Prove that

\[
\left\langle q\left|e^{-\beta \hat{H}}ight| q^{\prime}ightangle=\exp \left[-\beta \hat{H}\left(-i \hbar \frac{\partial}{\partial q}, qight)ight] \delta\left(q-q^{\prime}ight)
\]

where \(\hat{H}(-i \hbar \partial / \partial q, q)\) is the Hamiltonian operator of the system in the \(q\)-representation, which formally operates on the Dirac delta function \(\delta\left(q-q^{\prime}ight)\). Writing the \(\delta\)-function in a suitable form, apply this result to (i) a free particle and (ii) a linear harmonic oscillator.