Quantum operators — UPSC Physics Optional CSE 2022 Paper II

The raising (J_+) and lowering (J_-) operators are defined by J_+=J_x+iJ_y and J_-=J_x-iJ_y respectively. Prove the following identities:

(i) [J_z,J_\pm]=\pm\hbar J_\pm

(ii) J_-J_+=J^2-J_z^2-\hbar J_z