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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?

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You'd expand it just like you would a normal product - you just have to pay attention to the ordering of operators in the final product, in the case that they don't commute. Commutation only matters for multiplication, not arithmetic.

For your example, you'd get (dropping the hats, and keeping the order consistent):

$$(A_1 + A_2)(B_1 + B_2) = A_1 B_1 + A_1 B_2 + A_2 B_1 + A_2 B_2$$

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.

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Non commuting operators meant one need to be careful about switching the position of operators.

However, in physics, most of the operators were still Distributive. So just expand the operator out of the parenthesis without missing up the order, and it's done.

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