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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 521 \\
    Homework \#12, Rev. B, due Wednesday, 2016-03-02 }
\end{center}

\vspace{.3 in}

{\bf 0.} Griffiths [2ed]
Ch.~5 \#2,  %p203 reduced mass
      \#7,  %p210 Slater det
      \#15, %p223 energy of Fermi gas
      \#17, %p223 bulk modulus
      \#35, %p246 white dwarf
      \#36% %p247 black hole
.

{\bf 1. Davilov Parameters} [Danilov, Phys. Lett. {\bf 35B}, 579 (1971)] The
key properties of parity violation in elastic nn, np, and pp scattering
(neutrons and protons) can be deduced from basic symmetry and conservation
principles studied in class.

The neutron and proton are considered as 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 say the
nucleon has {\it isospin} $I=\frac12$.  The proton ($I_3=\frac12$) and neutron
($I_3=-\frac12$) are both represented by two-component {\it isospinors}
$\Upsilon=\scriptstyle{\pmatrix{1\cr 0}}$ and $\scriptstyle{\pmatrix{0\cr
    1}}$, respectively.  Their operators are Pauli matrices writen $\bs\tau
=(\tau_1,\tau_2,\tau_3)$ as opposed to
$\bs\sigma=(\sigma_x,\sigma_y,\sigma_z)$.  Because spinors and isospinors live
in different spaces, we use the symbols $I$ and $\tau$ for isospin, and use
generic indices $(1,2,3)$ instead of spatial coordinates $(x,y,z)$.  Addition
of isospin works exactly the same as spin---in particular, two-particle
systems have the symmetry of an $I=0$ singlet (for example, the $\omega$
meson) and $I=1$ triplet (the $\rho^+,\rho^0,\rho^-$ mesons, which are
identical except for charge).

{\bf \quad a)} For now, neglect the spatial part of the nucleon wavefunction
$\psi(\bs r, \bs S, \bs I)=\phi(\bs r)\chi_{m_S}\Upsilon_{m_I}$.  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 two isospin states by u and d.

{\bf \quad b)} Construct all 10 symmetric and 6 antisymmetric NN wavefunctions
by [anti]symmetrizing the 16 combinations of two single-particle
spin-isospinors.  Write them as products of singlet and triplet [iso]spin
states.  What is the particle exchange symmetry of each?  How many symmetric
and antisymmetric states are there in the NNN system (eg. p=uud, n=udd)?

{\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
wavefunctions (either particle 1,2 or $\bs R,\bs r$) have definite orbital
angular momentum with quantum numbers $\ell,m_\ell$.  Disregarding the radial
dependence (in low-energy scattering), show that the relative wave function
$Y_{\ell m}(\bh r)$ is even[odd] under $P_{\bs r}$ for even[odd] $\ell$, so
the simplest parity violating amplitudes are S-P ($\ell=0\to1$) 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 ${}^{2S+1}L_J
(I)$, where $S$,$L$,$J$, and [$I$] are the [iso]angular momenta of the
combined NN system.

Using conservation of total angular momentum, we find there are only five
allowed S-P transition amplitudes: $\lambda_s (\Delta I = 0,1,2)$, $\lambda_t
(I=0)$, and $\rho_t (I=0\to1)$, as illustrated below.

\begin{center}
\includegraphics[width=4in]{H12_fig1}
\end{center}

\newpage

{\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 by the properties of
physical mesons ($\pi$, $\rho$, $\omega$), which transmit the force.  Ignoring
the lightest meson, the long-range pion $\pi$, correlate the five Danilov
elastic transition amplitudes with the five meson exchange amplitudes
$h_\rho^{\Delta I=0,1,2}$ and $h_\omega^{\Delta I=0,1}$.

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