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\begin{document}

\csection{Ph 340 -- Problem Set 3 -- Spring 1997}

\begin{enumerate}
\item
Given
\begin{displaymath}
	\psi (x,t) = \int_{-\infty}^{\infty} \phi (k) \exp [i(kx -
		\omega t)] dk ,
\end{displaymath}
where $\omega = \omega (k)$ is real, evaluate
\begin{displaymath}
	\int_{-\infty}^{\infty} \psi \psi^* dx ,
\end{displaymath}
showing that it equals
\begin{displaymath}
	2 \pi \int_{-\infty}^{\infty} \phi (k) \phi^* (k) dk .
\end{displaymath}
Note that this result is independent of time.  Hint: Use
\begin{displaymath}
	\int_{-\infty}^{\infty} \exp [i(k - k' )x] dx = 2 \pi
		\delta (k - k' ) .
\end{displaymath}
\item
Given
\begin{displaymath}
	\psi (x,t) = \sum_m C_m f_m (x) \exp (-i \omega_m t)
\end{displaymath}
over the interval $[-L/2 , L/2]$, where $\omega_m$ is real and where
\begin{displaymath}
	\int_{-L/2}^{L/2} f_m (x) f_n^* (x) dx = \delta_{mn} ,
\end{displaymath}
Evaluate
\begin{displaymath}
	\int_{-L/2}^{L/2} \psi \psi^* dx ,
\end{displaymath}
showing that the result is independent of time.  Note that
$\delta_{mn}$ is $1$ if $m = n$ and $0$ otherwise.  This is called the
Kronecker delta.  The $f_m (x)$ are called a complete, orthonormal set
of functions over the interval $[-L/2 , L/2]$.  Fourier series is a
special case of this in which the $f$s are complex exponentials.
\end{enumerate}

\end{document}
