} \dot{θ}^{2} d t .

(1)

The integral includes all paths on the circle, not just paths whose chosen real lifts have the same endpoint difference. Operator evolution and the stated energy-state normalization give

K (θ_{f}, θ_{i}; T) = ⟨ θ_{f} ∣ e^{- i H T /ℏ} ∣ θ_{i} ⟩ = \frac{1}{2 π} n \in Z \sum e^{in (θ_{f} - θ_{i})} e^{- i ℏ n^{2} T / (2Λ)} .

(2)

The factor

1/ (2 π)

comes from the completeness relation; it makes the zero-time limit the periodic Dirac delta distribution. For real time the oscillatory sum is understood through

T \to T - i 0

, or as a distributional limit.

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