 Solution

Set $q=e^{-\beta \hslash \omega }$. The oscillator populations are $(1-q)q^n$, and differentiating the geometric series gives $\bar{n}=q/(1-q)$. The canonical oscillator coordinate variance follows from $2\bar{n}+1=(1+q)/(1-q)=\coth (\beta \hslash \omega /2)$, andIt approaches $\hslash /(2m\omega )$ at zero temperature and $k_{B}T/(m{\omega }^2)$ at high temperature, agreeing respectively with ground-state fluctuations and classical equipartition.