I've heard from a lot of people that the reason momentum and position have an uncertainty relation is because of the Fourier Transform. But is this in any way the case?
If it were I would expect all conjugate quantities related by the canonical commutation relation to be Fourier transforms, but I don't think is true. in fact, I'm not sure how to write the eigenstates of an arbitrary operator as a function that I could Fourier transform.
So if it is the case that it is just, somehow, a coincidence that momentum and position are Fourier transforms of each other, which also coincidentally implies an uncertainty relation, could someone elaborate on this "coincidence"?
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