Fourier transform as (continuous) change of basis

[TrickyDicky]

Trying not to get too confused with this but I'm not clear about switching from coordinate representation to momentum representation and back by changing basis thru the Fourier transform.
My concern is: why do we need to change basis? One would naively think that being in a Hilbert space where global sets of basis are available one shouldn't be required to change basis when performing a linear transformation.

I guess this is related to the noncommuting of x and p (HUP), and the Hilbert infinite -dimensional space topological structure but how exactly?

 

[waveandmatter]

I am not completely sure if i understand the question correctly.

One likes to change the basis from the position to the momentum basis, because the momentum basis is an eigen-basis ie the plane waves are eigenstates of propagation and it is thus easy to calculate their time-evolution.

So we change basis, propagate, and change back.

Edit: In the general case you do not have to change the basis in order to apply any operator, provided you know what the operator looks like in the basis you start with.