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Intuitively I've been able to understand a Fourier transform a change-of-basis formula - you're basically moving from position to momentum basis or from time to frequency - but what does it mean that these spaces are 'conjugate' to each other? Does this have to do with them being complete bases?

A related question comes from considering the electric field generated by a travelling electron, $\textbf{E}(\textbf{r},t)$. If we consider sending $\textbf{E}$ to position-frequency space $\tilde{\textbf{E}}(\textbf{r},\omega)$, I find it weird that there is no longer a time dependency. Have we 'smeared' the electron across its trajectory and computed some quasi-average electric field? Is there some other interpretation that might make more sense?

Qmechanic
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alexvas
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1 Answers1

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I quess that thiss follows up on a basic property of conjugation : it is an involution. The "double fourier transform" gives an identity on the level of functions (up to parity). Inverse Fourier transform is the same as fourier transform (up to a minus sign). This justifies using word "conjugated".

The second part I do not understand: what is so different in fact that if you exchange time variable by FT with the frequency, there is no longer a time variable?

$t$ does not appear in $E(r,\omega)$ this is just an expansion coefficient in time dependent basis. You have moved the time dependence from the coefficients $E(r,t)$ to the basis functions $exp(i \omega t).$