= Canonical oscillator coordinate variance
{title2=$\langle x^2\rangle_T=\hbar\coth(\beta\hbar\omega/2)/(2m\omega)$}
A canonical <quantum harmonic oscillator> has geometric number probabilities and mean occupation $(e^{\beta\hbar\omega}-1)^{-1}$. The position operator then gives the displayed variance. It interpolates between zero-point fluctuations $\hbar/(2m\omega)$ and the classical variance $k_BT/(m\omega^2)$. The kinetic and potential <energies> are equal at every <temperature>, although classical equipartition describes their values only at high <temperature>.
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