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\markboth{Homework \#5, Phys623, Spring 1998, Prof.~Yakovenko}
{Homework \#5, Phys623, Spring 1998, Prof.~Yakovenko}
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{\bf Homework \#5} --- Phys623 --- Spring 1998 \\
{\bf Deadline: 5 p.m., Monday, March 9, 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{enumerate}
\begin{center}
\section*{Helium (Chapter 13.2)}
\end{center}
\item {\bf [5 points]} Schwabl's Problem 13.1.
\item {\em Adapted from Qualifier, January 1993, II-4.}
Consider a Li$^+$ ion: an atom with 2 electrons and the nuclear
charge $Z=3$.
\begin{enumerate}
\item {\bf [3 points]} In the spirit of Sections 13.2.1--13.2.2,
write down the electronic wave functions for the lowest
and the first excited energy levels of this ion in terms
of the hydrogen one-electron wave functions. What are the
degeneracy, the orbital angular momentum $L$ and the total
spin $S$ of each state?
\item {\bf [3 points]} In the first-order perturbation theory,
the ground-state energy of the He atom is --5.5 Ry (Eq.\
(13.27a)). In the same approximation, find the ground-state
energy of the Li$^+$ ion. You don't need to repeat the
whole derivation from the textbook; just rescale the
answers using $Z=3$ instead of $Z=2$. \label{pertLi}
\item {\bf [3 points]} Following Section 13.2.3, find the
ground-state energy of the Li$^+$ ion using the
variational method. Compare result with that of Problem
\ref{pertLi}.
\item {\bf [5 points]} Suppose one of the electrons is replaced
by a muon, an electron-like particle with a much heavier
mass $m_\mu=207\,m_e$. Developing a sensible picture of
the Coulomb charge screening for this atom, find the wave
functions and the energies of the lowest and the first
excited energy levels.
\end{enumerate}
\begin{center}
\section*{The Hartree and Hartree-Fock Approximations (Chapter 13.3)}
\end{center}
\item This problem is not about the Hartree or Hartree-Fock
approximations. Its purpose is rather to show the necessity of
these approximations. Answering the following questions, {\bf
ignore completely the interaction between electrons.}
\begin{enumerate}
\item {\bf [2 points]} As you know, the energy levels of an atom
with a nuclear charge $Z$ go as $Z^2$ (Eq.\
(13.20)). While the Coulomb potential $Ze/r$ is
proportional only to the first power of $Z$, why the
energies are proportional to $Z^2$? \label{Z^2}
\item {\bf [2 points]} For a hydrogen-like atom, let us define
the radius of the $n$-th electron shell, $r_n$, as the
radius where the probability density of the radial wave
function $R_{n,n-1}(r)$ has a maximum. (The wave functions
$R_{n,l}(r)$ with $l