![]() Nevertheless, the general meaning of the energy-time principle is that a quantum state that exists for only a short time cannot have a definite energy. 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.For technical reasons beyond this discussion. Uncertainty principles like Heisenberg uncertainty inequality and qualitative uncertainty principle have been investigated for the. I commend to you my “An algebraic approach to Koopman classical mechanics”, in Annals of Physics 2020, on arXiv, and highlighted by Ann.Phys. 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.Ī 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. 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. ![]() 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. 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 =1 and its representations the Heisenberg Uncertainty Principle Hilbert spaces and all that, for that done more properly than a physicist such as I am needs.] If that’s only classical, then noncommutativity is classical. 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. ![]() ![]() 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. 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. 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. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |