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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 35C / a / Solution

One quantum state occupies volume

(2 π)^{d} / V

in wavevector space. Counting the shell of radius

k

and including spin degeneracy gives

g (E) d E = \frac{g _{s} V S _{d - 1}}{( 2 π ) ^{d}} k^{d - 1} d k .

Using

k = (E / A)^{1/ α} /ℏ

gives the stated power with

B = \frac{g _{s} S _{d - 1}}{α ( 2 π ) ^{d} ℏ ^{d} A ^{d / α}} .

Solved by gpt-5.6-sol high.

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