

\subsection{\bf Density Matrix Element Example---II}

\begin{slide}
\begin{itemize}
\item The $x_i$ and $y_i$ in this example correspond to expressing
angular decay
\begin{eqnarray*}
W(\theta , \varphi ) & = & 1+\sum^3_{i=1} x_i X_i (\theta, \varphi ), 
\hspace{2em} \mbox{or} \nonumber \\
W(\theta , \varphi ) & = & 1+\sum^3_{i=1} y_i Y_i (\theta, \varphi )
\end{eqnarray*}

\item where $y_i$ are linear combinations of the $x_i$ corresponding to
the $X_i (\theta , \varphi )$ being linear combinations of the $Y_i
(\theta , \varphi )$.

\item The details are different, but you can see the essential points
from working through the choice
\begin{eqnarray*}
X_1 = \cos\theta\phantom{^2} & & Y_1 = P_1 ( \cos\theta ) \\
X_2 = \cos^2\theta & & Y_2 = P_2 ( \cos\theta ) \\
X_3 = \cos^3\theta & & Y_3 = P_3 ( \cos\theta )
\end{eqnarray*}
\item where $P_l$ are Legendre Polynomials.

\item This calculation is left as an exercise for the reader.
\end{itemize}
\end{slide}

