When I learned the Uncertainty Principle, I noticed its similarity with the characteristics of the Fourier transform, but couldn't understand why. This paper says:<p>> In quantum mechanics, the wave function of position is the Fourier transform of the wave function of momentum.<p>That explains it!
The clearest, most intuitive explanation of the Fourier Uncertainty Principle that I have come across is by Grant Sanderson of 3Blue1Brown.<p><a href="https://www.youtube.com/watch?v=MBnnXbOM5S4" rel="nofollow">https://www.youtube.com/watch?v=MBnnXbOM5S4</a>
There is an approachable explanation in [1], chapter 16 ("Duration-bandwidth relationships and the uncertainty principle"), that says that the product of rise-time and bandwidth of a signal must be greater than some minimum.<p>[1] Siebert, W. M. (1986). Circuits, Signals, and Systems. McGraw-Hill.
I remember an undergraduate homework question that was just asking you to calculate the fourier transform of a gaussian of mean 0 and variance a. You get out a gaussian of mean 0 and variance 1/a.<p>I missed the significance of this, until we went over the homework with the TA and pointed out the implications of this result, heisenberg, etc.<p>It was very enjoyable that something I had previously taken as a sort of spooky truth of the quantum universe (Heisenberg's uncertainty principle) was actually just a pretty mechanically apparent consequence of some basic algebra on an EE homework.
>"The most popular use of Fourier uncertainty principles is as a description of the natural <i>tradeoff</i> between the <i>stability</i> and <i>measurability</i> of a system"<p>Related:<p><a href="https://en.wikipedia.org/wiki/Complementarity_(physics)" rel="nofollow">https://en.wikipedia.org/wiki/Complementarity_(physics)</a>
>Uncertainty principles are not formally defined<p>In physics it seem to be pretty straight forward. For instance <a href="https://www.wolframalpha.com/input/?i=uncertainty+principle" rel="nofollow">https://www.wolframalpha.com/input/?i=uncertainty+principle</a>
I have read a lot of documents on ncatlab and other places to try to pin down a coherent model of the physical role of planck's constant in fourier transforms on physical systems. I understand that it often serves as the scale factor for embedding the integers into the reals, but it's not totally clear to me what its role is in physical pontryagin duality/fourier transforms. It's some kind of volume in phase space, but where does that volume come from? For a constant like c, we have the narrative "c is the ratio of unit lengths in time and space", but I have not yet found a good narrative about the meaning of h that works for fourier transforms. Would appreciate any articles on the matter.
The variable n comes out of nowhere in theorem 3.3, and they do not refer to it in the proof itself as far as I can tell. Is this just an editing error (I think the formula 3.4 needs the variable n if f is multidimensional and we are integrating over R^n, but since f is in L^1(R) I'm not sure what it signifies. I am however worried that there's something I'm missing).
At least in some contexts, I never really agreed with calling it "uncertainty"; a frequency cannot exist in less time than the time needed to measure it. You're not really uncertain about it, it does not exist at all. Like looking at a single pixel's color and saying you're uncertain about the picture.
"if i want a good look at big things, i need a big window so i can see as much of them as possible, but if i use a big window, then i don't know where exactly things are in that big window."
Here is my question to those who understand this "paper":<p>How does the discovery described in this paper help engineer something the world has never seen before?<p>As an engineer, I'm always looking for some new thing to make. What does this paper make more possible to make that was less possible to make before?
<a href="https://m.youtube.com/watch?v=D1WfID6kk90" rel="nofollow">https://m.youtube.com/watch?v=D1WfID6kk90</a><p>take a time series dataset like an audio file or stock ticker price over time ... give your self a healthy period of time ... for example a second of broadcast quality audio gives you 44,100 data points spread across that time period stored as information ... importantly this time series audio curve wobbles up and down as it's recorded over time ... in order to justify taking 44,100 audio samples per second (on the X axis) you must balance that by breaking up the granularity of your measurement of the up and down wobble (Y axis) by devoting two bytes (a bit depth of 16 bits) of memory storage per data point which gives you 2 raised to the 16 power distinct gradations of resolution<p>above defines the time domain representation of the one second of audio data ... now feed this dataset into a Fourier transform which will output the same information you started with but now in the frequency domain ... it will give you not 44,100 points in time but instead 44,100 distinct frequencies ... super cool side note you can feed this new frequency domain representation of the dataset into an inverse Fourier transform to rescue back the original time series audio<p>If instead of a second of audio we start with a fraction of that number this reduction of recording duration will compromise the frequency resolution of the data in the frequency domain giving it less granularity hence larger increments to the next frequency