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\begin{center}
  {\bf University of Kentucky, Physics 404G \\
    Homework \#10, Rev. A, due Sunday, 2021-12-05 }
\end{center}

\vspace{.3 in}

{\bf 1. Rotating room}---Experience first-the inertial forces in a rotating reference frame!

{\bf \quad a)} Compare the radius of curvature of a ball rolling on the table with calculations.

{\bf \quad b)} Calculate the centrifugal acceleration as a function of radius by measuring the angle of a pendulum hanging from the ceiling, and compare with calculations.


{\bf 2. Nuclear Magnetic Resonance} (NMR) is a high-precision spectroscopy technique developed in 1946 by Edward Purcell and Felix Bloch.  This technology is used in Magnetic Resonance Imaging (MRI) machines.  In NMR, the spin $\bs s$ of an atomic nucleus precesses about a constant magnetic field $B_{0}$ similar the precession of a spinning top.  This characteristic resonance called the {\it Larmor frequency} $\omega_{L}=\gamma B$ is proportional to $B_{0}$ and to the gyromagnetic ratio $\gamma$ of the magnetic moment $\bs\mu$ to the angular momentum $\bs s$ of the nucleus, ie. $\bs\mu=\gamma\bs s$.

{\bf \quad a)} Show that $\gamma=e/2m$ for a charged point particle in a circular orbit, independent of radius $r$.  This does not hold for a composite particle; we define its {\it g-factor} by $\gamma=g\cdot e/2m$, ie. how much larger than for a point particle of the same mass and charge.  A pointlike quantum mechanical particle with spin $s=\hbar/2$ has $g=2$.  For example, the electron has $g_{e}=2.002319$, which is slightly larger than 2 due to vacuum polarization.  The neutron also has spin $s=\hbar/2$, but $g_n = -3.826$ due to its internal quark structure.  Thus its magnetic moment is $\mu_{n}=g(e/2m)(\hbar/2) = (g/2)\mu_{N} = -1.91~\mu_{N}$. The unit of magnetic moment used in nuclear physics is the {\it nuclear magneton} $\mu_{N}=e\hbar/2m_{p}$, which equals the magnetic moment of a pointlike proton with orbital angular momentum $\ell=\hbar$ (p-orbital).
              
{\bf \quad b)} Solve the classical equation of motion (the Bloch equation) $\dot{\bs s} = \bs \tau = \bs \mu_n\times \bs B = \gamma_n \bs s \times \bs B$ for the {\it Larmor precession} of a neutron in a constant magnetic field $\bh z B_0$, with initial spin $\bs s_0$ at $t=0$.  Compare this with the precession of a spinning top.

{\bf \quad c)} The mechanical equations of motion are simplified in a rotating reference frame, where \scalebox{.8}{$\pmatrix {\bh x' & \bh y'}=\pmatrix {\bh x & \bh y} \pmatrix{\cos\omega t & - \sin\omega t \cr \sin\omega t & \cos\omega t}$}.  \href{http://bit.ly/2TNaBQ8}{[Rabi, Ramsey, Schwinger, Rev.  Mod.  Phy.  {\bf 26}, 167, 1954]}.  Since the operator $\bs\omega dt \times$ generates this rotation, the time derivative becomes $\dot{\bs s} = \dot{\bs s}' + \bs\omega \times \bs s$, where $\dot{\bs s}'$ is of components in the rotating frame.  Substitute $\dot{\bs s}$ into the Bloch equation and show that the form remains the same except for the replacement of $\bs B$ with the effective field $\bs B'=\bs B+\bs \omega/\gamma_n$.  Note this field is zero if the frame is rotating at the Larmor frequency $\bs\omega_L = -\gamma_n \bs B$, and thus the spin remains constant $\bs s'=\bs s_0$.  Reconcile this picture with the solution in the static frame.

{\bf \quad d)} The $z$-component of spin $s_{z}$ does not change in a constant magnetic field $\bh z B_0$ (called a {\it holding field} because it preserves the spin state).  To transition the spin from up to down, we must use an oscillatory (RF) field $B_1(\bh x\cos\omega t + \bh y\sin\omega t)$.  In the rotating frame with angular velocity $\bh z\omega$, show that the total field is $\bs B'=\bh z' (B_0+\omega/\gamma_n)+\bh x' B_1$, which is constant.  Let $\theta$ be the angle between $\bs B'$ and $\bh z$, and let the initial spin be $\bs s_0=\bh z$.  Show that the $z$-component of the spin varies as $s_z(t) = s_0(\cos^2\theta + \sin^2\theta \cos \gamma_nB't) = s_0(1-2\sin^2\theta\sin^2(\gamma_n B' t/2))$, which oscillates at the {\it Rabi flopping frequency} $\omega_{R}=\gamma_nB'$.  Plot the amplitude of oscillation as a function of the RF frequency $\omega$ and note the resonance at $\omega=\omega_L$.  [bonus: Plot the 3-d trajectory of $\bs s$ in the lab frame over half a Rabi cycle.]

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