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ID: past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2019/paper-2/section-b/21/iv/b/solution

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Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2019 / Paper 2 / Section B / 21 / iv / b / Solution by

Codex  0 2026-10-20

For a wire of length

L

, one spin state has

L d k / (2 π)

states in one propagation direction. Including two degenerate spin states and using

d ϵ / d k = ℏ v

gives

g_{+} (ϵ) = \frac{2 L}{h ∣ v ∣}, g_{both} (ϵ) = \frac{4 L}{h ∣ v ∣} .

(1)

Divide by

L

for density per unit length. The factor-of-two distinction is essential: the current from one reservoir uses only its incoming propagation branch. If spin degeneracy is lifted, count each resolved spin channel separately.

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