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\begin{document}

\csection{Problem Set 4}

\begin{enumerate}
\item
If $\phi_i$ are the (column) eigenvectors of a Hermitian operator, $A$,
show that the sum
\begin{displaymath}
	\sum_{i} \phi_i \phi'_i
\end{displaymath}
is the identity matrix.  (Hint: Work in a reference frame in which $A$
is diagonal.)  Also, as an example of this, show {\em numerically}
that this result holds for the eigenvectors of the harmonic oscillator.
Any set of eigenvectors exhibiting this property is said to be
{\em complete}.
\item
Given the potential energy for a particle of mass $m = 1$,
\begin{displaymath}
	U = -(6 + 5x^3 ) \exp (-x^2 ) :
\end{displaymath}
\begin{enumerate}
\item
Compute all the bound state eigenvalues and list them.
\item
Show that these vectors are orthonormal.
\item
Compute the dipole moment matrix elements $\psi'_i x \psi_j$ for
$i = 1$ and $j$ for all the bound states.  How do the results differ
from those of {\em anho\_demo.r}?
\end{enumerate}
\item
Consider the 2-D harmonic oscillator with Hamiltonian
$H = (p_x^2 + p_y^2 + x^2 + y^2 )/2$ and wave functions
$\psi_{ij} (x,y)$.  The indices $i = 1,2,3,...,~j = 1,2,3,...$
indicate the state of the $x$ and $y$ harmonic oscillator components
obtained by separating variables in Cartesian coordinates.
\begin{enumerate}
\item
Compute (numerically or otherwise) the matrix elements
\begin{displaymath}
	X_{ijkl} = \psi'_{ij} x \psi_{kl} =
	\int dx \int dy \psi_{ij}^{*} x \psi_{kl}
\end{displaymath}
\begin{displaymath}
	Y_{ijkl} = \psi'_{ij} y \psi_{kl} =
	\int dx \int dy \psi_{ij}^{*} y \psi_{kl}
\end{displaymath}
for the transitions from the ground state, $k = l = 1$, to the
states $(i,j) = (1,2), (2,1), (2,2)$.  Explain your results, i. e.,
if one or more of these transitions is forbidden, explain why.
(You may find {\em xydipole.r} useful.)
\item
Find the eigenvalues of the Hamiltonian $H = H_0 + 0.1 x^2 y^2$,
where $H_0$ is the 2-D harmonic oscillator Hamiltonian described
above.  Use first order perturbation theory to find the energies
of the states $(i,j) = (1,1),(2,1),(1,2),(3,1),(2,2),(1,3)$.
(Hint: Try modifying {\em xydipole.r} to do the integrals.)
\end{enumerate}
\end{enumerate}

\end{document}
