 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-81/3/c/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

K_{0} (x; β) = (\frac{m}{2 π ℏ ^{2} β})^{1/2} exp (- \frac{m x ^{2}}{2 ℏ ^{2} β}) .

The two image endpoints have displacements

2 r L

and

2 r L - 2 q

from the starting point. Inserting their kernels into the thermal trace of an interval image kernel gives

Z = (\frac{m}{2 π ℏ ^{2} β})^{1/2} \int_{0}^{L} d q r \in Z \sum [e^{- 2 m (r L)^{2} / (ℏ^{2} β)} - e^{- 2 m (r L - q)^{2} / (ℏ^{2} β)}] .

The prefactor comes from the normalization of the free-particle propagator; it cannot be dropped from a thermal trace.

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