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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 521 \\
    Homework \#15, Rev. B, due Monday, 2016-04-25 }
\end{center}

\vspace{.3 in}

{\bf 0.} Griffiths [2ed]
Ch.~9 \#9.1, %p343 H dipole elements n=1-2
      \#9.2, %p343 check unitarity of TDPT
      \#9.6, %p345 time indep 2nd order
      \#9.7, %p348 Rabi flopping vs exact
      \#9.8, %p356 thermal stim emission
      \#9.9%, %p346 vacuum energy
.

{\bf 1.} The {\bf Lyman-alpha} ($L_\alpha$) transition ($n=2\to1$) is the
highest energy, shortest wavelength line of the hydrogen spectrum.

{\bf \quad a)} Calculate the energy and wavelength of this transition.

{\bf \quad b)} Which $n=2\to1$ substates satisfy the electric dipole selection
rules?

{\bf \quad c)} Calculate the lifetime and line width of this transition.

{\bf \quad d)} Calculate the effects of collisional broadening [use
$\sigma_{col}=\pi(\langle r^2\rangle_{100}+\langle r^2\rangle_{211})$, see
Griffiths problem \#4.13], Doppler broadening, and recoil line shift at STP
(20$^\circ$C, 1~atm), as fractions of the natural wavelength and line width.
[Note, however, that UV light attenuates rapidy in air!]

{\bf 2. Lasers} require a {\it population inversion} of occupation numbers
$N_b > N_a$ to achieve a higher rate of {\it stimulated emission} than of
absorption of photons in the transition $E_b\to E_a$.  An {\it optical cavity}
with mirrors at each end reflects the laser beam back and forth to create a
standing wave, enhancing the photon density and narrowing the line width.  A
partial silvered mirror on one end allows a fraction of the photons to exit as
the laser beam, contributing to attenuation of photons in the cavity.  Lasing
occurs if there is a net gain of photons in the laser due to stimulated
emission.

{\bf \quad a)} Integrate the spectral density of the Breit-Wigner resonance
\[\rho(\omega) = \rho(\omega_0) \frac{
  (\Gamma/2)^2}{(\omega-\omega_0)^2+(\Gamma/2)^2}
\]
to obtain the energy density $u=\int_0^\infty\rho(\omega) d\omega \approx
\int_{-\infty}^\infty\rho(\omega) d\omega $ in the cavity.  Convert this to
$N_\gamma$, the number of photons in the beam, which occupies a volume $V$ in
the cavity.

{\bf \quad b)} Show that $\dot{N}_\gamma$, the rate of change of the number of
photons in the cavity [due to stimulated emission, absorption, and
attenuation] is positive if the population inversion is greater than the {\it
  critical density}
\[
\Delta n_c \equiv \frac{N_b - N_a}{V}
= \frac{\omega^2\ \Gamma \tau_s}{2\pi c^3 \tau_\gamma}
= \frac{2\pi \Gamma \tau_s}{ \lambda^2 c \tau_\gamma},
\]
neglecting the degeneracy of states, where $\tau_s$ is the lifetime of
spontaneous emission, and $\tau_\gamma$ is the characteristic lifetime of
photons in the laser cavity.

{\bf \quad c)} Calculate the population inversion density needed for a Ruby
laser [the first pulsed laser] and a He-Ne laser [the first CW laser], using
the following values [Tipler \& Llewellyn, 5ed]:

\begin{center}
\begin{tabular}{rcc}
  &Ruby laser & He-Ne laser \\ \\
  $\lambda$ & 694.3~nm & 632.8~nm \\
  $n$ (refractive index) & 1.76 & 1.00 \\
  $\tau_s$ & 3~ms & 100~ns \\
  $\tau_\gamma$ & 29~ns & 330~ns \\
  $\delta\nu=\Gamma/2\pi$ & 330~GHz & 0.9~GHz. \\
\end{tabular}
\end{center}


{\bf 3. [bonus]} Design an efficient, economic Lyman-alpha laser.

\end{document}
