Q:

A quick question on scalars and commutators.

I just had a quick question about the factoring of scalars within the commutator.
Say we have operators $\hat{A}$ and $\hat{B}$, and scalars $a$ and $b$. If we take the commutator: $$[a\hat{A}, b\hat{B}]$$That is just equivalent to $ab(\hat{A}\hat{B} - \hat{B}\hat{A})$, right? Probably a dumb question, but I just wanted some clarification.

A:

We have 
\begin{align*}
  [\def\A{\hat A}\def\B{\hat B}a\A, b\B] &= (a\A)(b\B) - (b\B)(a\A)\\
   &= ab\A\B - ba\B\A \\
   &= ab(\A\B - \B\A)\\
   &= ab[\A, \B]
\end{align*}
for a commutative ring of scalars. So the answer is: Yes.

