 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-3/14e/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

The Bose--Einstein and Fermi--Dirac occupation factors are

\frac{1}{e x p (( E - μ ) / ( k _{B} T )) \mp 1} .

In the stated nonrelativistic dilute regime,

E - μ

is dominated by

m c^{2}

, so the exponential is large and either denominator is asymptotic to

exp ((E - μ) / (k_{B} T))

. Integrating the resulting Maxwell--Boltzmann occupation over momentum states gives

n = \frac{4 π g _{s}}{h ^{3}} \int_{0}^{\infty} p^{2} e^{- [E (p) - μ] / (k_{B} T)} d p .

For

E = m c^{2} + p^{2} / (2 m)

, integration by parts or differentiation of the supplied Gaussian integral gives

\int_{0}^{\infty} p^{2} e^{- p^{2} / (2 m k_{B} T)} d p = \frac{π}{4} (2 m k_{B} T)^{3/2} .

Consequently

n = g_{s} (\frac{2 πm k _{B} T}{h ^{2}})^{3/2} e^{(μ - m c^{2}) / (k_{B} T)} .

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