Role of Angular Momentum in Defining Vector Operator ##\mathbf{V}##

[blue_leaf77]

A vector operator $\mathbf{V}$ is defined as one satisfying the following property:

$[V_i,J_j] = i\hbar \epsilon_{ijk}V_k$

where $\mathbf{J}$ is an angular momentum operator. My question is what is the role of $\mathbf{J}$, does it have to be the total angular momentum from all angular momenta appearing in the system, i.e. if we don't consider spin then $\mathbf{J}=\mathbf{L}$, if we consider it then $\mathbf{J} = \mathbf{L}+\mathbf{S}$?

 

[DrDu]

Yes, exactly.

 

[blue_leaf77]

But using L instead of the total angular momentum J even in the presence of spin doesn't prevent the relation $[x_i,L_j] = i\hbar \epsilon_{ijk}x_k$ to prevail.

 

[DrDu]

Of course, but if you take V=S or V=J, you will need J=L+S to rotate also the spin part.