Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 4 / 34C / iv / Solution

For occupation numbers zero or one, the energy and particle number are additive over modes. The grand canonical partition function factorizes:

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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