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  {\bf University of Kentucky, Physics 420 \\
    Homework \#1, Rev. A, due Monday, 2015-09-14 }
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\vspace{.3 in}

{\bf 1. Planck's law}---Derive Wien's displacement law $\lambda_{max}T=b$, and
the Stephan Boltzman law $I=\sigma_{SB}T^4$ from Planck's law.  Calculate the
numerical value of the constants $b$ and $\sigma_{SB}$.  In Mathematica, you
can use the functions {\tt D}, {\tt NSolve}, and {\tt Integrate}.

{\bf 2. Compton Edge}---Similar to atomic spectra measured with a diffraction
grating, nuclear spectra (gamma emission due to nuclear transitions from one
state to another) can be observed by histogramming the photon energy detected
directly in a High Purity Germanium (HPGe) detector.  However, some of the
photons do not deposit all of their energy, but Compton scatter off of
electrons in the HPGe crystal and exit the detector.  Explain the shape of
Compton background in this
\href{https://upload.wikimedia.org/wikipedia/commons/8/88/60Co_gamma_spectrum_energy.png}{$^{60}$Co
  $\gamma$-spectrum} at Wikipedia:
\href{https://en.wikipedia.org/wiki/Cobalt-60}{Cobalt-60}, which has two
lines, one at 1.17~MeV and the other at 1.33~MeV.  Calculate the two {\it
  Compton Edges}, below which background count rates become much higher.
                                                                               
{\bf 3. Bohr's Oscillator}---Follow the same steps used by Bohr quantize a
3-dimensional ideal frictionless harmonic oscillator of mass $m$ and natural
angular frequency $\omega$, (ie. spring constant $k=m\omega^2$).  If you start
with quantization of angular momentum, show that the correspondence principle
holds; or vice versa.

{\bf 4. COW Experiment}---A beam of neutrons of de Broglie wavelength 2~\AA\
passes through a Mach-Zehnder interferometer, with lattice spacing
$d$=3.135~\AA\ in the $\langle 111\rangle$ planes in Si.  Calculate the angle
of incidence that matchs the Bragg diffraction angle so that exiting split
beam is symmetric with the blade, as shown below.  Assuming a 30~mm separation
of each of the three blades, calculate the phase difference in the two paths,
due to gravitational retardation in the top path (see
\href{http://www.phys.utk.edu/neutron-summer-school/lectures/wietfeldt.pdf}{Neutron
  interferometry}, page 41).

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