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Subsections

A. Notation and Phase Conventions

Spherical Harmonics

The phase convention used here is
$\displaystyle Y_L^m ( \theta ,\phi ) =
\sqrt {{ 2L+1 \over 4 \pi} {(L-m)! \over (L+m)!} } (-1)^m e^{im\phi}
P_L^m ( \cos \theta )$      

for $ m \geq 0 $, and YL-m = (-1)m YLm * to give negative m values.

Angular Momentum Coupling Coefficients

The notation $\langle \ell_1 m_1 \ell_2 m_2 \vert LM\rangle $ has been used for the Clebsch-Gordon coupling coefficient for coupling states $\ell_1 m_1 $ and $\ell_2 m_2 $ together to form LM. The

$\displaystyle \left ( \begin{array}{ccc}a&b&c\\  \alpha&\beta&\gamma \end{array...
...1)^{a - b -\gamma} \over \hat c}
\langle a\alpha b \beta \vert c -\gamma\rangle$      

represents the Wigner 3-j symbol, and
$\displaystyle \hat x \equiv \sqrt{2x+1} .$      

The 9-j coupling coefficient is used in two forms related by
$\displaystyle \left \{ \begin{array}{ccc}a&b&c\\  d&e&f\\  g&h&i \end{array}\ri...
...hat h ~
\left ( \begin{array}{ccc}a&b&c\\  d&e&f\\  g&h&i \end{array}\right ) .$      

The binomial coefficient is
$\displaystyle \left ( \begin{array}{c}x \\  y \end{array}\right ) = {x! \over y!(x-y)!} .$      


next up previous
Next: B. Coupled Channels Codes Up: Coupled Reaction Channels Calculations Previous: Bibliography
Prof Ian Thompson 2004-05-09