Solution
= Solution
On the tensor-product Hilbert space, <addition of angular momentum> gives
$$
\mathbf J=\mathbf J_1\otimes I_2+I_1\otimes\mathbf J_2.
$$
Suppressing the identity operators, its components and <spin ladder operator>[ladder operators] are
$$
J_x=J_{1x}+J_{2x},
\quad
J_y=J_{1y}+J_{2y},
\quad
J_z=J_{1z}+J_{2z},
$$
$$
J_\pm=J_x\pm iJ_y=J_{1\pm}+J_{2\pm}.
$$