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\markboth{Homework \#12, Phys623, Spring 1998, Prof.~Yakovenko}
{Homework \#12, Phys623, Spring 1998, Prof.~Yakovenko}
\begin{document}
\thispagestyle{empty}
\noindent
\begin{minipage}[t]{3.5in}
{\bf Homework \#12} --- Phys623 --- Spring 1998 \\
{\bf Deadline: 5 p.m., Monday, May 4, 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*{Scattering Theory (Chapter 18, see also Chapters XVII and
XVIII from Landau and Lifshitz)}
\end{center}
\bigskip
\begin{enumerate}
\item {\bf [7 points]} Find the scattering length and the total
scattering cross section of very slow particles by the potential
$U(r)=\beta/r^4,\;\beta>0$.
\underline{Directions:} As explained on pp.\ 338 and 344 of
Schwabl, at low energies (small $k$), only the scattering in the
$l=0$ channel should be taken into account. In that limit, the
phase $\delta_0(k)$, which appears in the partial wave matrix
element
\begin{equation}
S_0(k)=\exp(2i\delta_0(k)),
\label{S0}
\end{equation}
is proportional to $k$:
\begin{equation}
\delta_0(k)=-ka,
\label{ka}
\end{equation}
where $a$ is called the {\em scattering length}.
In the context of the current problem, you can use a different,
but equivalent definition of the scattering length given
below. When particles are very slow ($E\rightarrow0$), their
wave function can be written asymptotically as:
\begin{equation}
\psi\approx1-a/r,
\label{1-a/r}
\end{equation}
where $a$ is the scattering length. The first term in Eq.\
(\ref{1-a/r}) represents the plane wave in the limit
$k\rightarrow0$, and the second term describes the scattered
wave with $l=0$. It follows from Eq.\ (\ref{1-a/r}) that the
amplitude of scattering is $$f=-a,$$ and the total cross section
is
$$\sigma=4\pi a^2.$$
Thus, all you need is to solve the Schr\"{o}dinger equation with
the potential $U(r)$ and energy $E=0$, get asymptotic expression
(\ref{1-a/r}) for the wave function, and extract
$a$ {\bf(see Hints)}. \label{slow}
\item \begin{enumerate}
\item {\bf [5 points]} As discussed in Problem
\ref{slow}, the scattering of slow (low energy) particles
is characterized by a scattering length $a$. In the
presence of absorption, the scattering length $a$ is
complex and has a negative imaginary part (see Sec.\
18.6).
Find the behavior of the elastic and the inelastic
scattering cross sections, $\sigma_{\rm el}$ and
$\sigma_{\rm inel}$, in the limit of small energy of the
scattering particle in terms of the complex scattering
length $a$ {\bf(see Hints)}. Which of the two cross
sections dominates in that limit? \label{1/v}
[{\em This is the so-called $1/v$ law due to Hans Bethe.}]
\item {\bf [7 points]} {\em Adapted from Qualifier, Fall 1994, II-3.}
Absorption of particles can be modeled by introducing a
complex potential into the Schr\"{o}dinger
equation. Indeed, if the imaginary part of the potential
is negative, the particle's probability $|\psi(t)|^2$
decreases in time $t$.
Calculate the cross sections of the elastic and inelastic
scattering of slow particles with the wave vector
\begin{equation}
k=\sqrt{2mE}/\hbar \ll 1/R
\label{kR}
\end{equation}
by a complex potential well:
\begin{equation}
U(r)=\left\{\begin{array}{ll}
V_1-iV_2,&\quad r0, \\
0, &\quad r>R,
\end{array} \right.
\end{equation}
under condition that
\begin{equation}
\kappa=\sqrt{2m|V_{1,0}|}/\hbar \ll 1/R.
\label{shallow}
\end{equation}
(The well is shallow) {\bf(see Hints)}. \label{complex}
\end{enumerate}
\item {\em Adapted from Qualifier, January 1998, Fall 1984, Fall 1983,
Spring 1981, II-3.}
A charged particle is scattered elastically on the electric
potential $\varphi(\vec{r})$ produced by an atom, which has $Z$
electrons and $Z$ protons in the nucleus.
\begin{enumerate}
\item {\bf [7 points]} In the Born approximation, find a general
expression for the differential cross section of
scattering in terms of a given charge distribution of
electrons $n(\vec{r})$. The nucleus is treated as a point
charge $Ze$ {\bf(see Hints)}. \label{Coulomb}
[{\em The following quantity
\begin{equation}
F(\vec{q})=\int n(\vec{r}) e^{-i\vec{q}\vec{r}} d^3r,
\label{F}
\end{equation}
is called the \em atomic form-factor.}]
\item {\bf [3 points]} How does the answer to question
\ref{Coulomb} simplify when $qa_0\gg1$, where $a_0$ is the
width of the electron charge distribution in the atom?
\item {\bf [5 points]} Using your solution of Problem
\ref{Coulomb}, find a general expression for the
differential cross section of scattering in the forward
direction ($\theta=0$) in terms of $n(\vec{r})$. {\bf [5
points]}
\item {\bf [5 points]} Using your solution of Problem
\ref{Coulomb}, calculate differential and total cross
sections of scattering on the hydrogen atom in the ground
state. When calculating the latter quantity, assume that
the energy of the scattering particle is sufficiently
high.
\end{enumerate}
\item {\em Adapted from Qualifier, Fall 1996, II-3.}
Consider {\em inelastic} scattering of fast charged particles on
an atom that has $Z$ electrons and $Z$ protons in the
nucleus. The scattering particles and the atom interact via the
Coulomb potential. Neglect recoil of the atom.
\begin{enumerate}
\item {\bf [7 points]} Find a general expression for the
differential cross section of scattering with the
excitation of the atom from the state $|0\rangle$ (say,
ground state) to the state $|n\rangle$, where $n$ denotes
all quantum numbers of the excited state.
Assume that the scattering particles have high velocity
and use an approximation analogous to the Born
approximation, that is use the Fermi Golden Rule to
calculate the scattering rate. [{\em In this context the
approximation was actually developed by Hans Bethe.}]
Make sure that your result agrees with your solution of
Problem \ref{Coulomb} in the case where the final state of
the atom $|n\rangle$ coincides with its initial state
$|0\rangle$, that is the scattering is
elastic. \label{inelastic}
\item {\bf [7 points]} Using your solution of Problem
\ref{inelastic}, calculate differential and total cross
sections of scattering on the hydrogen atom with its
excitation from the 1s state to the 2s state. When
calculating the latter quantity, assume that the velocity
of the scattering particle is sufficiently high.
Compare the total cross section of the 1s$\rightarrow$2s
scattering with the total cross section of the
1s$\rightarrow$1s scattering found in Problem
\ref{Coulomb}.
\end{enumerate}
\item Electron and positron may annihilate and emit two photons. Thus,
the positronium (a hydrogen-like atom that consists of electron
and positron) has a finite lifetime $\tau=1/W_{\rm pos}$, where
$W_{\rm pos}$ is the positronium annihilation rate per unit
time.
Find a relation between the lifetime of the positronium in its
ground state and the cross section $\sigma$ of annihilation of
slow positrons colliding with electrons.
Assume that the radius of interaction, which is responsible for
the annihilation, is very small compared to the size of the
positronium, and the interaction can be treated as a
perturbation. Specific form of this interaction is not
important. The energy of the emitted photons, $2mc^2$, is much
greater than the binding energy of the electron and positron in
the positronium and the kinetic energy of the colliding
electrons and positrons.
Solve the Problem in two steps as follows:
\begin{enumerate}
\item {\bf [3 points]} As a warmup, get an approximate solution
of the problem by dimensional analysis. Take into account
that, according to Bethe's law (Problems \ref{1/v}),
$\sigma\propto 1/v$, where $v$ is the relative velocity of
colliding electrons and positrons. Complete this formula
to the appropriate dimensionality by using the positronium
lifetime $\tau$ and the Bohr radius of positronium $a$.
\item {\bf [7 points]} Now, actually solve the Problem using
Fermi's Golden rule. You need to calculate the ratio of
$\sigma$ to $W_{\rm pos}$. The interaction and the final
states (the two high-energy photons) are the same in both
cases. Only the initial states are different in the two
cases.
\end{enumerate}
\end{enumerate}
\vfill
\section*{\centerline{Hints}}
\begin{description}
\item[\ref{slow}] When solving the Schr\"{o}dinger equation, it is
convenient to change the variable $r$ to the variable $\xi=1/r$.
\item[\ref{1/v}] Substitute Eqs.\ (\ref{S0}) and (\ref{ka}) into Eq.\
(18.38), (18.39), and (18.41), and take the limit where $k$ is
small. My definition (\ref{S0}) of the complex $\delta_0$
includes both Schwabl's real variables $s_0$ and $\delta_0$.
\item[\ref{complex}] Under condition (\ref{shallow}), one can use the
Born approximation to find the (complex) amplitude of the elastic
scattering $f_0$ in the limit of small $k$.
Once the complex amplitude of the elastic scattering $f_0$ is
known, one can recover the complex phase $\delta_0$ from the
equation $S_l=1+2ikf_0$ and Eq.\ (\ref{S0}) and find the cross
sections using the answer to Problem \ref{1/v}.
\item[\ref{Coulomb}] Use the Coulomb law to relate the density of
charge and the electric potential $\varphi(\vec{r})$. Also,
\begin{equation}
\int\frac{e^{-i\vec{q}\vec{r}}}{|\vec{r}-\vec{r}_a|}d^3\vec{r}
=\frac{4\pi}{q^2}e^{-i\vec{q}\vec{r}_a}.
\end{equation}
\end{description}
\end{document}