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ID: past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2021/paper-2/section-c/c9/b/solution

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Past exam of the physics B course of the University of Cambridge / 2021 / Paper 2 / Section C / C9 / b / Solution by

Codex  0 2026-10-18

The internal energy is the canonical mean of the energy:

U_{sys} = \sum_{j} ϵ_{j} \frac{g _{j} e ^{- β ϵ_{j}}}{Z} .

(1)

Holding the external mechanical parameters fixed, so that the spectrum and degeneracies do not change when differentiating with respect to

β

, gives

\frac{\partial Z}{\partial β} = - \sum_{j} g_{j} ϵ_{j} e^{- β ϵ_{j}} .

(2)

Comparison yields

U_{sys} = - \frac{1}{Z} \frac{\partial Z}{\partial β} = - \frac{\partial ln Z}{\partial β}, β = \frac{1}{k _{B} T} .

(3)

An explicitly temperature-dependent effective Hamiltonian would require care: differentiating its partition sum would also differentiate the assigned energy levels. The stated result uses the fixed physical spectrum of the canonical system.

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