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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 520 \\
    Homework \#1, Rev. C, due Wednesday, 2017-09-05 }
\end{center}

\vspace{.3 in}

{\bf 0.} Griffiths [2ed]
Ch.~1 \#1, %p12, std dev
      \#3, %p12, gaussian
      \#13, %p21, Buffon's needle
      \#18. %p23, thermal de broglie wavelength
.

{\bf 1. Maxwell-Boltzman distribution}---The velocity spectrum of an ideal gas
follows the Boltzman distribution $f(\bs v)= e^{-\frac12 mv^{2}/kT}$, assuming
that the distribution of velocity states is uniform in $(v_{x},v_{y},v_{z})$
space and that the kinetic energy $T=\frac12mv^{2}$ is thermally distributed.

{\bf \quad a)} Integrate the distribution $\int d^{3}v f(\bs v)$ over $\theta$
and $\phi$ to obtain the Maxwell distribution $f(v)$ of the scalar velocity of
the atoms in the gas.  Normalize the distribution and calculate
$\langle v\rangle$.

{\bf \quad b)} Show that if the density of atoms in a box is $\rho$, then the
number of atoms escaping through a hole of area $A$ per time is
$\rho \langle v\rangle A/4$.  Thus show that the average forward spectral
intensity of black body radiation is $I=u c/4$, where $u$ is the spectral
energy density.

{\bf \quad c)} [bonus] An {\it effusive source} of gas is a nozzle in which
the atomic mean free path is much longer the the width of the nozzle.  Show
that the velocity distribution is
$f(\bs v) \propto v^{3} \cos\theta\, e^{-\frac12 mv^{2}/kT}$.  The
forward-peaked velocity distribution makes an effusive source ideal for atomic
beams.

{\bf 2. Planck's law}---The spectral energy density of thermal radiation from
a black body is $u(\nu,T) = 8\pi h \nu^3/(c^3(e^{\hbar \nu/kT}-1))$.  You
can solve the following problems using the functions {\tt D}, {\tt NSolve},
and {\tt Integrate} in Mathematica.

{\bf \quad a)} Show that
\href{https://en.wikipedia.org/wiki/Planck\%27s_law}{Plank's law} obeys the
following {\it scaling law}: that $u(\nu,T)/T^3$ depends only on
$x=\hbar \nu/kT$, not $\nu$ or $T$ individually, and thus, the graph of the
spectral intensity $u(\nu)$ has the same shape for different temperatures.
How should the $\nu$- and $u$- axes scale with temperature to keep the graph
invariant?

{\bf \quad b)} Show
\href{https://en.wikipedia.org/wiki/Wien\%27s_displacement_law}{Wien's
  displacement law}, that $\lambda_{max}T=b$ by calculating the value of $x$
for which $u(\nu)$ attains its maximum at constant temperature $T$. Use this
value to obtain the constant $b$.

{\bf \quad c)} Integrate $\int_0^\infty u(\nu,T) d\nu$ to obtain the total
energy density over all wavelengths.  Using 1c), calculate the constant
$\sigma_{SB}$ from the
\href{https://en.wikipedia.org/wiki/Stefan\%E2\%80\%93Boltzmann_law}{Stephan-Boltzman
  law} $I=\sigma_{SB}T^4$.

{\bf 3. Bohr's Oscillator.}  A 3-dimensional ideal frictionless harmonic
oscillator of mass $m$ and natural angular frequency $\omega$ behaves very
similar to a hydrogen atom, except the central force law is $F=-kr$, with
spring constant $k=m\omega^2$.

{\bf \quad a)} Use Bohr's quantization of angular momentum $L=\hbar n$ to
calculate the energy levels $E_n$ of the stationary states of this `atom', and
calculate spectrum of emitted wavelengths.

{\bf \quad b)} Show that Bohr's correspondence principle holds in this
system.

{\bf \quad c)} [bonus] The classical Larmor formula for the power radiated by
a point charge $q$ under acceleration $a$ is
$P=(qa)^{2}/6\pi\epsilon_{0} c^{3}$.  Calculate the number of orbits an
electron would make in the $n$\ts{th} orbital before decaying to the
$(n-1)$\ts{th} orbital.

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