A. Notation and Phase Conventions

Spherical Harmonics

The phase convention used here is

$$Y_L^m ( \theta ,\phi ) = \sqrt {{ 2L+1 \over 4 \pi} {(L-m)! \over (L+m)!} } (-1)^m e^{im\phi} P_L^m ( \cos \theta )$$

for $m \geq 0$, and Y_L^-m = (-1)^m Y_L^{m *} 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

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

represents the Wigner 3-j symbol, and

$$\hat x \equiv \sqrt{2x+1} .$$

The 9-j coupling coefficient is used in two forms related by

$$\left \{ \begin{array}{ccc}a&b&c\\ d&e&f\\ g&h&i \end{array}\ri... ...hat h ~ \left ( \begin{array}{ccc}a&b&c\\ d&e&f\\ g&h&i \end{array}\right ) .$$

The binomial coefficient is

$$\left ( \begin{array}{c}x \\ y \end{array}\right ) = {x! \over y!(x-y)!} .$$