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\sect{Fluctuations of harmonic oscillator}


A particle ${\left(x,p\right)}$  of mass ${\mathsf {m}}$ is bounded
by a harmonic potential of frequency ${\Omega}$, and experiences a
damping with a coefficient eta. It is subject to an external force
${f\left(t\right)}$.


\begin {itemize}

\item[(a)]
Write the generalized susceptibility that describes the response of
${x}$ to the driving by ${f\left(t\right)}$.

\item[(b)]
Using the ${FD}$ relation deduce what is the power spectrum of
the ${x}$ fluctuations.


\item[(c)]
What are the fluctuations of the velocity?

\item[(d)]

Show that in the limit ${\eta-->0}$ the second moments ${\langle
x^{2}\rangle}$ and ${\langle v^{2}\rangle}$ are as expected from the
canonical formalism.

\end {itemize}




