\documentclass[twoside]{article}
\textwidth 6.5in
\oddsidemargin 0in
\evensidemargin 0in
\topmargin 0in
\advance\topmargin by -0.5in
\textheight 9in
\renewcommand{\thefootnote}{\arabic{footnote})}
\newcounter{continue}
\newcounter{item}
\renewcommand{\labelenumi}{\bf\arabic{enumi}.}
\renewcommand{\labelenumii}{\bf(\alph{enumii})}
\def\bfs{\mbox{\boldmath$\sigma$\unboldmath}}
\def\bfa{\mbox{\boldmath$\alpha$\unboldmath}}
\pagestyle{myheadings}
\markboth{Homework \#6, Phys623, Spring 1998, Prof.~Yakovenko}
{Homework \#6, Phys623, Spring 1998, Prof.~Yakovenko}
\begin{document}
\thispagestyle{empty}
\noindent
\begin{minipage}[t]{3.5in}
{\bf Homework \#6} --- Phys623 --- Spring 1998 \\
{\bf Deadline: 5 p.m., Monday, March 16, 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!}
\centerline{Equation numbers with the period, like (3.25), refer to the
equations of Schwabl.}
\centerline{Equation numbers without period, like (5), refer to
the equations of this homework.}
\begin{center}
\section*{The Zeeman Effect and the Stark Effect (Chapter 14)}
\end{center}
\noindent
\S76 ``An atom in an electric field'' and \S113 ``An atom in a
magnetic field'' from the book by Landau and Lifshitz are recommended.
\bigskip
\begin{enumerate}
\item {\bf [5 points]} Calculate the Zeeman effect for the singlet
($F=0$) and the triplet ($F=1$) hyperfine-split states of the
hydrogen atom ground state (see Section 12.4.2). Assume that a
magnetic field is sufficiently small, so that the Zeeman energy
splitting is small compared to the hyperfine energy
splitting. Because the magnetic moment of proton is much smaller
that the magnetic moment of electron, take into account only the
interaction of magnetic field with the electron spin and neglect
the interaction of magnetic field with the proton spin.
\item {\bf [5 points]} Calculate the Zeeman effect for positronium in
the case where the Zeeman energy splitting is much greater than
the fine energy splitting. Argue that the orbital Zeeman effect
vanishes for positronium to the first order in the magnetic
field, so only the spin Zeeman effect need to be considered.
\item {\em Adapted from Qualifier, Spring 1991, II-2.}
\begin{enumerate}
\item {\bf [5 points]} Using perturbation theory in the lowest
non-vanishing order in $B$, calculate the change of the
ground state energy of the He atom in a magnetic field
$B$. Use the variational wave function found in Section
13.2.3. Is this change of the energy positive or negative?
(Chapter 7.2 may be useful.) \label{giamag}
From this energy shift, find magnetic susceptibility of
the He atom in the ground state. Is the susceptibility
paramagnetic or diamagnetic?
\item {\bf [5 points]} Estimate magnetic susceptibility of the
He atom in the $^3P_0$ state. Compare result with the
magnetic susceptibility of the ground state found in
Problem \ref{giamag}.
\end{enumerate}
\item \begin{enumerate}
\item {\bf [2 points]} From Eq.\ (14.29) obtain a formal
expression for the polarizability $\alpha$ of the hydrogen
atom in the ground state in terms of a sum over the states
$n=2,3,\ldots$
\item {\bf [5 points]} Find a lower bound on the polarizability
by keeping only the term with $n=2$ in the sum. Express
your answer as $C_la^3<\alpha$, where $C_l$ is a numerical
constant that you need to calculate. (Notice that
polarizability, as well as magnetic susceptibility, has
the dimensionality of $length^3$ in the CGS
system. Explain this!)
\item {\bf [7 points]} Find an upper bound on the polarizability
by replacing $E_n$ by $E_2$ in the denominator and taking
the sum exactly. (It may be useful to treat the sum as a
sum over all intermediate states except $n=1$.) Express
your answer as $\alpha