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 Incoming links: Quantum-oscillator energy and heat capacity

Past exam of the thermal and statistical physics course of the University of Cambridge / 2014 / Paper 1 / 1 / a / Solution 2026-10-20

For one oscillator, set

x = ℏΩ/ (k_{B} T)

. Its quantum-oscillator energy and heat capacity are

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

At low temperature,

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.

Figure 1.
Original quantum-oscillator energy and heat-capacity curves, with zero-point energy, low-temperature freeze-out and classical high-temperature limits
. Details.

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