[concreteblock]

I may have missed this, but doesn't this just raise the question:

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?

[gyodx]

The Fourier transform of the position wave for a particle yields the space frequency wave of the particle in the same way that the Fourier transform of a picture gives you the spatial frequency of that picture.

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.

[concreteblock]

First part makes sense to me. But second part doesn't. I remember learning and successfully using the p=h* f/c formula in high school physics, but what is the justification for this?

And if the formula only holds for photons, why can we say that frequncy = constant * momentum for other particles?

[dirkt]

> Why is the fourier transform of position equal to momentum?

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.

[goldenkey]

Energy and time are conjugate variables, not energy and position. And as we would expect, energy is the fourier transform of phase. This is why E=hf. Frequency just measures the rate of phase change akin to how momentum measures the rate of position change. As far as current knowledge goes, phase is not an observable that we can measure. Phase shift is actually a gauge symmetry in electromagnetism.