= Diagonal thermal kernel of a quantum harmonic oscillator
{title2=$K_\beta(x,x)=\sqrt{m\omega/(2\pi\hbar\sinh z)}e^{-(m\omega/\hbar)x^2\tanh(z/2)},\quad z=\beta\hbar\omega$}
The diagonal matrix element $K_\beta(x,x)=\langle x|e^{-\beta H}|x\rangle$ 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> is
$$
p_\beta(x)=\sqrt{\frac{m\omega\tanh(z/2)}{\pi\hbar}}
\exp\!\left[-\frac{m\omega}{\hbar}x^2\tanh(z/2)\right],
\qquad
\langle x^2\rangle=\frac{\hbar}{2m\omega}\coth(z/2).
$$
At high <temperature>, the <variance> approaches $1/(\beta m\omega^2)$ as in the classical <canonical ensemble>. At zero <temperature>, it approaches $\hbar/(2m\omega)$, the <ground state> <variance>. Dividing $K_\beta(x,x)$ by $K_\beta(0,0)$ cancels its entire fluctuation prefactor.
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