The recoupling of two subsystems [math]\psi[/math] with angular momenta [math]j_1[/math] and [math]j_2[/math] to a new system[math] \Psi[/math] with total angular momentum [math]J[/math] is written as
[math]\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}[/math] = expansion of the systems total angular momentum in terms of the uncoupled original basis states of each individual constituent
[math]C^{j_1,j_2,J}_{m_1,m_2,M}[/math] : Clebsch-Gordon Coefficient
[math]C^{1,\frac{1}{2},\frac{3}{2}}_{1,\frac{1}{2},\frac{3}{2}}=1[/math]
[math]C^{1,\;\;\frac{1}{2},\frac{3}{2}}_{1,-\frac{1}{2},\frac{1}{2}}= \frac{1}{\sqrt{3}}[/math]
[math]C^{1,\;\;\frac{1}{2},\frac{3}{2}}_{0,\frac{1}{2},\frac{1}{2}}= \frac{2}{\sqrt{3}}[/math]
[math]C^{1,\;\;\frac{1}{2},\frac{1}{2}}_{1,-\frac{1}{2},\frac{1}{2}}= \frac{2}{\sqrt{3}}[/math]
[math]C^{1,\;\;\frac{1}{2},\frac{1}{2}}_{0,\frac{1}{2},\frac{1}{2}}= -\frac{1}{\sqrt{3}}[/math]
[math]\sigma \propto |M_{if}|^2[/math]
[math]M_{fi} = \lt \Psi_f | H_{int} | \Psi_i\gt [/math]
[math]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[/math]