= Solution
For one oscillator, set $x=\hbar\Omega/(k_BT)$. Its <quantum-oscillator energy and heat capacity> are
$$
\boxed{U=\hbar\Omega\left(\frac12+\frac1{e^x-1}\right),\qquad
C_V=k_B\frac{x^2e^x}{(e^x-1)^2}.}
$$
At low <temperature>, $U\to\hbar\Omega/2$ and $C_V\sim k_Bx^2e^{-x}\to0$: excitations freeze out but <zero-point energy> remains. At high <temperature>, $U\sim k_BT$ and $C_V\to k_B$, recovering <equipartition>. The zero-point term does not contribute to <heat capacity>.
\Image[/past-exam-of-the-thermal-and-statistical-physics-course-of-the-university-of-cambridge/2014/paper-1-tsp2014-oscillator.png]
{title=Original quantum-oscillator energy and heat-capacity curves, with zero-point energy, low-temperature freeze-out and classical high-temperature limits}
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