Diagonal thermal kernel of a quantum harmonic oscillator ($K_{\beta }(x,x)=\sqrt{m\omega /(2\pi \hslash \sinh z)}{e}^{-(m\omega /\hslash )x^2\tanh (z/2)},z=\beta \hslash \omega$)

The diagonal matrix element $K_{\beta }(x,x)=⟨x|e^{-\beta H}|x⟩$ combines the equal-endpoint Euclidean oscillator path with its endpoint-independent Dirichlet oscillator fluctuation determinant. It is not itself a normalized probability density: its integral is the canonical partition function $[2\sinh (z/2){]}^{-1}$. The normalized position probability density isAt high temperature, the variance approaches $1/(\beta m{\omega }^2)$ as in the classical canonical ensemble. At zero temperature, it approaches $\hslash /(2m\omega )$, the ground state variance. Dividing $K_{\beta }(x,x)$ by $K_{\beta }(0,0)$ cancels its entire fluctuation prefactor.