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\begin{center}
  {\bf University of Kentucky, Physics 521 \\
    Homework \#15, Rev. A, due Wednesday, 2018-02-07 }
\end{center}

\vspace{.3 in}

{\bf 0.} Griffiths [2ed]
Ch.~4 \#34, %p189 *apply S+/- to each state
      \#36;~ %p189 S=1+2, probabilities
%      \#55  %p196 J=L+S
Ch.~5 %\#2,  %p203 reduced mass
      \#7,  %p210 Slater det
      \#15, %p223 energy of Fermi gas
      \#17, %p223 bulk modulus
      %\#29, %p243 BEC
      \#35, %p246 white dwarf
      \#36. %p247 black hole

{\bf 1. Davilov Parameters} [Danilov, Phys. Lett. {\bf 35B}, 579 (1971)] The
parity violating elastic nucleon scattering amplitudes can be classified using
symmetry and conservation principles studied in class.  The neutron (n) and
proton (p) are two states of a single particle, the {\it nucleon} (N), just as
their constituent up (u) and down (d) particles are both quarks (q).  In
analogy with the two-state system spin, we assign the nucleon {\it
  isospin} $T=\frac12$.  The proton ($T_3=\frac12$) and neutron
($T_3=-\frac12$) are represented by two-component {\it isospinors}
$\Upsilon=\scriptsize{\pmatrix{1\cr 0}}$ and $\scriptsize{\pmatrix{0\cr 1}}$,
respectively, with Pauli matrix operators writen
$\bs\tau =(\tau_1,\tau_2,\tau_3)$ as opposed to
$\bs\sigma=(\sigma_x,\sigma_y,\sigma_z)$.  Addition of isospin is just
like spin: the two-particle system forms an $T=0$ singlet (for example, the
$\omega$ meson) and an $T=1$ triplet (the $\rho^+,\rho^0,\rho^-$ mesons).

{\bf \quad a)} For now, neglect the spatial part $\phi(\bs r)$ of the nucleon wavefunction
$\psi(\bs r, \bs S, \bs T)=\phi(\bs r)\chi_{m_S}\Upsilon_{m_T}$.  Write down
the spin-isospinor $\chi\Upsilon$ for the four different spin and isospin
combinations of N.  Denote the two spin states by $\uparrow$ and $\downarrow$,
and the isospin up and down states by p and n, respectively.

{\bf \quad b)} Construct all 10 symmetric and 6 antisymmetric NN wavefunctions
by [anti]symmetrizing the 16 combinations of two single-particle
spin-isospinors.  Show that each of these states can be written as a linear
combination of states with definite total spin and isospin, ie., the product
of a spin singlet or triplet multiplied by an isospin singlet or triplet).
What is the particle exchange symmetry of each?  {\bf [bonus}: How many
completely symmetric and antisymmetric states are there in the NNN system?
In the three-quark system, they are p, n, $\Delta^{-,0,+,++}$.]

{\bf \quad c)} Transform the two-particle spatial wave function
$\phi_1(\bs r_1)\phi_2(\bs r_2)$ into CM-relative coordinates
$\phi_{\bs R}(\bs R)\phi_{\bs r}(\bs r)$ about the center of mass and reduced
mass.  Show that $P_{12}\phi=P_{\bs r}\phi$, where $P_{12}$ is the normal
particle exchange operator and $P_{\bs r}$ is parity (spatial inversion)
acting on the relative coordinate: $\bs r\to-\bs r$.  As free particles, the
two spatial (either particle $\bs r_{1}, \bs r_{2}$ or relative $\bs R,\bs r$)
wavefunctions have definite orbital angular momentum with quantum numbers
$\ell,m_\ell$.  Show that the relative angular wave function
$Y_{\ell m}(\bh r)$ is even[odd] under $P_{\bs r}$ for even[odd] $\ell$, so
that the simplest parity violating amplitudes (that couple even to odd parity)
are S-P ($\ell=0\leftrightarrow 1$) transitions.

{\bf \quad d)} The total wave function $\psi=\phi\,\chi\Upsilon$ must be
antisymmetric under particle exchange $P_{12}$ by the Pauli exclusion
principle.  Mix even/odd combinations of $Y_{\ell m}(\bh r)$ from part c) and
$\chi\Upsilon$ from part b) to construct all S and P states.  Use linear
combinations of these to create eigenfunctions of total angular momentum
$\bs J=\bs L + \bs S$.  Denote each state in spectroscopic notation
${}^{2S+1}L_J (T)$, where $S$,$L$,$J$, and [$T$] are the total [iso]angular
momentum quantum numbers of the combined NN system.  Use conservation of
total angular momentum $\bs J$, to show there are only five allowed S-P
transition amplitudes: $\lambda_s (\Delta T = 0,1,2)$, $\lambda_t (T=0)$, and
$\rho_t (T=0\leftrightarrow 1)$, as illustrated below.

\begin{center}
  \includegraphics[height=1in]{H15_fig1}
  \hspace{1in}
  \includegraphics[height=1in]{H15_fig2}
\end{center}

{\bf \quad e) [bonus]} In the DDH meson exchange model [Desplanques, Donoghue,
Holstein, Ann. Phys. {\bf 124}, 449 (1980)], the spin and isospin dependence
of the parity violating NN interaction is characterized through the exchange
of physical mesons ($\pi$, $\rho$, $\omega$), which transmit the force,
exchanging not only energy, momentum, spin and isospin also.  Ignoring the
lightest meson, the long-range pion $\pi$, correlate the five Danilov elastic
transition amplitudes with the five meson exchange amplitudes
$h_\rho^{\Delta T=0,1,2}$ and $h_\omega^{\Delta T=0,1}$.

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