View Single Post
P: 32
Commutators of vector operators

I would think that you should define $[{\bf{\hat S}},{\bf{\hat T}}] = {\bf{\hat S}} \cdot {\bf{\hat T}} - {\bf{\hat T}} \cdot {\bf{\hat S}}$, and therefore start your derivation with
$$[{\bf{\hat A}} \cdot {\bf{\hat B}},{\bf{\hat C}}] = ({\bf{\hat A}} \cdot {\bf{\hat B}}){\bf{\hat C}} - {\bf{\hat C}}({\bf{\hat A}} \cdot {\bf{\hat B}}).$$
But from there, I'm not sure how you can safely proceed, if you're being rigorous with your dots and parens. For instance — and correct me if I'm wrong on this — but I don't think $({\bf{\hat A}} \cdot {\bf{\hat C}}){\bf{\hat B}}$ is equal to ${\bf{\hat A}}({\bf{\hat C}} \cdot {\bf{\hat B}})$, so your next step seems iffy.