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\sect{The spreading of a free particle}
Given a free classic particle ${H=\frac{p^{2}}{2m}}$, that has been
prepared in time ${t=0}$ in a state represented by the probability
function
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\[\rho_{t=0}\left(X,P\right)\propto
exp\left(-a\left(X-X_{0}\right)^{2}-b\left(p-p_{1}\right)^{2}\right)\]
\begin {itemize}
\item[(a)]
Normalize ${\rho_{t=0}\left(X,P\right)}$.
\item[(b)]
Calculate ${\langle X\rangle,\, \langle P\rangle,\,
\sigma_{X},\, \sigma_{P},E}$
\item[(c)]
Express the random variables ${\hat{X}_{t}, \hat{P}_{t}}$ with
${\hat{X}_{t=0}, \hat{P}_{t=0}}$
\item[(d)]
Express ${\rho_{t}\left(X,P\right)}$ with
${\rho_{t=0}\left(X,P\right)}$. (Hint: 'variables replacement').
\item[(e)]
Mention two ways to calculate the sizes appeared in paragraph
b in time ${t}$. use the simple one to express
${\sigma_{x}\left(t\right), \sigma_{p}\left(t\right)}$ with
${\sigma_{x}\left(t=0\right), \sigma_{p}\left(t=0\right)}$ (that
you've calculated in b).
\end {itemize}