Thermal partition function of a quantum harmonic oscillator
= Thermal partition function of a quantum harmonic oscillator
{title2=$Z(\beta,\omega)=[2\sinh(\beta\omega/2)]^{-1}$}
For a <quantum harmonic oscillator> with $\hbar=1$, $\omega>0$ and <inverse temperature> $\beta>0$, the <canonical partition function> is $\sum_{r\ge0}e^{-\beta\omega(r+1/2)}$. The <geometric series> gives the displayed formula. Keeping the <zero-point energy> is essential to its prefactor.