 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/ii/paper-4/34c/iv/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Z = \prod_{r} (1 + e^{- (ϵ_{r} - μ) / (k T)}) .

Dividing the product of Boltzmann weights by this product gives

\overline{ρ}_{i} = r \prod \frac{e ^{- n_{r} (ϵ_{r} - μ) / (k T)}}{1 + e ^{- (ϵ_{r} - μ) / (k T)}} .

The marginal mean occupation is therefore

\overline{n}_{r} = \frac{1}{e ^{(ϵ_{r} - μ) / (k T)} + 1} .

This is the Fermi-Dirac distribution of a noninteracting Fermi gas: the exclusion principle allows only zero or one fermion in each single-particle state, including its spin label.

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