Question

Let A and B be vector operators. This means that they have certain
nontrivial commutation relations with the angular momentum opera￾tors. Use those relations to prove that A·B commutes with Jx, Jy, and
Jz.

Answer 1

We have nontrivial commutation relations. Then consider the following provisions. We have the scalar product of vectors A and B, that is, it will be the following expression: A·B=xAxB+yAyB+zAzB.

Let's look at the switch: [A·B, Jx]=xAxBJx-JxxAxB=0, because the product of X will be a number, and it can be rearranged in any order.

Let's look at the switch: [A·B, Jy]=yAyBJy-JyyAyB=0, because the product of Y will be a number, and it can be rearranged in any order.

Let's look at the switch: [A·B, Jz]=zAzBJz-JzzAzB=0, because the product of X will be a number, and it can be rearranged in any order.

In this way, A·B and Jx, A·B and Jy, A·B and Jz commute.