Thermal partition function of a quantum harmonic oscillator ($Z(\beta ,\omega )=[2\sinh (\beta \omega /2){]}^{-1}$)

For a quantum harmonic oscillator with $\hslash =1$, $\omega >0$ and inverse temperature $\beta >0$, the canonical partition function is ${\sum }_{r\ge 0}{e}^{-\beta \omega (r+1/2)}$. The geometric series gives the displayed formula. Keeping the zero-point energy is essential to its prefactor.

For one quantum harmonic oscillator of frequency $\Omega$, $Z=e^{-x/2}/(1-e^{-x})$ with $x=\hslash \Omega /(k_{B}T)$. Differentiating $\ln Z$ gives $U=\hslash \Omega [1/2+1/(e^x-1)]$ and $C_V=k_B{x}^2{e}^x/(e^x-1{)}^2$. Zero-point energy leaves $U\to \hslash \Omega /2$ as $T\to 0$ but contributes no heat capacity. Excitation freeze-out makes $C_V$ exponentially small, while high temperature gives $U\sim k_{B}T$ and $C_V\to k_B$.