What is the physical meaning of commuting in quantum mechanics?

Question

In quantum mechanics, if two observables commute, then perfect knowledge can be gained about both observables simultaneously.

But what does the commutator actually, physically represent?

Like observables correspond to things that can be observed, is there a similar physical meaning or classical analogy for commutators?

Answer 1

I like to think of the commutator as (twice) the "amount of disturbance" the measure of one observable induces on the other. For instance, the commutator of the position and momentum operators $[X, P] = i\hbar$ somehow encodes the uncertainty principle $\Delta x \Delta p \ge \hbar/2$