Fourier transform as (continuous) change of basis

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SUMMARY

The discussion centers on the necessity of changing basis in quantum mechanics through the Fourier transform, specifically transitioning between position and momentum representations. It highlights that the momentum basis serves as an eigen-basis, facilitating easier calculations of time evolution due to the nature of plane waves as eigenstates of propagation. The conversation also touches on the implications of the Heisenberg Uncertainty Principle (HUP) and the topological structure of Hilbert spaces, clarifying that changing basis is not always required if the operator is known in the initial basis.

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TrickyDicky
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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?
 
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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.
 
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