 Solution

ID: past-exam-of-the-thermal-and-statistical-physics-course-of-the-university-of-cambridge/2014/paper-1/1/a/solution

x = ℏΩ/ (k_{B} T)

U = ℏΩ (\frac{1}{2} + \frac{1}{e ^{x} - 1}), C_{V} = k_{B} \frac{x ^{2} e ^{x}}{( e ^{x} - 1 ) ^{2}} .

U \to ℏΩ/2

and

C_{V} \sim k_{B} x^{2} e^{- x} \to 0

: excitations freeze out but zero-point energy remains. At high temperature,

U \sim k_{B} T

and

C_{V} \to k_{B}

, recovering equipartition. The zero-point term does not contribute to heat capacity.

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