Angular Momentum Operator - Solved Problems

Problem 1: Basic Commutator

Question: Compute the commutator [ L_x, L_y ].

Solution:

From the standard angular momentum commutation relations:

[ L_x, L_y ] = iħ L_z

Answer: [ L_x, L_y ] = iħ L_z

Question: Evaluate [ L², L_x ], where L² = L_x² + L_y² + L_z².

Solution:

Since L² is a scalar operator under rotations, it commutes with all components of L:

[ L², L_x ] = 0

Answer: [ L², L_x ] = 0

Question: Find [ L_z, L_± ], where L_± = L_x ± iL_y.

Solution:

Use known commutators:

Then:

[ L_z, L_± ] = [ L_z, L_x ± iL_y ] = ±ħ L_±

Answer: [ L_z, L_± ] = ±ħ L_±