Consider two arbitrary operators $\hat{A}(\hat{R})$ and $\hat{B}(\hat{R})$ that are both functions of the position operator $\hat{R}$ (and do not depend on any other operators). Does it follow that $$[\hat{A},\hat{B}] = 0?$$ And if so, then why (what is the proof)?
The context of my question is the derivation of the Goeppert-Meyer transformation (from the velocity-gauge of the minimal coupling Hamiltonian to the length-gauge Hamiltonian). There, the following transformation is used: $$\hat{T} = exp(-\frac{i}{\bar{h}}q\hat{R}\cdot\hat{A}(t)).$$ Then, it is given that $\hat{T}$ commutes with the potential energy operator $\hat{V}(\hat{R})$, such that $$[\hat{T}(\hat{R}),\hat{V}(\hat{R})] = 0.$$ The only justification given for this is that both operators depend on $\hat{R}$.
