 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-3/35a/a/solution

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

For an ideal gas of identical bosons, the Bose-Einstein distribution gives the mean occupation of a discrete one-particle state

r

n_{r} = \frac{1}{e ^{β (E_{r} - μ)} - 1}, β = \frac{1}{k _{B} T},

where

μ

is the chemical potential. If the spectrum is sufficiently dense to use a density of states

g (E)

, the discrete sum becomes

N = \int_{0}^{\infty} \frac{g ( E )}{e ^{β (E - μ)} - 1} d E, E_{tot} = \int_{0}^{\infty} \frac{E g ( E )}{e ^{β (E - μ)} - 1} d E .

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