# Product of non-commuting operators

I want to expand the product: $$\left(\hat{A}_{1}+\hat{A}_{2}\right)\left(\hat{B}_{1}+\hat{B}_{2}\right)$$ $$\hat{A}_1$$ and $$\hat{B}_1$$ are operators both working on the same particle, and do not mutually commute, whereas $$\hat{A}_2$$ and $$\hat{B}_2$$ work on a different particle. How can I expand this product?

$$(A_1 + A_2)(B_1 + B_2) = A_1 B_1 + A_1 B_2 + A_2 B_1 + A_2 B_2$$
But because you can commute the operators acting on different particles in this case (that's not always true - with fermions, swapping operators picks up a minus sign, $$A_1 B_2 = - B_2 A_1$$), the order of the two middle terms in that expansion doesn't matter. So you could write $$A_1 B_2 + B_1 A_2$$ if you'd like operators on particle 1 to always be written before those acting on particle 2.