
\sect{Baruch's  D29.}

Consider a damped harmonic oscillator
with mass $M$ whose coordinate $x(t)$ satisfies
\[\ddot{x}+\gamma \dot{x} +\Omega^2 x=A(t)\]
where the power spectrum of $A(t)$, in the quantum case, is
\[\phi_A(\omega)=\frac{\hbar\omega\gamma}{M}\coth
\frac{\hbar\omega}{2k_BT}\,.\]
\begin{itemize}
\item[(a)] Deduce the power spectrum of the velocity
$\phi_v(\omega)$.
\item[(b)] For $\gamma\rightarrow 0$ show that
$\phi_v(\omega)\rightarrow \delta(\omega-\Omega)$ and that
$\langle \half M v^2\rangle = \frac{1}{4} \hbar\Omega
\coth\frac{\hbar\omega}{2k_BT}$. Explain why is this the expected
result.
\item[(c)] Bonus: Evaluate $\langle \half M v^2\rangle$  with
$\gamma \neq 0$.\\
\end{itemize}

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