I have to ask a question that looks very simple, and perhaps is very simple, but for some reason I can't answer it in a way that I'm satisfied with. If A and B are bounded self-adjoint operators that correspond to two different equivalence classes of measuring devices in the real world, and [A,B]≠0, then what device or procedure in the real world does AB correspond to? I thought it would be something like "measure B first and write down the result, then measure A and multiply the two results". But I see several reasons why that can't be correct: 1) The average result of that procedure isn't equal to the expectation value of AB. 2) If the procedure ends with a measurement of A, the system should end up in an eigenstate of A, but a measurement of AB should leave the system in an eigenstate of AB. It's actually pretty embarassing that I don't know this.