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\begin{center}
  {\bf University of Kentucky, Physics 521 \\
    Homework \#17, Rev. A, due Wednesday, 2018-03-28 }
\end{center}

\vspace{.3 in}

{\bf 0.} Griffiths [2ed]
Ch.~7 \#1, %p298 gaussian trial functions
%      \#3, %p298 delta in inf sq well
%      \#5, %p298 perturbation overestimates
      \#6, %p303 ionization energy
%      \#8, %p308 direct,exchange integrals
      \#10, %p309 vibrations of H2 ion
      \#14, %p310 yukawa potential
%      \#15, %p310 two-state system  LONG
      \#16, %p311 electron in magnetic field
      \#19. %p313 muon catalyzed fusion

{\bf 2. Hyperfine Zeeman splitting in deuterium} is similar to hydrogen,
except that the nucleus is a spin $I=1$ deuteron instead of a spin $I=\frac12$
proton.  Thus the 1s ground state has 6-fold degeneracy, including hyperfine
structure.  Since the deuteron is a loosely bound proton--neutron isosinglet
($T=0$, note the new notation) dominated by $L=0$, its magnetic moment is
approximately the sum of its constituents
$\mu_d=g_I\mu_N=0.857~\mu_N\approx \mu_p+\mu_n = 2.79~\mu_N-1.91~\mu_N$.

{\bf \quad a)} Using the Pauli exclusion principle, show that the deuteron must
\mbox{be a spin triplet ($S=1$)}.

%different notation for 'b'
{\bf \quad b)} Show that the hyperfine perturbation to the Hamiltonian is
$\mathcal{H}'_{\rm hf}=b \bv I\cdot\bv S$ and calculate $b$ for deuterium.
Show that the perturbation for the Zeeman effect is
$\mathcal{H}'_{\rm Z} = (g_S \mu_B \bv S/\hbar + g_I \mu_N \bv I/\hbar) \cdot
\bv B_{ext}$, ignoring $\bv L$.  Compare the magnitude of $g_S$ and $g_I$ for
the deuterium.  Why can we ignore fine structure when considering the
degeneracy of the 1s states?

{\bf \quad c)} Using degenerate perturbation theory, calculate the hyperfine
energy shift for deuterium in a weak external magnetic field
$B_{ext}\ll B_{int}$.  What are the eigenstates and good quantum numbers which
break the degeneracy of the Bohr levels?  Which quantum numbers are still
degenerate?  Calculate the Land\'e factor $g_F$ for the Zeeman shift
$(g_F\mu_B \bv F/\hbar) \cdot \bv B_{ext}$.  At what field
$B_{int}=b\hbar/g_F\mu_B$ are the two perturbations are approximately equal?
Plot the Zeeman shift as a function of $B_{ext}$.

{\bf \quad d)} In a large external field, $B_{int}\ll B_{ext}$, we must first
break the degeneracy according to the Zeeman shift, which dominates hyperfine
structure.  Calculate the perturbed energies as a function of $B_{ext}$.  What
are the good quantum numbers and corresponding eigenstates?  What degeneracies
remain?  Use these states to calculate the hyperfine energy shift (constant
for $B_{ext}$).

{\bf \quad e)} In intermediate fields where both perturbations are of the same
order, we must diagonalize $\mathcal{H}'_{\rm hf}+\mathcal{H}'_{\rm Z}$
together.  Calculate this $6\times 6$ matrix in the $nlm_lm_sm_I$ basis and
diagonalize it to obtain the exact energy shifts.  Plot the energies as a function
of $B_{ext}$.  Show that the small- and large-field limits match (b) and (c)
respectively.  Repeat the calculation and diagonalization of matrix elements
in the $nljFM_F$ basis to show that the result is independent of basis.

% {\bf \quad f) [bonus]} An Atomic Beam Source (ABS) polarizes deuterons by
% passing deuterium through a Stern-Gerlach gradient field to select the
% $m_s=\frac12$ electronic state.  The atomic beam then passes through an RF
% field, which shifts the energy levels via $B_{ext}'$ in the rotating frame.
% A gradient static field in the RF field forces an adiabatic transition between

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