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\markboth{Homework \#1, Phys623, Spring 1998, Prof.~Yakovenko}
{Homework \#1, Phys623, Spring 1998, Prof.~Yakovenko}
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\noindent
\begin{minipage}[t]{3.5in}
{\bf Homework \#1} --- Phys623 --- Spring 1998 \\
{\bf Deadline: Friday, February 6, 1998.} \\
Return homework in class, by e-mail, or \\
put in the box on the door of Phys 2314.
\end{minipage}
\hfill
\begin{minipage}[t]{2.9in}
\raggedleft
Victor Yakovenko, Assistant Professor \\
Office: Physics 2314 \\
Phone: (301)--405--6151 \\
E-mail: yakovenk@physics.umd.edu
\end{minipage}
\medskip
\centerline{\bf Do not forget to write your name and the homework
number!}
\begin{center}
\section*{Time-Independent Perturbation Theory (Chapters 11.1 and 11.4)}
\end{center}
\begin{enumerate}
\item {\bf [3 points]} Schwabl's Problem 11.4.
\item {\bf [5 points]} Schwabl's Problem 11.1.
\item {\bf [5 points]} Schwabl's Problem 11.3.
\item {\bf [3 points each part]} Schwabl's Problem 11.9. \label{secular}
As a specific example of the situation discussed in this
problem, consider the Hamiltonian of a three state system
\begin{equation}
\hat{H}=\hat{H}_0+\lambda\hat{H}_1
\end{equation}
with
\begin{equation}
\hat{H}_0=\left(\begin{array}{ccc}
0&0&0\\
0&0&0\\
0&0&1
\end{array} \right),\qquad
\hat{H}_1=\left(\begin{array}{ccc}
0&0&1\\
0&0&1\\
1&1&0
\end{array} \right).
\end{equation}
\begin{enumerate}
\stepcounter{enumii}
\item Find the exact eigenvalues and eigenvectors of $\hat{H}_0$ and
of $\hat{H}=\hat{H}_0+\lambda\hat{H}_1$. Find small
$\lambda$-approximations for the exact eigenvalues and
eigenvectors of $\hat{H}=\hat{H}_0+\lambda\hat{H}_1$ in the
lowest nonvanishing order of $\lambda$. \label{exact}
\item For $\hat{H}=\hat{H}_0+\lambda\hat{H}_1$, write and solve the
modified secular equation derived in Schwabl's Problem
11.9. Compare the answer with the result obtained in the
previous Problem \ref{exact}.
\end{enumerate}
\item {\em Adapted from Qualifier, January 1997, II-1, August 1995,
II-2, and August 1986, II-1.}
A diatomic molecule with moment of inertia $I$ is constrained to
rotate freely within the $xy$-plane with angular momentum $L=m\hbar$,
where $m$ is an integer. The molecule has a permanent electric dipole
moment {\bf d}, which is parallel parallel to its axis and whose
amplitude $d$ is independent of the rotational motion or the external
conditions. A weak uniform electric field $\cal E$ is applied along
the $x$-axis.
\begin{enumerate}
\item {\bf [3 points]}
\begin{enumerate}
\item Write down the unperturbed (${\cal E}=0$) wave functions and
energies.
\item Then, write down the Hamiltonian in the presence of
the electric field (${\cal E}\neq0$).
\item Also write down the full Hamiltonian (${\cal E}\neq0$) as a
matrix with respect to the unperturbed states.
\end{enumerate}
\item {\bf [7 points]} Using perturbation theory up to the
second order in $\cal E$,
\begin{enumerate}
\item Find the shifts and, perhaps, the splittings of the
energy levels of the system. Discuss the difference
between the cases of $m=0$, $m=1$, and $m>1$. [Hint:
You may need to use the results of Problem
\ref{secular}] Discuss qualitatively a possibility
of energy splitting of the levels with $m>1$ in the
higher orders of perturbation theory. \label{shifts}
\item Find the expectation value $\langle d_x\rangle$ of
the $x$-component of the dipole moment and deduce
the electric polarizability $\alpha=\langle
d_x\rangle/\cal E$.
\item Formulate the conditions of applicability of your
calculation.
\end{enumerate}
\item {\bf [5 points]} Consider a classical rotor with moment of
inertia $I$, intrinsic dipole moment $d$, and suppose that
in the absence of an electric field its angular velocity
is $\omega_0$. Using energy conservation, determine its
angular velocity $\omega$ in the presence of a {\em weak}
electric field and then calculate the time-averaged
electric dipole moment $\langle d_x\rangle$. [Hint: note
that $dt=d\theta/\omega$.] Compare the classical result to
the quantum result in the limit of large
$m$. Qualitatively explain the difference in the signs of
the energy shifts found in Problem \ref{shifts} in terms
of the classical picture of the spinning dipole.
\item {\bf [5 points]} Let's consider the quantum
two-dimensional rotator again. Consider now the case of a
strong electric field such that $Id{\cal
E}/\hbar^2\gg1$. Find approximately the wave functions and
the energy levels in the lower part of the energy
spectrum. Formulate the conditions of applicability of
your results.
\end{enumerate}
\item Consider a quantum system which, in a basis truncated to two
levels, can be described by the following matrix Hamiltonian:
\begin{equation}
\hat{H}=\left(\begin{array}{cc}
V_{11}&V_{12} \\
V_{21}&V_{22}
\end{array} \right),
\label{H}
\end{equation}
where $V_{12}=V_{21}^*$ is complex, whereas $V_{11}$ and
$V_{22}$ are real. These matrix elements can be parameterized as
$V_{11}=E_0+U$, $V_{22}=E_0-U$, and $V_{12}=\Delta e^{i\phi}$
with $\Delta>0$.
\begin{enumerate}
\item {\bf [5 points]} Find exactly the eigenvalues and
eigenvectors of $\hat{H}$.
\item {\bf [2 points each part]}
\begin{enumerate}
\item Plot qualitatively the eigenvalues as functions of
$U$ when $U$ changes from $U\ll-\Delta$ to
$U\gg\Delta$.
\item Do the energy levels cross or repulse in the region
$U\approx0$ where they would cross if $\Delta$ were
zero?
\end{enumerate}
\item {\bf [2 points each part]}
\begin{enumerate}
\item Describe the evolution of the eigenvectors as
functions of $U$ when $U$ changes from $U\ll-\Delta$
to $U\gg\Delta$.
\item When $U=0$, what are the eigenvectors and the
probabilities to find the system in the basis states
in which Eq.\ (\ref{H}) is written?
\item When $|U|\gg\Delta$, what are the eigenvectors to
the zeroth order in $\Delta$? Are the eigenvectors
the same for $U\ll-\Delta$ and $U\gg\Delta$? Pay
attention to the phases of the eigenstates.
\item When $|U|\gg\Delta$, find corrections to the first
order in $\Delta$ to the eigenvalues and
eigenvectors.
\item For which values of $U$ the degenerate and
non-degenerate versions the perturbation theory are
appropriate?
\end{enumerate}
\end{enumerate}
\item {\em Qualifier, August 1987, II-1.}
A free particle of mass $m$, wave number $k$, momentum $\hbar
k$, and kinetic energy
$$\epsilon(k)=\hbar^2k^2/2m$$
moves one-dimensionally along the $x$-axis under the influence
of the weak periodic potential
\begin{equation}
U(x)=2V\cos(qx),
\label{V}
\end{equation}
where $V\ll\epsilon(q/2)$.
\begin{enumerate}
\item {\bf [3 points]} While the spectrum of the problem is
actually continuous, for solving the following problems,
it may be convenient to discretize it by imposing the
periodic boundary conditions $\psi(x)=\psi(x+L)$, where a
big distance $L$ will be taken to infinity at the end of
calculations. This results in quantization the wave vector
values: $k_n=2\pi n/L$ (see pages 229--230 of Schwabl for
a more detailed explanation). We may also assume that $L$
is selected so that $q$ also belongs to the set of values
$2\pi n/L$.
Find the matrix elements of the perturbation (\ref{V}) in
the basis of unperturbed energy eigenstates.
\item {\bf [5 points]} The perturbation (\ref{V}) modifies the
energy relation of the particle:
\begin{equation}
E(k)=\epsilon(k)+E^{(2)}(k).
\label{E2}
\end{equation}
Calculate the energy shift $E^{(2)}(k)$ using the
second-order perturbation theory in (\ref{V}). Show that
$E^{(2)}(k)$ diverges near $k=\pm q/2$ and interpret this
result.
\item {\bf [7 points]} Show that $U(x)$ mixes a state of the
wave vector $k=q/2 +\delta k$ close to $q/2$ with the
state of the wave vector $k=-q/2 +\delta k$ close to
$-q/2$. For a small $\delta k$, the unperturbed energies
of these states are approximately equal to
\begin{equation}
\epsilon(k)\approx\epsilon(q/2)\pm v\delta k,
\label{D}
\end{equation}
where $v=d\epsilon/dk$ taken at $k=q/2$.
Using these two states as a truncated basis, represent the
Hamiltonian as a $2\times2$ matrix and find its
eigenvalues $E(k)$. Sketch $E(k)$ versus $k$ and comment
on the appearance of an energy gap and on its magnitude.
\end{enumerate}
\end{enumerate}
\end{document}