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\begin{flushright}
{\bf Prof. Aephraim Steinberg}
\end{flushright}

{\Large  \bf \noindent Physics 485/1485: Problem Set \#1 \\
assigned 23 September, 1999 \\
(due 5 October, 1999)}


\vspace{0.5 in}

\begin{enumerate}



\item
%problem 1

Let us calculate the cross-section for a Lorentz atom.  Recall the 
equation of motion
$\ddot{x}=-\omega_0^2 x - \gamma\dot{x}-eE/m$.\\

\begin{enumerate}
\item
%problem 1a

For a coherent drive $E(t) = E_0 \cos\omega_0 t$ exactly on
resonance, solve for the position of the electron as a function of
time, in terms of $E_0$. 
This determines the polarizability ({\it i.e.}, the index 
of refraction) of
the atoms.

\item
%problem 1b

Since the force applied to the electron is $-eE$, the instantaneous 
power transferred to the atom is $-eE(t)\dot{x}(t)$.  Integrating
over many periods of the oscillations at $\omega_0$, what is the
average power absorbed?


\item 
%problem 1c

Setting this equal to $I\sigma$, what do you get
 for the cross-section $\sigma$?  (You will need 
to remember that the energy density of an electromagnetic wave is 
$\frac{\epsilon_0}{2}E_0^2$, and pay
attention to units when you convert this to an intensity.)  Consider 
the Lyman-$\alpha$ transition in Hydrogen, for
which the linewidth $\gamma / 2\pi = 100$ MHz.  What numerical value 
do you find
for the cross-section?  How does this compare to the physical size of 
an atom?  How
does it compare to the wavelength of Ly$\alpha$ radiation?

\item
%problem 1d

Consider a Lorentz atom set in motion at $t=0$.  As it begins to
decay and emit energy, how does the energy emission rate depend on
the damping constant $\gamma$?\\
In the light of thermodynamic arguments, how then should the 
{\em absorption}
rate of this atom in a thermal radiation field depend on $\gamma$?\\
How would you reconcile this with the on-resonance cross-section
derived in part 1(c)?

\end{enumerate}


\item
% problem 2
 
Remember from class that for an atom with two levels
$\ket{g}$ and $\ket{e}$ with energies $E_{g}$ and $E_{e}$
respectively,  $\bra{g}x\ket{g}
=\bra{e}x\ket{e}=0$ and  $\bra{e}x\ket{g}\equiv x_0 = 
\left[\bra{g}x\ket{e}\right]^{*}$.  \\
\begin{enumerate}

\item
% problem 2a

For an atom  in a state $\ket{\Psi} = c_g\ket{g}+c_e\ket{e}$
at $t=0$, what is 
the expectation
value of the dipole moment $-ex$ as a function of time?  (Recall
that $c_{g}$ and $c_{e}$ may be complex.)

\item
% problem 2b

Sketch the amplitude of $\expect{x}$ versus the expectation value
of the atom's energy. \\
Make the comparable sketch for the amplitude and energy of a 
Lorentz-model oscillator.\\
Compare and contrast.

\item
% problem 2c

Suppose the Lorentz model were ``true,'' {\it i.e.}, that atoms {\it were} in
fact little harmonic oscillators.  Of course, the correct description
of these oscillators would still be quantum mechanical, so the atom
could be
in the ground state \ket{0}, or excited states 
\ket{1},\ket{2},\ldots; 
these states are 
related by raising and lowering operators $a^\dagger_{HO}$ and 
$a_{HO}$ with the well-known matrix elements $a^\dagger_{HO}\ket{n}
=\sqrt{n+1}\ket{n+1}$ and $a_{HO}\ket{n} = \sqrt{n}\ket{n-1}$.  
Recall that
$x = \sqrt{\hbar/2m\omega}(a_{HO}+a^\dagger_{HO})$.  \\
A ``quasiclassical,'' or coherent, state of the harmonic oscillator
can be represented as an eigenstate $\ket{\alpha}$ of the lowering
operator:
\begin{equation}
a_{{HO}} \ket{\alpha} = \alpha \; \ket{\alpha}\;.
\end{equation}
(Note that $a_{HO}$ is non-Hermitian, and that $\ket{\alpha}$ is {\em not}
an eigenstate of $a^{\dagger_{HO}}$.)\\
What are the expectation value of the dipole moment $-ex$ and
of the energy $\left(a^{\dagger}_{HO}a_{HO} + 1/2\right)\hbar 
\omega_{0}$ for an
atom in a state $\ket{\alpha}$?  Does this more closely resemble
the classical Lorentz-model atom or the quantum two-level atom?


\item
%problem 2d

Consider again the 1s-2p transition in 
Hydrogen.  Calculate
the transition frequency $\omega/2\pi$ and the wavelength 
$\lambda=2\pi c/\omega$.  
For a harmonic oscillator at this frequency,
what would \expect{-ex} be in an equal superposition of the ground 
and first excited states? 
Is this a reasonable quantity to expect for a real 
(quantum-mechanical) Hydrogen atom, and if so, why?

\item
%problem 2e

The spontaneous decay rate of Sodium in its first excited
state  is $\gamma/2\pi = 9.89$ MHz.  It decays by emitting
a photon with a wavelength of $589$ nm.
Given that the cross-section for absorption is $3\lambda^{2}/2\pi$, how
many photons per second would a single ground-state atom
absorb from a $589$ nm beam
with an intensity of $I$ mW/cm$^{2}$?  (Recall problem 1.)\\
How many photons per second will a single excited-state atom
be stimulated to emit by the same beam (remember the discussion in
class)?\\  
Combining the spontaneous emission, absorption, and emission, 
write down the rate equation for $n_e$, the population in the excited
state.  Write down the steady state solution for $n_e$ as a function
of intensity $I$.  How does it behave for low intensities?  Does this
fit the expectations we have from the Lorentz model?  At what 
intensity does this begin to change?\\
  This is known as the saturation
intensity.  Note that it is easily achievable with lasers; since
this calculation assumed the light to be on resonance, the same
effect would be exceedingly difficult to achieve with broadband
light (in effect, it would require a light source with a temperature
on the order of the order of the excitation energy, around an electron
Volt, or $12,000$ K). 

\end{enumerate}

\end{enumerate}

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