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\markboth{Homework \#9, Phys623, Spring 1998, Prof.~Yakovenko}
{Homework \#9, Phys623, Spring 1998, Prof.~Yakovenko}
\begin{document}
\thispagestyle{empty}
\noindent
\begin{minipage}[t]{3.5in}
{\bf Homework \#9} --- Phys623 --- Spring 1998 \\
{\bf Deadline: 5 p.m., Monday, April 13, 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*{Interaction with Electromagnetic Field (Chapter 16)}
\end{center}
\bigskip
The absorption and emission of electromagnetic radiation by a quantum
system often can be analyzed in the so-called ``semiclassical
approximation'', in which the electromagnetic field is treated
classically, whereas the electron is treated quantum-mechanically, via
its wave function. In this homework we take this semiclassical point
of view.
\begin{enumerate}
\item {\em Adapted from Qualifier, January 1998, August 1991, II-2.}
A charged particle of mass $m$ moves in a one-dimensional
attractive potential $U(x)=-\lambda\delta(x)$, where $\delta(x)$
is the Dirac delta-function, and $\lambda>0$. Initially, the
particle is in the ground state of the potential with the energy
$E_0<0$. Then, a small ac electric field ${\cal E}(t)={\cal
E}_0\sin(\omega t)$ of the frequency $\omega>|E_0|/\hbar$ is
turned on. The Hamiltonian of the perturbation is
\begin{equation}
V=-2ex{\cal E}_0\sin(\omega t),
\label{V}
\end{equation}
where $e$ is the electron charge. This perturbation may cause a
transition of the particle from the bound state to an unbound
state (``ionization'' of the ``atom'').
\begin{enumerate}
\item {\bf [5 points]} Calculate the matrix element of
perturbation (\ref{V}) between the bound ground state and
an unbound states excited state. Choose the wave functions
of the excited states to have certain parities and use the
parity selection rule.
\item {\bf [5 points]} Using the Fermi golden rule, calculate the
ionization rate of the ``atom''. In other words, calculate the
probability of transition of the electron from the ground state
to an unbound state per unit time. Make sure the dimensionality
of your final result is 1/time.
\item {\bf [3 points]} Sketch how the ionization rate depends on the
frequency $\omega$.
\end{enumerate}
\item {\em Adapted from Qualifier, Fall 1988, II-2.}
An atom in the ground state is subject to a weak,
time-dependent, and uniform in space electric field $\bfE(t)$. A
simplest way to introduce the electric field into the electron
Hamiltonian is to select the gauge with the scalar potential
$\phi({\bf r},t)=-{\bf r}\cdot\bfE(t)$ and zero vector potential
${\bf A}=0$. Then, the perturbation of the electron potential
energy is
\begin{equation}
\hat{V}=e\phi({\bf r},t)=-e{\bf r}\cdot\bfE(t).
\label{V'}
\end{equation}
\begin{enumerate}
\item {\bf [5 points]} In the lowest order of the time-dependent
perturbation theory, show that the time-dependent
expectation value of the electron dipole moment,
\begin{equation}
{\bf d}(t)=\langle\psi(t)|e{\bf r}|\psi(t)\rangle,
\label{mu}
\end{equation}
depends on the values of the electric field at all
previous moments of time:
\begin{equation}
d(t)=\int_0^\infty\alpha(\tau){\cal E}(t-\tau)\, d\tau.
\label{alphatau}
\end{equation}
In this case, the vectors {\bf d} and $\bfE$ are parallel,
but in a more general case the polarizability $\alpha$
would be a tensor. Find an expression for the
polarizability $\alpha(\tau)$ in terms of nonvanishing
matrix elements of perturbation (\ref{V'}).
\item {\bf [5 points]} Calculate the frequency-dependent
polarizability of the atom, $\alpha(\omega)$, which is a
Fourier transform of $\alpha(\tau)$:
\begin{equation}
\alpha(\omega)=\int_0^\infty \alpha(\tau)\,e^{i\omega\tau}d\tau.
\label{alpha}
\end{equation}
To make integral (\ref{alpha}) convergent, assume that the
frequency $\omega$ has an infinitesimal positive imaginary
part $\epsilon$: $\omega=\omega+i\epsilon$. The
frequency-dependent polarizability $\alpha(\omega)$
describes response of the atom to a periodic-in-time
electric field, such as the electric field of an
electromagnetic wave.
\item {\bf [2 points]} Compare $\alpha(\omega=0)$ with the
expressions for the static polarizability you encountered
in previous homework on the time-independent perturbation
theory, e.g.\ Eq.\ (14.29). Do the two expressions agree?
\item {\bf [5 points]} Calculate the imaginary part
$\alpha''(\omega)$ of $\alpha(\omega)$ {\bf(see Hints)}.
Using the Thomas-Reiche-Kuhn sum rule (Problem 16.10),
calculate the integral
\begin{equation}
\int_{-\infty}^{+\infty}\omega\alpha''(\omega)\,d\omega
\end{equation}
and show that it is independent of the nature of the
considered system, thus represents a sum rule.
\label{Im}
\item {\bf [5 points]} What is the physical meaning of the real
and imaginary parts of the polarizability? Which of them
determines
\begin{enumerate}
\item velocity of light in hydrogen gas,
\item absorption of light in hydrogen gas?
\end{enumerate}
Explain why.
\item {\bf [3 points]} Compare expressions for the
frequency-dependent polarizability of a one-dimensional
harmonic oscillator and an atom. Contemplate the
similarities and differences between the two expressions.
\end{enumerate}
\item {\em Adapted from Qualifier, Spring 1986, II-2.} \label{cross-section}
A plane electromagnetic wave specified by the vector potential,
\begin{equation}
{\bf A}({\bf r},t)={\bf A}_0(\omega)e^{i({\bf k}{\bf r}-\omega t)}
+{\bf A}_0^*(\omega)e^{-i({\bf k}{\bf r}-\omega t)},
\quad\quad \phi=0, \quad\quad {\rm div}{\bf A}=0,
\label{A}
\end{equation}
interacts with an electron that belongs to an atom.
\begin{enumerate}
\item {\bf [5 points]} Using the Fermi golden rule, derive an
expression for the transition rate $R_{i\to
f}(\omega)$ per unit time from an electron state
$|i\rangle$ to an electron state $|f\rangle$ induced by
the field (\ref{A}) in terms of the wave functions and the
energies of the states $|i\rangle$ and $|f\rangle$.
Compare this expression with Eq.\ (16.85). Do they agree?
To answer this question, you need to connect the number of
photons $n$ in the quantized theory of electromagnetic
radiation with the intensity $A_0$ in the classical
theory.
\item {\bf [5 points]} The energy flux of the electromagnetic
wave (\ref{A}) is defined as the amount of energy crossing
a unit area in a unit time:
\begin{equation}
F(\omega)=\frac{c}{4\pi}\langle|\bf E\times B|\rangle=
\left[\frac{\mbox{Energy}}{{\rm Area}\times{\rm Time}}\right],
\label{FE}
\end{equation}
where the angular brackets denote averaging over time.
Show that $F(\omega)$ is given by the following formula
in terms of the amplitude $A_0(\omega)$ in Eq.\ (\ref{A}):
\begin{equation}
F(\omega)=\frac{\omega^2|A_0(\omega)|^2}{2\pi c}.
\label{Ip}
\end{equation}
\item {\bf [5 points]} \label{R/F} The absorption cross
section of electromagnetic radiation by the electron,
$\sigma_{i\to f}(\omega)$, is defined as the energy
absorption rate $\hbar\omega R_{i\to f}(\omega)$ divided
by the energy flux $F(\omega)$ (\ref{Ip}):
\begin{equation}
\sigma_{i\to f}(\omega)=
\frac{\hbar\omega R_{i\to f}(\omega)}{F(\omega)}
=\left[\frac{\frac{\rm Energy}{\rm Time}}
{\frac{\rm Energy}{{\rm Area}\times{\rm Time}}}\right]
=[\rm Area].
\label{i->f}
\end{equation}
The advantage of $\sigma_{i\to f}(\omega)$
compared to $R_{i\to f}(\omega)$ is that the
former does not contain the external radiation intensity,
thus characterizes the atom itself.
Obtain an expression for the absorption cross section
$\sigma^{(d)}_{i\to f}(\omega)$ in the
long-wavelength approximation (in the dipole
approximation). Compare this expression with
$\alpha''(\omega)$ found in Problem \ref{Im}.
\item {\bf [5 points]} Let us consider the electron in a ground
state $|0\rangle$. The total absorption cross section is
the sum of $\sigma_{0\to f}(\omega)$ over all
final states: $\sigma(\omega)=\sum_f\sigma_{0\to
f}(\omega)$. Using the Thomas-Reiche-Kuhn sum rule
(Problem 16.10), prove in the dipole approximation that
\begin{equation}
\int_0^\infty\sigma^{(d)}(\omega)\,d\omega=\frac{2\pi^2e^2}{mc},
\label{sigma}
\end{equation}
where $m$ is the mass of electron. Note that Eq.\
(\ref{sigma}) is universal in the sense that it does not
contain any information about the details of the
considered system (the type of the atom). It does not
contain even $\hbar$, which means that this result should
be valid also classically!
\end{enumerate}
\item {\bf [7 points]} The hydrogen atom in the ground state of the
energy $E_0<0$ is irradiated by an electromagnetic wave of a
frequency $\omega>|E_0|/\hbar$. This perturbation may cause
ionization of the atom: an electron transition from the bound
state to a unbound state with the wave vector {\bf q}.
Using the results of Problem \ref{cross-section}, calculate the
differential cross section $d\sigma$ of the electron excitation
to the solid angle $d\Omega$ of the directions of the wave
vector {\bf q} and the total cross section of ionization,
$\sigma$.
\item \begin{enumerate}
\item {\bf [5 points]} Let us consider a continuous distribution
in the frequency $\omega$ of the plane waves of type
(\ref{A}). The energy flux in a small window of
frequencies, $\Delta\omega$ can be written as
$F(\omega,\Delta\omega)=
\int_\omega^{\omega+\Delta\omega}I(\omega)\,d\omega$,
where $I(\omega)$ is the spectral density of the energy
flux.
Using your solution of Problem \ref{R/F} and Eq.\
(\ref{i->f}), show that the transition rate $R_{i\to f}$
between two discrete electron states $|i\rangle$ and
$|f\rangle$ is proportional the spectral density of the
energy flux $I(\omega)$ taken at the frequency of the
atomic transition $\hbar\omega=E_f-E_i$:
\begin{equation}
R_{i\to f}=B(\omega)I(\omega), \qquad {\rm where}\quad
\label{RB}
\hbar\omega=E_f-E_i.
\end{equation}
Find an expression for the coefficient $B$, which is
called the Einstein coefficient $B(\omega)$. (Different
books have slightly different definitions of $B$.)
\item {\bf [3 points]} Compare your expression for $B$ with
expression (16.74) for the rate of spontaneous transitions
from an excited state $|f\rangle$ to a lower state
$|i\rangle$. The spontaneous transition rate (16.74) is
called the Einstein coefficient $A$, by definition. Find a
ratio of $A$ to $B$.
\item {\bf [5 points]} Substituting the electron wave functions
of the hydrogen atom into Eq.\ (16.74), calculate the
lifetime of the 2p state of the hydrogen atom in
seconds. Compare your result with the number given in
Schwabl on page 305.
This number also determines the excitation rate of the
1s$\to$2p transitions via Eq.\ (\ref{RB}) and the relation
between $A$ and $B$ found in the previous problem.
\end{enumerate}
\end{enumerate}
\vfill
\section*{\centerline{Hints}}
\begin{description}
\item[\ref{Im}]
\begin{equation}
{\rm Im}\frac{1}{x-i\epsilon}=\pi\delta(x),
\label{delta}
\end{equation}
where $\delta(x)$ is Dirac's delta-function. Prove Eq.\
(\ref{delta}) starting from a finite $\epsilon$ and taking the
limit $\epsilon\to0$.
\end{description}
\end{document}
\item Compare the imaginary part of $\alpha(\omega)$ with the
absorption cross section of photons $\sigma_{i\to
f}(\omega)$ obtained in Problem
\ref{cross-section}. Calculate
\begin{equation}
\int_{-\infty}^{+\infty}\omega\alpha''(\omega)\,d\omega,
\end{equation}
where $\alpha''(\omega)$ is the imaginary part of
$\alpha(\omega)$ {\bf(see Hints)}. Show that this
integral is independent of the nature of the considered
system, thus it represents a sum rule. {\bf [5 points]}
\label{Im}
\item {\bf [5 points]} Using the results of Problem
\ref{cross-section}, calculate the cross section of the
1s$\to$2p transition in the hydrogen atom,
$\sigma^{(d)}_{{\rm1s}\to\rm2p}(\omega)$. \label{1s2p}
\item {\bf [3 points]} Let us consider a continuous distribution
in the frequency $\omega$ of the plane waves of type
(\ref{A}). The energy flux in a small window of
frequencies, $\Delta\omega$ can be written as
$F(\omega,\Delta\omega)=
\int_\omega^{\omega+\Delta\omega}I(\omega)\,d\omega$,
where $I(\omega)$ is the spectral density of the energy
flux.
Using your solution of Problem \ref{1s2p} and Eq.\
(\ref{i->f}), express the transition rate
$R_{{\rm1s}\to\rm2p}$ in terms of the spectral
density of the energy flux $I\left(\frac{E_{\rm 2p}-E_{\rm
1s}}{\hbar}\right)$.