 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-3/35c/a/solution

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

In two dimensions, each allowed wavevector occupies area

(2 π)^{2} / A

k

-space. Including the two spin states, the number of states in an annulus

(k, k + d k)

d N = 2 \frac{A}{( 2 π ) ^{2}} 2 πk d k .

Since

E = \frac{ℏ ^{2} k ^{2}}{2 m}, k d k = \frac{m}{ℏ ^{2}} d E,

g (E) = \frac{d N}{d E} = \frac{A m}{π ℏ ^{2}}

for

E \geq 0

f (E) = \frac{1}{e ^{β (E - μ)} + 1}, β = \frac{1}{k _{B} T} .

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