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\begin{document}

\csection{Ph 340 -- Problem Set 10 -- Spring 1997}

\begin{enumerate}
\item
Spontaneous symmetry breaking.  The RLaB program {\it adwell\_demo.r}
solves the scaled Schr\"{o}dinger equation for the eigenvalues and
eigenvectors of the problem with two separated potential wells.  The
potential takes the form
\begin{displaymath}
	U(x) = -6 \{ \exp [-(x - s/2)^2 ] /a + a \exp [-(x + s/2)^2 ] \}
\end{displaymath}
where $s$ and $a$ are constants.  The parameter $s$ defines the
spacing of the two wells and $a$ is an ``asymmetry parameter'' --- if
$a = 1$ the potential is symmetric in $x$ and the solutions must be
eigenfunctions of the parity operator.  If $a \ne 1$, this restriction
does not apply.
\begin{enumerate}
\item
For $s = 2, 4, 6$ determine how different from unity $a$ must be to
result in significant asymmetry in the ground state and first excited
state eigenfunctions.
\item
Compare the energy difference between the ground state and first
excited state in each of the above cases to the difference in depths
between the two potential wells in this problem.
\item
Qualitatively, how does $<x>$ change for the lowest two eigenstates as
$a$ is increased from unity?
\end{enumerate}
\end{enumerate}

\end{document}
