Fourier analysis and the Uncertainty Principle

As I mentioned here, I was one of many contributors to the Cosmic Shambles “Nine Lessons and Carols for Socially Distanced People” extravaganza just before Christmas. My bit was a short physics chat trying to give some insight into how the uncertainty principle arises in quantum physics, from the point of view of waves, beats, and Fourier analysis. Immediately afterwards, Steve Pretty performed a musical piece which echoed some of those ideas, particularly things drifting in and out of phase.

Trent at Cosmic Shambles has been making various segments of the whole (25 hour, in fact, Robin always overruns) show available to watch in your own time. He has just added my bit (and Steve’s, and also Zahara, Diesel, Pete Etchelss and Dave Coplin). Since I was on at about 4:30 am GMT this might increase the audience slightly.

The show was also being illustrated live by Matt Kemp, and the illustrations (including one with me and Steve) are being auctioned off for charity here. (Deadline 8 Jan.)

Hope some of you enjoy it, and the other goodies available from the Cosmic Shambles bunch. If so, then please support them if you can via their Patreon – they do great stuff and times are tough with all live gigs cancelled.

About Jon Butterworth

UCL Physics prof, works on LHC, writes (books, Cosmic Shambles and elsewhere). Citizen of England, UK, Europe & Nowhere, apparently.
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3 Responses to Fourier analysis and the Uncertainty Principle

  1. I’ll take a look Peter, thanks. This was supposed to be a 7min thing for the general viewer so obviously isn’t complete. It says nothing about statistics or issues around measurement in QM since, as I’m sure you’re aware, the whole wave/Fourier analysis discussion is purely classical. I think it still gives some valid and hopefully interesting insight though, even if it’s one you were already familiar with.

    • Peter Morgan says:

      I suppose most people must think “the whole wave/Fourier analysis discussion is purely classical”, so it will be an uphill battle to persuade anyone otherwise, but it can be much more. We can even construct an isomorphism between a random field and the quantized electromagnetic field, instead of working with quantization (and its quasi-inverse, the correspondence principle,) if we get our heads around Koopman’s formalism. There are tradeoffs, unsurprisingly, but the tradeoffs are different from those we encounter when we use de Broglie-Bohm-type approaches and I think we can learn something from this different perspective even if we don’t use it. It’s quite sad that Koopman constructed a Hilbert space formalism for classical mechanics in 1931 and it’s only in the last 20 years that a small group of people have started to try to really run with it.
      Mathematically, without any mention of quantum or classical, the fourier transform is a transformation from a basis of improper eigenfunctions of the operator “multiply by x” to a basis of improper eigenfunctions of the operator “differentiate with respect to x”, and for those operators we have the Heisenberg algebra [∂/∂x,x]=1 and its representations; the Heisenberg Uncertainty Principle; Hilbert spaces; and all that [one can see the Kisil reference, [17], for that done more properly than a physicist such as I am needs.] If that’s only classical, then noncommutativity is classical.
      No worries if you decide I haven’t closed the case and don’t reply: most people seem to think that I haven’t and I certainly think there are gaps, so the question now may be more whether someone else will tell a fuller story than I am capable of.
      It’s great that people keep doing the wave/Fourier analysis thing, because each time there’s a different vibe and eventually people will find ways to do it better, making slightly more contact between signal analysis done classically and done QMically.

  2. Peter Morgan says:

    I suppose you must know that there is a criticism of this kind of account of the relationship between fourier analysis and the Heisenberg Uncertainty Principle [which is rolled out quite often on YouTube, four examples being: 3Blue1Brown,; Minute Physics,; Sixty Symbols,; The Science Asylum,, that they do not include anything about statistics/probability and that they do not substantively enough address the question of measurement incompatibility/noncommutativity.
    A somewhat better account, with statistics and probability front and center (but not yet, as far as I know, done nicely on YouTube, which is a challenge I’d like to see someone take up,) can be given if we work with Koopman’s Hilbert space formalism for classical mechanics. I commend to you my “An algebraic approach to Koopman classical mechanics”, in Annals of Physics 2020, on arXiv, and highlighted by Ann.Phys. as I’m hopeful part of this will make its way into the undergraduate syllabus fairly soon because of its very pragmatic focus on QM as a signal analysis formalism, which makes it possible to make QM seem much less weird and closer to CM than if we focus on QM as a particle properties formalism.

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