Canonical oscillator coordinate variance ($⟨x^2{⟩}_T=\hslash \coth (\beta \hslash \omega /2)/(2m\omega )$)

A canonical quantum harmonic oscillator has geometric number probabilities and mean occupation $(e^{\beta \hslash \omega }-1{)}^{-1}$. The position operator then gives the displayed variance. It interpolates between zero-point fluctuations $\hslash /(2m\omega )$ and the classical variance $k_{B}T/(m{\omega }^2)$. The kinetic and potential energies are equal at every temperature, although classical equipartition describes their values only at high temperature.