Prove that $J_z=J_{1z}+J_{2z}$

Question

This is probably a standard question but I couldn't found it anywhere online, so I thought it might be a good ideal to add it in Physics exchange.

Modern Quantum Mechanics Second Edition J.J. Sakurai Jim Napolitano Equation 3.8.36

$$(J_z-J_{1z}-J_{2z})|j_1,j_2; jm\rangle =0$$

However, the textbook doesn't exactly explained that where did this expression come from, i.e. although that $J\equiv J_1\otimes 1+1\otimes J_2$, it's not necessarily such that $J_z= J_{1z}\otimes 1+1\otimes J_{2z}$. Especially, no relationship was given in the textbook such that $J_z-J_{1z}-J_{2z}$ was understood.

Could you show that why $(J_z-J_{1z}-J_{2z})|j_1,j_2; jm\rangle =0$ ?

Answer 1

It may not be obvious, but it is definitional. For both J1 and J2, we define the z axis to point in the same direction (is it's an external axis) and from your definition of J it follows that $J_z=J_{z1}+J_{z2}$. Does that help?