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!


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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