The Commutators of the Angular Momentum Operators

however, the square of the angular momentum vector commutes with all the components.

This will give us the operators we need to label states in 3D central potentials.

Lets just compute the commutator.

Since there is no difference between , and , we can generalize this to

where is the completely antisymmetric tensor and we assume a sum over repeated indices.

The tensor is equal to 1 for cyclic permutations of 123, equal to -1 for anti-cyclic permutations, and equal to zero if any index is repeated. It is commonly used for a cross product. For example, if

then

where we again assume a sum over repeated indices.

Now lets compute commutators of the operator.

We can generalize this to

commutes with every component of .

Jim Branson 2013-04-22