A Fourier transform is the decomposition of a position space function into a basis of plane waves, each of which has a well defined momentum.
$$
f(x) \sim \int \text{d}p\; F(p) e^{\text{i}px}
$$
This relies on the quantum mechanical idea that waves can have a well defined momentum.
The Fourier transform is a well defined mathematical operation. Transforming basis from position to momentum in wave mechanics has the same form as a Fourier transform.
I think there's something really deep behind that because position and momentum appear as conjugate variables in many situations. But at the very least, it has to be that way given the axioms of quantum mechanics.
Maybe one insight is that a plane wave (which you could consider a basic building block of a wave mechanical theory) has momentum and position in a conjugate relationship i.e. $e^{ipx}$.
Actually you should consider exp(ikx). Then connect wave number and momentum using quantum mechanics.
There's a lot of empirical evidence motivating the ideas of wave mechanics. In the end, our physical theories are motivated by the empirical evidence we have experience with.
I suspect you're trying to find an answer to an ill-posed question...
I think the fundamental question here is why we can identify momentum with spatial frequency.
"The terms "momentum" (symbol p, also a vector) and "wavevector" are used interchangeably due to the De Broglie relation p = ħk, meaning they are equivalent up to proportionality"
it says: "The momentum representation of a wave function is very closely related to the Fourier transform and the concept of frequency domain. Since a quantum mechanical particle has a frequency proportional to the momentum (de Broglie's equation given above)
, describing the particle as a sum of its momentum components is equivalent to describing it as a sum of frequency components (i.e. a Fourier transform).[3] This becomes clear when we ask ourselves how we can transform from one representation to another."
Discussion between bdforbes and P.A.M
Discussion between bdforbes and P.A.M