Why is the fourier transform of position equal to momentum?
I.e why is position conjugate to momentum?
More generally, why would the fourier transform of an observable be another observable?
We can then relate the spatial frequency of a photon to its momentum by the formula p = h * f / c, where h and c are the Planck constant and the speed of light in a vacuum, respectively. From this we see that the momentum of a photon is a function of frequency, which, from the properties of the Fourier transform, we know to be the conjugate pair of position.
And if the formula only holds for photons, why can we say that frequncy = constant * momentum for other particles?
That comes out of the way you get the probability distribution of the position and the probability distribution of the momentum out of the wave function.
If you do the math, then one turns out to be the fourier transformation of the either. (And of course the article skipped over this, because it needs quite a bit of math).
Of course this then begs the question why there are wave functions, and why you get the position and momentum from the wave function in that particular way. And I guess the only answer I have to that is "that's how quantum mechanics works, and it matches what we observe".
> More generally, why would the fourier transform of an observable be another observable?
Again, that comes out of how you calculate particular observables from the wave function. Like above, if you do this for the energy probability distribution and the time probability distribution, they turn out to be related.
And it's not the case for any observables you can compute. For example, the energy and position are NOT fourier transforms of each other.