[femto]

You can also think about the Fourier Transform in terms of its physical properties.

For example, the Fourier transform is behind quantum uncertainty (dp.dx>h). Think of it this way: the inverse Fourier transform of a frequency impulse (zero extent) is a sine wave of infinite duration. Truncate the infinite sine wave and its spectrum ceases being an impulse, broadening into the shape of the windowing function used to truncate the sine wave. That is, an attempt to constrain/define time leads to a broadening in frequency, and vice versa. The uncertainty principle naturally arises from using the Fourier Transform in an environment where "you can't have infinities".

This is true of any two variables which are related by a Fourier Transform. Yes, position and momentum are related by a Fourier transform (as are energy and time).

The thinking also works for the other extreme: if you consider how a time impulse (zero extent) related to its Fourier transform, a flat spectrum of infinite extent on the frequency axis.

[jjoonathan]

http://www.youtube.com/watch?v=Znby3t3AS5s

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Uncertainty is not so odd as long as we're aware

position and momentum are a Fourier transform pair

So anything that tightens our precision on the one

means certainty about the other value gets undone

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Still, I'm not sure I approve of your suggestion to use this physical property of the universe as a basis for intuition. The reasons why quantum mechanics "works" are far, far more difficult to understand, internalize, and accept than the concepts behind the DFT which only really requires an understanding of first-semester linear algebra. In other words, I expect that the set of people who understand QM at this level but do not understand the DFT is nearly empty.

If you actually meant to go the other way (use the properties of the FT/DFT to gain an understanding of QM) then I completely agree with everything you said, of course.

[VLM]

"That is, an attempt to constrain/define time leads to a broadening in frequency, and vice versa."

That is also a really well phrased one line point of commonality to talk to a telecom / RF / EE type person about communications bandwidth theory. If you just wedge in signal to noise ratio / bit error rate, and look at what you phrase "time definition" and "frequency broadening" in the right way, then you pretty much have Shannons famous paper.

The FT shows up all over the place in science and anywhere you find it, you can analogize it into a totally different field of study. One "design pattern" hundreds of "implementations".

[femto]

That would be a nice topic for a website: to encapsulate and present the duality present in any number of fields.

A major barrier to understanding any new field is being able to strip away the jargon and recast the ideas into a familiar form. One can envisage website, where you tick the field(s) "A" that you want to learn about, tick the field(s) "B" that you already know about, and it recasts field(s) "A" in terms of the terminology of field(s) "B".

Make it have a "plug-in" structure, so supporting a new field is a matter of writing a mapping of field specific concepts to a set of "design patterns". Make it open source, so experts can jump in and contribute mappings for a wide variety of fields. Getting fancy, mappings could be written in terms of any two fields (so the expert does not have to learn a set of web-site specific design patterns), and the software could construct a graph to allow conversions between arbitrary fields.