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]
[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]