Scalar operators and commutators

Question

Given a scalar operator $S$ and vector operators $V_1, V_2$, show that the commutator $[S,V_1\times V_2]= [S,V_1]\times V_2+V_1\times [S,V_2]$. I don't quite understand what a scalar operator is. But the question would be trivial if by scalar it means that, for example, $V_1S\times V_2=V_1\times SV_2$. Thanks in advance.

Answer 1

Since this probably arises in the context of a physics problem, a scalar operator is an operator that transforms as a scalar under symmetry operations (i.e. changes of frame of reference). Whereas a vector operator will transform as a vector.

But I think that as far as this problem is concerned, you only need to know that a vector operator has three components, so if $V_1=(V_{1,x},V_{1,y},V_{1,z})$ and likewise for $V_2$, such that

$V_1 \times V_2 = \left(V_{1,y}V_{2,z} - V_{1,z}V_{2,y} \; ; \; V_{1,z}V_{2,x} - V_{1,x}V_{2,z} \; ; \; V_{1,x}V_{2,y} - V_{1,y}V_{2,x} \right) \; $

is the outer vector product.

On the other hand, a scalar operator just has one component, the scalar operator itself.

The rest of the exercise just amounts to checking out that the right hand side expression is indeed equal to the left hand side expression, which you can verify by simply working them out separately. I'll make an edit later if you're still struggling.