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\begin{document}

\section{Problem Set 1}

\begin{enumerate}
\item
Compute the probability flux for a plane wave $\psi = \exp [i(px -
Et)/ \hbar ]$.  Does this result make sense?
\item
We assume a potential energy well of the form
\begin{displaymath}
	U = \left\{ \begin{array}{ll}
		0 , & |x| < a/2 \\
		U_0 , & |x| > a/2
	\end{array} \right.
\end{displaymath}
for a particle of mass $m$.  The wavefunctions are either symmetric
or antisymmetric about $x = 0$ and bound state energies can range over
$0 < E < U_0$.
\begin{enumerate}
\item
Show that symmetric solutions must satisfy the condition
\begin{displaymath}
	\tan (\kappa \epsilon^{1/2} ) - (1/ \epsilon - 1)^{1/2} = 0 ,
\end{displaymath}
where $\epsilon = E/U_0$ and $\kappa = a(2mU_0 )^{1/2} / 2 \hbar$.
\item
Tabulate the values of $\epsilon$ that satisfy the above equation
for $\kappa = 0.1,0.3,1,3,10$.  Use RLaB with the rfile {\bf symmwell.r},
available in the demonstrations.
\item
Find the corresponding condition for antisymmetric solutions and tabulate
values of $\epsilon$ for the above values of $\kappa$.
\end{enumerate}
\item
Modify {\bf shoot\_demo.r} to find the allowed energies and wavefunctions
for the potential $U = \kappa [1 - \exp (-x^2 )]/2$, with $\kappa = 1,3$.
\end{enumerate}

\end{document}
