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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 36B / b / Solution

The classical one-particle phase-space integral is

Z_{1} = \frac{1}{h ^{3}} \int_{R^{3}} d^{3} x \int_{R^{3}} d^{3} p e^{- β (p c + U (x))} .

The momentum integral is

4 π \int_{0}^{\infty} p^{2} e^{- β c p} d p = \frac{8 π}{β ^{3} c ^{3}} .

Hence

Z_{1} = \frac{8 π ( k _{B} T ) ^{3}}{h ^{3} c ^{3}} \int_{R^{3}} e^{- U (x) / (k_{B} T)} d^{3} x = \frac{( k _{B} T ) ^{3}}{π ^{2} ( ℏ c ) ^{3}} \int e^{- U / (k_{B} T)} d^{3} x .

Solved by gpt-5.6-sol high.

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