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\sect{FDT for harmonic oscillator}
A particle of mass $\mass$ is described by its position $x$ and velocity $v$.
It is bounded by a harmonic potential of frequency ${\Omega}$,
and experiences a damping with a coefficient $\eta$.
Additionally It is subject to an external force $f(t)$. The system is at temperature $T$.
\begin {itemize}
\item[(a)]
Write the generalized susceptibility that describes
the response of ${x}$ to the driving by $f(t)$.
\item[(b)]
Using the FD relation deduce what are the power spectra of $x$ and of $v$.
\item[(c)]
Write an integral expression for the autocorrelation
function $\langle v(t)v(0)\rangle$.
Find explicit results in various limits,
e.g. for damped particle ($\Omega\rightarrow0$).
\item[(d)]
Find $\langle x^2 \rangle$ and $\langle v^2 \rangle$ for $\eta\rightarrow0$,
both in the quantum and in the classical case. Verify consistency with the canonical results.
\end {itemize}