Past exam of the thermal and statistical physics course of the University of Cambridge / 2014 / Paper 1 / 1 / a

For one oscillator, set $x=\hslash \Omega /(k_{B}T)$. Its quantum-oscillator energy and heat capacity areAt low temperature, $U\to \hslash \Omega /2$ 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.