Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 35B / c

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- Solution c

Solution

The number flux through the surface is

J_{z} = \frac{g _{s}}{( 2 π ℏ ) ^{3}} \int_{p_{z} > p_{0}} \frac{p _{z}}{m} \frac{d ^{3} p}{e ^{β (p^{2} /2 m - E_{F})} + 1}, p_{0} = 2 m (E_{F} + V_{0}) .

Throughout this tail the occupation is Boltzmann to leading order. The two transverse Gaussian integrals contribute

2 πm / β

, while

\int_{p_{0}}^{\infty} \frac{p _{z}}{m} e^{- β (p_{z}^{2} /2 m - E_{F})} d p_{z} = \frac{1}{β} e^{- β V_{0}} .

Therefore

J_{z} = \frac{g _{s} m}{4 π ^{2} ℏ ^{3}} (k_{B} T)^{2} e^{- V_{0} / (k_{B} T)} .

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