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.(adsbygoogle = window.adsbygoogle || []).push({});

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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# Fourier transform as (continuous) change of basis

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