If the initial state is \(\psi=\alpha_+ \psi_+ +\alpha_- \psi_-\), the effect of a rotation can only be to turn this into \(\psi'=\alpha'_+ \psi_+ +\alpha'_- \psi_-\). \[\begin{aligned} \psi&=(\alpha^...If the initial state is \(\psi=\alpha_+ \psi_+ +\alpha_- \psi_-\), the effect of a rotation can only be to turn this into \(\psi'=\alpha'_+ \psi_+ +\alpha'_- \psi_-\). \[\begin{aligned} \psi&=(\alpha^1_+ \psi^1_+ +\alpha^1_- \psi^1_-) (\alpha^2_+ \psi^2_+ +\alpha^2_- \psi^2_-) \nonumber\\ &= \alpha^1_+\alpha^2_+ \psi^1_+ \psi^2_++ \alpha^1_+\alpha^2_- \psi^1_+ \psi^2_- + \alpha^1_-\alpha^2_+ \psi^1_- \psi^2_+ \alpha^1_-\alpha^2_- \psi^1_- \psi^2_-.\end{aligned} \nonumber \]
