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From Wikipedia:

The Gibbs Phenomenon is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity. The nth partial sum of the Fourier series has large oscillations near the jump, which might increase the maximum of the partial sum above that of the function itself. The overshoot does not die out as the frequency increases, but approaches a finite limit. This sort of behavior was also observed by experimental physicists, but was believed to be due to imperfections in the measuring devices.

Two questions on this effect:

  1. Does it carry over from Fourier Series into Fourier Transforms? I have done a search on Google, but no immediate answer jumps out at me, apologies if I missed something obvious, or skipped over something too math heavy for me to follow.

  2. If there is a Fourier Transform version of the Gibbs effect, how would that affect, in physical terms, and taking for simplicity (and my chances of understanding any answer), the transformation of a 1D electron wavefunction from the position space representation into it's momentum state equivalent.

EDIT I completely failed to spot this answer: Why no Gibbs Phenomenon in QM, as provided in Qmechanic's comment below. END EDIT

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The Fourier transform of a waveform has to contain an infinite range of frequencies in order to represent the full information content of the original. It is of course possible to create a waveform that only contains a finite range of frequencies (or rather - one for which the high frequencies vanish exponentially). The Gaussian is one such shape - its FT is another Gaussian, and limiting yourself to a few times the width of that Gaussian and reconstructing the wave from those limited frequency components will give no practical error.

The problem arises when there are very high frequency components present - for example, any waveform which is truncated with a top hat function in time domain will have very high frequencies in the Fourier domain.

So the answer to your first question is "yes". As for the second question - maybe the best way to think about this is in terms of the uncertainty principle: the better defined the position of the particle (the more sharply truncated its wavefunction), the greater your uncertainty in the momentum (high frequency components in the Fourier domain). There is no "ringing" as such that I can think of, because there is no way to limit the momentum values that the particle could have when observed.

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  • $\begingroup$ thanks for the quick reply, your answer was very helpful and I need to study more for sure, but I had assumed the problem , if any, was avoided because of the Gaussian, anyway, it's not a problem in practical terms. $\endgroup$ – user81619 Jun 22 '15 at 18:48

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