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%{Phys 623, Introduction to Quantum Mechanics II, Spring 1998, Dr.~Yakovenko}
%{Phys 623, Introduction to Quantum Mechanics II, Spring 1998, Dr.~Yakovenko}
\begin{document}
\begin{center}
{\sl Phys 623, Introduction to Quantum Mechanics II, Spring 1998,
Dr.~Yakovenko}
\bigskip%\bigskip
\large {\bf Midterm Examination} \\
Wednesday, March 11, 1998, 3:00--4:15 p.m.
\end{center}
%\bigskip
\begin{enumerate}
\item {\bf [5 points]} Find an upper bound for the ground state energy
of the one-dimensional attractive delta function potential
$V(x)=-\lambda\delta(x)$ using the variational method with a
Gaussian trial function. Compare with the true ground state
energy.
$$\int_{-\infty}^{\infty}e^{-\alpha x^2}dx = \sqrt{\pi/\alpha}$$
\item {\bf [5 points]} One-dimensional attractive potential
\begin{equation}
V(x)=-\frac{V_0a^4}{(x^2 + a^2)^2}
\label{WKB}
\end{equation}
has some bound states. New bound states appear when the depth of
the well, characterized by the parameter $V_0$, increases. Using
the WKB approximation, find the values of $V_0$ that correspond
to appearance of new bound states in the spectrum of the
problem.
$$\int_{-\infty}^{\infty}\frac{dx}{x^2+a^2} =\pi $$
\vfill \hfill {\em Please turn over}
\newpage
\item Consider a spin-1/2 particle of mass $m$ bound in a
three-dimensional harmonic oscillator of frequency $\omega$. The
particle is subject to a small perturbation described by the
Hamiltonian
\begin{equation}
\hat{H}'=\lambda\, {\bf r}\cdot\mbox{\boldmath$\sigma$\unboldmath},
\label{H'}
\end{equation}
where {\bf r} is the 3D vector coordinate of the particle, and
$\mbox{\boldmath$\sigma$\unboldmath}=(\sigma_x,\sigma_y,\sigma_z)$
are the Pauli matrices acting on the spin of the particle.
\begin{enumerate}
\item {\bf [5 points]} Using perturbation theory, calculate the
change in the ground state energy of the particle to order
$\lambda^2$. Does this correction remove the Kramers
degeneracy of the ground state (of the up and down spin
states)?
\item {\bf [3 points]} Is Hamiltonian (\ref{H'}) invariant under
the parity operation? Under the time reversal? Under
combined parity and time reversal?
Would perturbation (\ref{H'}) remove the Kramers
degeneracy in any order of $\lambda$?
\item {\bf [3 points]} Is Hamiltonian (\ref{H'}) invariant under
rotation of {\bf r}? Under rotation of spin? Under
combined rotation of {\bf r} and spin? Is the total
angular momentum $j$ ($\hat{\bf J}=\hat{\bf S}+\hat{\bf
L}$) a good quantum number?
What are the possible values of $j$ for the first excited
energy level? Describe qualitatively what kind of the
energy splitting of the six states of the first excited
energy level is compatible with the symmetry of the
problem. \label{symmetry}
\item {\bf [2 points]} Does perturbation (\ref{H'}) split the
initially degenerate first excited energy level to order
$\lambda$? \label{lambda}
\item {\bf [5 points]} Taking into account the matrix elements
of perturbation between the first excited and the ground
energy levels, write down the secular equation matrix that
need to be diagonalized to answer question \ref{lambda} to
order $\lambda^2$. Do not attempt to diagonalize the
matrix, but reduce it to two $3\times3$ matrices. Do you
think the eigenvalues of these matrices would be
compatible with the symmetry requirements formulated in
part \ref{symmetry}?
\end{enumerate}
\end{enumerate}
\end{document}