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I'm going through my quantum mechanics course and have become a little stuck on something very basic, how can we prove that the operator $\hat{X}$ commutes with $\hat{Y}$?

I can simply get to the step where $$[\hat{X},\hat{Y}] = \hat{X}\hat{Y} - \hat{Y}\hat{X}$$

But can't see how I can end up with $[\hat{X},\hat{Y}] = 0$ which is what is necessary for them to commute?

Any help would be massively appreciated!

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They are postulated to commute. You cannot prove it, because it is one of the axioms of the theory. The reason for this axiom is that the classical coordinates $x(t)$ and $y(t)$ have vanishing Poisson bracket: $$\{x,y\}=0$$ (which I invite you to prove) and therefore the quantisation rule is to take $[X,Y]=0$. Again, this can be motivated but not proven. It is a postulate of QM.

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