Forest AngMomRecoupling

The recoupling of two subsystems $\psi$ with angular momenta $j_1$ and $j_2$ to a new system$\Psi$ with total angular momentum $J$ is written as

$\Psi^{J}_{M} = \sum_{m_1,m_2} C^{j_1,j_2,J}_{m_1,m_2,M} \psi^{j_1}_{m_1} \psi^{j_2}_{m_2}$

$C^{j_1,j_2,J}_{m_1,m_2,M}$ : Clebsch-Gordon Coefficient

$C^{1,\frac{1}{2},\frac{3}{2}}_{1,\frac{1}{2},\frac{3}{2}}=1$

$C^{1,\;\;\frac{1}{2},\frac{3}{2}}_{1,-\frac{1}{2},\frac{1}{2}}= \frac{1}{\sqrt{3}}$

$C^{1,\;\;\frac{1}{2},\frac{1}{2}}_{1,-\frac{1}{2},\frac{1}{2}}= 1$

$\sigma \propto |M_{if}|^2$

$M_{fi} = \lt \Psi_f | H_{int} | \Psi_i\gt$

$A = \frac{\sigma_{\frac{1}{2}} - \sigma_{\frac{3}{2}}}{\sigma_{\frac{1}{2}} + \sigma_{\frac{3}{2}}} = \frac{\frac{1}{3} - 1}{\frac{1}{3} + 1} = -1/2$