Angular Momentum Operator

The Angular Momentum Operator might refer to any of:

• Orbital Angular Momentum Operator $L=(L_x,L_y,L_z)$
• Spin Angular Momentum Operator $S=(S_x,S_y,S_z)$
• Total Angular Momentum Operator $J=L+S$.

Properties

All of the above satisfy the Canonical Commutation Relations:

• $[L_i,L_{i+1}]=i\hslash L_{i+2}$
• $[L_i,X_{i+1}]=i\hslash X_{i+2}$
• $[L_i,P_{i+1}]=i\hslash P_{i+2}$ where the indices are taken cyclically on $\{x,y,z\}$. Equivalently, $L\times L=i\hslash L$, where $\times$ is the Cross product.