= Quantum-oscillator energy and heat capacity
{title2=$C_V=k_Bx^2e^x/(e^x-1)^2$}
For one <quantum harmonic oscillator> of <frequency> $\Omega$, $Z=e^{-x/2}/(1-e^{-x})$ with $x=\hbar\Omega/(k_BT)$. Differentiating $\ln Z$ gives $U=\hbar\Omega[1/2+1/(e^x-1)]$ and $C_V=k_Bx^2e^x/(e^x-1)^2$. <Zero-point energy> leaves $U\to\hbar\Omega/2$ as $T\to0$ but contributes no <heat capacity>. Excitation freeze-out makes $C_V$ exponentially small, while high <temperature> gives $U\sim k_BT$ and $C_V\to k_B$.
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