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\begin{document}

\csection{Ph 340 -- Problem Set 2 -- Spring 1997}

\begin{enumerate}
\item
If $\tilde{f} (k) = \exp (- \mu^2 k^2 )$, where $\mu$ is a constant,
we showed that the inverse Fourier transform yields $f(x) = (
\pi^{1/2} / \mu ) \exp (-x^2 / 4 \mu^2 )$.  Show that the forward
Fourier transform recovers the above form for $\tilde{f} (k)$.
\item
If $f(x) = \exp [i \kappa x - (x - a)^2 / b^2 ]$, where $\kappa$,
$a$, and $b$ are constants, find the Fourier transform of $f(x)$.
Note that this function represents a wave packet of central wavenumber
$\kappa$, width $b$, and location $a$.  What role do each of these
constants play in determining the form of $\tilde{f} (k)$?
\item
Evaluate
\begin{equation}
	f(x) = \int_{-\infty}^{\infty} \exp (-x'^2 )
		\frac{\partial}{\partial x'} \delta (x - x' ) dx' .
\end{equation}
Hint: Try integration by parts.
\item
Using the default dispersion relation in {\tt packet\_demo.r},
describe what happens at large time to wave packets with different
initial widths.  (Hint: Use the default values of {\tt dx}, {\tt dt},
and {\tt k0}, but increase {\tt nx} and {\tt nt}.  Then vary {\tt
width}.
\end{enumerate}

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