 Solution

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

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

p^{2} / (2 m) + V (q)

⟨ q_{f} ∣ e^{- i H t /ℏ} ∣ q_{i} ⟩ = \int_{q (0) = q_{i}}^{q (t) = q_{f}} D q exp [\frac{i}{ℏ} \int_{0}^{t} (\frac{m}{2} \overset{q}{˙}^{2} - V (q)) d t^{'}],

with its measure defined by time slicing. Wick rotation gives the Euclidean path integral and the thermal trace.

(0, L)

ψ_{n} (q) = \frac{2}{L} sin \frac{nπ q}{L}, E_{n} = \frac{ℏ ^{2} π ^{2} n ^{2}}{2 m L ^{2}}, n = 1, 2, \dots .

There is no

n = 0

state: the corresponding sine is identically zero. Taking the trace in this orthonormal basis gives the canonical partition function

Z (β) = n = 1 \sum \infty exp (- \frac{β ℏ ^{2} π ^{2} n ^{2}}{2 m L ^{2}}) .

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