Addition of angular momentum
= Addition of angular momentum
{wiki}
For two subsystems, the total operator is $\mathbf J=\mathbf J_1\otimes I+I\otimes\mathbf J_2$, and $J_\pm=J_{1\pm}+J_{2\pm}$. The coupled basis diagonalizes $\mathbf J^2$ and $J_z$.