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\begin{center}
  {\bf University of Kentucky, Physics 404G \\
    Homework \#5, Rev. A, due Thursday, 2021-10-07 }
\end{center}

\vspace{.3 in}

{\bf 1.} It has been claimed that World War II was won by the \href%https://web.archive.org/web/20180219122811/
{http://www.radartutorial.eu/08.transmitters/Magnetron.en.html}{\bf Magnetron}, a vacuum device which converts electrical energy into microwave radiation.  It enabled short-wavelength tactical radar, which used a frequency too high for electrical circuits of the time.  Radiation is emitted from several cylindrical microwave cavities surrounding the central bore of a solid copper waveguide under vacuum.  A central {\it thermionic cathode} of radius $a$ at negative high voltage $V$ emits electrons, which accelerate to the grounded {\it anode} (inner cylindrical face of the cavity) of radius $b$.  A longitudinal magnetic field $\bs B$ sweeps the electrons in a spiral path around cylindrical void, exciting resonant currents in the outer microwave cavities as they sweep by.  The electric and magnetic fields are perpendicular in this {\it crossed-field device}.

{\bf \quad a)} In L16 we developed the equation of motion for an electron in a magnetic field, $\dot\eta=-i\omega\eta$, using complex coordinates $\xi=x+iy$ and $\eta=\dot\xi=v_{x}+iv_{y}$, where $\omega=qB/m$ is the {\it cyclotron frequency}.  Show that the equation of motion for an electron in constant electric $\bs E=\bh x E_{0}$ and magnetic $\bs B=\bh z B_{0}$ fields is $\dot\eta = -i\omega(\eta+iv_{d})$, where $v_{d}=E_{0}/B_{0}$.  Solve this equation to obtain the {\it cycloid} motion of the electron in these fields.  What is the physical interpretation of $iv_{d}$?

{\bf \quad b)} Express the force $\bs F=q\bs E$ due to the electric field $\bs E={\bh \rho V}/[\rho\ln(b/a)]$ in terms of $\xi$ to obtain the equations of motion $\dot\xi=\eta$ and $\dot\eta=\lambda/\xi^{*} - i\omega\eta$.  What is the value of the constant $\lambda$ when $a=0.5~\q{cm}$, $b=2.5~\q{cm}$, and $V=-4~\q{kV}$ is applied to the {\it cathode}?  Integrate the equations in Matlab for an electron emitted essentially at rest from the cathode, and $B=1~\q{T}$.  Before you start, what will the motion look like qualitatively?  Determine the {\it Hull cut-off} field $B_{c}$, such that an electron barely grazes the surface of the anode and returns to the cathode. {[bonus: Plot $B_{c}(V)$.]}

{\bf \quad c)} The electrons from part a) excite eight cylindrical resonant cavities of radius $r=1~\q{cm}$ distributed around the central cavity, each one separated from the anode (at $\rho=b$) by a $w=4~\q{mm}$ long, $t=5~\q{mm}$ wide channel cut in the solid cylindrical copper block of thickness $d$.  Calculate the resonant frequency using the inductance $L=N^{2}\mu_{0}\pi r^{2}/d$ and capacitance $C=\epsilon_{0}w d/t$.  Optimize both the voltage $V$ and magnetic field $B$ such that the electrons just graze past the resonators at the resonant frequency.  {\it Hint: use conservation of energy to determine the required voltage and then determine the magnetic field. Warning: Don't try this at home without also including the space-charge effect and its associated ``pinwheel''!}


{\bf 2.} Discovery of the {\bf Higgs boson} in 2012 confirmed the last piece of the Standard Model of particle physics.  But this elusive particle was not actually observed: only the two photons [or other particles] into which it decays after $1/\Gamma\sim$ \href{https://www.nature.com/articles/nphys3005#:~:text=The\%20Higgs\%20boson\%20lifetime\%20is,and\%20a\%20\%CF\%84\%CF\%84\%20lepton\%20pair.}{$10^{-22}$~s}.  The observed signature was a resonance in the histogram of the mass or frequency $E=mc^{2}=\hbar\omega$ reconstructed from the energy of the two photons, according to the damped oscillations of its quantum mechanical wavefunction $\Psi(t)=\Psi_{0}e^{-\Gamma t/2}e^{i\omega_{1}t}$, with probability $P(t)=|\psi(t)|^{2}=e^{-\Gamma t}$ decaying at the rate $\Gamma$.

{\bf \quad a)} Calculate the initial velocity $v_{0}=\Im\dot\Psi(0)$ of $\Im\Psi(t)=\Psi_{0}e^{-\Gamma t/2}\sin(\omega_{1}t)$.  Using the Fourier method, treat the system as a damped oscillator driven with the impulse $f_{0}(t)=v_{0}\delta(t)$, acting at $t=0$ to kick it with the initial velocity, not interfering afterwards.  Using the principle of supperposition, formally integrate all components to express the resulting impulse response $\Im\Psi(t)$ as an integral over the amplitude function $A(\omega)$.  This shows that the Fourier transform of the free decay wavefunction $\Im\Psi(t)$ directly encodes the resonance curve $A(\omega)$ of the associated driven oscillator.  This so-called {\it Breit-Wigner resonance} relates the spectral linewidth to the decay rate of the wave function, for anything from laser transitions to short-lived high-energy particles.

{\bf \quad b)} Show that the Full Width at Half Maximum (FWHM) of the resonance in $|A|^{2}(\omega)$ is $\Gamma=2\beta$.  Estimate the frequency $\omega_{1}$ and lifetime $1/\Gamma$ of the Higgs boson from the associated resonance in the {\it missing mass} spectrum in Fig. 4 of the \href{https://arxiv.org/pdf/1207.7214.pdf}{discovery paper}.  Calculate the $Q$ value of this resonance.  Note that $Q$-values upwards of $10^{11}$ have been achieved in atomic physics.

%https://www.rp-photonics.com/q_factor.html
%https://www.nature.com/articles/nphys3005 lifetime
%https://pdg.lbl.gov/2019/tables/rpp2019-sum-gauge-higgs-bosons.pdf
%https://atlas.cern/updates/physics-briefing/precise-measurement-higgs-boson-mass
%https://www.wearethemighty.com/soldiers-marching-over-bridges-myth -> Tacoma Narrows
%https://mythbusters.fandom.com/wiki/Break_Step_Bridge_(Episode)

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