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Schrödinger equation emerges mathematically from classical mechanics (2012)
- danbruc 3y agoNot a physicist and only read the abstract, but that does not sound right. One frequently hears that one can recover classical mechanics from quantum mechanics in the limit of Planck's constant becoming zero but not even that seems to be [completely] true [1] as a quick search shows. The other way around, as this paper claims, seems even more unlikely. As they mention a couple of mathematical tools that went into this analysis, maybe they accidentally introduced the relevant differences between classical and quantum mechanics with them. Or maybe just reading the abstract is not good enough and they claim something different than what I think they claim after reading the abstract. If they actually claim that one can recover important aspects of quantum mechanics from classical mechanics without introducing additional concepts or assumptions, then I am highly skeptical. [1] https://physics.stackexchange.com/questions/32112/classical-limit-of-quantum-mechanics https://physics.stackexchange.com/questions/32112/classical-...
- alecst 3y agoSchrodinger's reasoning was remarkable. The high-level description of classical mechanics was formulated by Hamilton, who was starting from optics. He saw a mathematical analogy between the equations for light and the equations for mechanics. The principle of least time (Fermat's principle) for light became the principle of least action for mechanics. But the principle of least time does not predict diffraction, just the geometric path of a light ray. It fails when the wavelength of the light is large compared to whatever it's interacting with. At the time, the equations for mechanics were clearly failing for small systems. Here's where Schrodinger had his incredible insight: what if mechanics broke in the same way as optics? Could matter itself display a kind of "diffraction" when its "wavelength" was similar in size to the objects it was interacting with? Could this explain the success of de Broglie's work, which treated small particles like waves? Guided by that, he was able to add "diffraction" to the equations of matter and come up with the Schrodinger equation. It's worth reading the original paper if you have a physics background -- probably grad-level (just being realistic.) I've been wanting to write a blog post about this because the physics lore is something like "Schrodinger just made a really good guess" but that totally undersells the depth of his reasoning.
- ahartmetz 3y agoYeah, I've been taught that Schrödinger basically made a guess as well... Very interesting how it really happened.
- djmips 3y agoNow all I understand is that he made an educated guess. ;-)
- kergonath 3y agoThe good thing is that the least action principle is fundamental and very flexible. All the physics are encapsulated in the Lagrangian. So you can come up with any crazy Lagrangian you want, plug it into Hamilton’s principle or Euler-Lagrange equations and see what you get. That way, you can build a whole theory from an insight somewhat easily as the fundamental framework is already in place.
- dataflow 3y agoFor those of us who didn't major in physics... where did the whole "action" thing (let alone the thesis that it's minimized) itself even come from? The whole notion of "action" feels entirely foreign and unintuitive for someone who's just studied Newtonian mechanics. At least I've never managed to find a real world feel for what it is, unlike with force or energy.
- nickpsecurity 3y agoI looked into it. There’s a Wikipedia article for action: https://en.m.wikipedia.org/wiki/Action_(physics) https://en.m.wikipedia.org/wiki/Action_(physics) I bet a lot of movie fans regretted stumbling onto this page. No, I wanted action!
- Koshkin 3y ago> No, I wanted action! But really what you want is to minimize it.
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- transfire 3y agoIs my impression correct — if you introduce fundamental (quantized) randomness, classic physics turns into quantum physics. Or is that an over simplification?
- kavalec 3y agoSo what happens if we follow this same path starting with general relativity?
- graycat 3y agoA lot of good physics students here! When I was a student of physics and math for physics, the math was solid but often the physics had to be just swallowed whole. This thread has some explanations missing from the physics sources I had! I'm busy with my startup, but I'd like to see how the physics of Lagrange, Hamilton, Schrödinger, etc., e.g., least action, quantum mechanics, really work and to compare them with Newton's calculus of variations (the shape of the wire that would let a bead slide down in least time), deterministic optimal control (e.g., the book by Athans and Falb), Kuhn-Tucker optimization conditions, Lagrangians in optimization, etc.
- Cleonis 3y agoI created demonstrations with interactive diagrams. http://cleonis.nl/physics/phys256/calculus_variations.php http://cleonis.nl/physics/phys256/calculus_variations.php The following case is used as motivation for developing Calculus of Variations: the shape of a soap film stretching between two coaxial rings. (The name of the solution is 'catenoid'; a surface of revolution.) Then the discussion moves to the Catenary problem: to calculate the shape of a hanging chain. The two problems have the same solution; the curve is the hyperbolic cosine. Demonstration of Hamilton's stationary action: http://cleonis.nl/physics/phys256/energy_position_equation.php http://cleonis.nl/physics/phys256/energy_position_equation.p... The diagrams have sliders. Moving the sliders sweeps out variation of a trial trajectory. The diagram shows how the kinetic energy and the potential energy respond to sweeping out variation.
- whatshisface 3y ago>In the case of the Schrödinger equation, this is done by extending the metaplectic representation of linear Hamiltonian flows to arbitrary flows; for the Heisenberg group this follows from a careful analysis of the notion of phase of a Lagrangian manifold, and for the uncertainty principle it suffices to use tools from multivariate statistics together with the theory of John's minimum volume ellipsoid. Thus, the mathematical structure needed to make quantum mechanics emerge already exists in classical mechanics. If they have to "extend," introduce the notice of "phase" and then recover the uncertainty principle from that, the quantum mechanics was not there to begin with. "A bucket of water emerges mathematically from a bucket."
- amai 3y agoThis paper smells like crack pot stuff. That is probably why it collected only two citations in more than 10 years. It also mentions the experiment from Couder et. al. in the summary, which has been debunked several years ago: https://www.quantamagazine.org/famous-experiment-dooms-pilot-wave-alternative-to-quantum-weirdness-20181011/ https://www.quantamagazine.org/famous-experiment-dooms-pilot... Behind sophisticated math it hides a beginners understanding of physics. Classical mechanics emerges from Quantum Mechanics in the same way as wave optics emerges from ray optics. https://physics.stackexchange.com/questions/397694/what-makes-a-theory-quantum/397974#397974 https://physics.stackexchange.com/questions/397694/what-make... If it would be otherwise, you would also argue, that wave optics emerges from ray optic. The experimental evidence is clear against such an interpretation.
- ngcc_hk 3y agoThat emerge has a second phase - as one side is 2nd order and the other side is first order, time and space are not of the equal footing. To be compatible with special relativity where time and space are on equal footing one has to … this line of thinking generate the quantum field theory. Still, if I remember correctly he is more onto differential equation. Later a physics PhD wants deeper or via different path or many lathes. Instead of light know the shortest time, light just goes all paths and we integrate the result. Philosophically it is the integration first approach of Leibniz vs the differentiation first of newton. Or that in his theology God see all paths and find the best for us. Except it is not God. And all pathes are going (except due to phase only some will be observed. Btw, these two line of thinking is so different one can easily see - if you see tangent line/plane etc you see it is possibly Newtonian approach. See general relativity or certain formation of the Qft. If you see integration and many paths, you use Leibniz approach. See Qft in its current form.
- hxypqr 3y agoThis is not guesswork, if one evening you lie in the garden feeling bad because of a breakup or other reasons and watch the shadow of the lights, you can get similar results. Introducing Fourier transform into optics can indeed explain some phenomena, I can't recall the specifics, but it is related to the shape formed between the light and the fence.
- hughw 3y agoI have not yet read the linked paper, but seismologists have used the Schroedinger wave equation in seismic imaging since at least the 1970s [1], certainly a "classical" system. [1] https://pubs.geoscienceworld.org/geophysics/article-abstract/36/3/467/71259/Toward-a-unified-theory-of-reflector-mapping https://pubs.geoscienceworld.org/geophysics/article-abstract...
- ndsipa_pomu 3y agoOkay, the abstract clearly had english words in there, but I've got no idea what they mean. Does anyone have an overview that would make sense to a non-expert?
- leeoniya 3y agoTurbo Encabulator? https://en.wikipedia.org/wiki/Turbo_encabulator https://en.wikipedia.org/wiki/Turbo_encabulator https://www.youtube.com/watch?v=Ac7G7xOG2Ag https://www.youtube.com/watch?v=Ac7G7xOG2Ag
- stephenhandley 3y agoThis is the first I had heard of "Turbo encabulator". Couldn't resist https://chat.openai.com/share/f32f5b22-0e3c-4228-ba8b-b90684fd8957 https://chat.openai.com/share/f32f5b22-0e3c-4228-ba8b-b90684...
- denton-scratch 3y agoYup. "Metaplectic" is a new one on me. Wikipedia isn't much help: https://en.wikipedia.org/wiki/Metaplectic_group https://en.wikipedia.org/wiki/Metaplectic_group
- svnt 3y agoI will try: A group is a collection of objects that you can transform via certain defined methods and they respond in known ways. A metaplectic group is like a mirrored version of a group you already know, with a few other changes in the behavior. Internally similar enough that in seeking to understand it you’re not starting from scratch; the group feels familiar.
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- lr1970 3y agoA Big missing part is the wave function and superposition principle that Classical Physics cannot emulate.The paper is at best a mathematical curiosity.
- aleph_minus_one 3y ago> A Big missing part is the wave function and superposition principle that Classical Physics cannot emulate. I am not a physicist, but doesn't the Schrödinger equation decribe the wave function?
- Racing0461 3y agoI believe that's that he's saying. thats why we needed the s equation.
- bluish29 3y agoI don't know if there is a rule about science papers links, but I think using the journal paper link [1] is more suitable. The paper is open access, so no need for research gate. [https://iopscience.iop.org/article/10.1088/1742-6596/361/1/012015 https://iopscience.iop.org/article/10.1088/1742-6596/361/1/0...]
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- JadeNB 3y agoI'd say the gold standard is DOI, though, for many journals, that's easily derived from the URL: https://doi.org/10.1088/1742-6596/361/1/012015 https://doi.org/10.1088/1742-6596/361/1/012015
- ben_WG 3y agoThe Schrödinger equation emerges from classical mechanics most closely (well ok that's a bit subjective) from the Hamilton Jacobi frame work, and it was indeed here that Schrödinger saw, in hindsight, because in the beginning he pretty much guessed it, the biggest connection to classical dynamics. This is also related to the optic-mechanical relation that abstracts mechanics to the point it becomes comparable to optics. Hamilton Jacobi theory: https://en.m.wikipedia.org/wiki/Hamilton%E2%80%93Jacobi_equation https://en.m.wikipedia.org/wiki/Hamilton%E2%80%93Jacobi_equa... Optic-mechanical analogy: https://en.m.wikipedia.org/wiki/Hamilton%27s_optico-mechanical_analogy https://en.m.wikipedia.org/wiki/Hamilton%27s_optico-mechanic... Schrödinger equation from HJ theory: https://www.reed.edu/physics/faculty/wheeler/documents/Quantum%20Mechanics/Miscellaneous%20Essays/Schrodinger's%20Argument.pdf https://www.reed.edu/physics/faculty/wheeler/documents/Quant...
- patcon 3y ago> in the beginning he pretty much guessed it Ah, you've given me a thought I'm grateful for. Thanks! I'm someone who's had a gut feeling about something in some random niche of science for several years. I've spent that time slowly gathering evidence from the literature to validate my hunch. It feels less like a "guess" and more like a high dimensional observation (of a form that's hard to cite or trace origins for) that first needs to be re-grounded in "real research". Though maybe it DID feel like a guess to Schrödinger...! but if he didn't say it that way, I'd assume it's not quite so accurate a framing :) though it is an entertaining way to communicate it, and I appreciate that it lends a sense of serendipity and happenstance and luck, which is perhaps the most important thing to telegraph about how science happens... to take a swing at the false inevitability and certainty that has its hooks in our histories!
- danbruc 3y agoI've spent that time slowly gathering evidence from the literature to validate my hunch. That is most likely the wrong way to go about this, you should probably look for evidence that your hunch is wrong, that it is in conflict with established physics.
- danilor 3y agoIf I wanted to know what the community thought of a particular paper, is there a place where I can find a discussion of it? I thought maybe researchgate was the place, but I usually don't see discussion on the paper submission there. I know sometimes you can find the peer reviewer comments before the paper got published, but what I mean is comments from other scientists.
- lamontcg 3y agohttps://pubpeer.com/ https://pubpeer.com/ but it is hit or miss. It has a browser extension so it'll alert you if it finds URLs to papers with discussions.
- bowsamic 3y agoAs a physicist, no, that's not a thing, at least not that I know of. Beyond whatever you can find from a simple google search is unknown to us Best you can do is look for papers that cite this paper.
- bjelkeman-again 3y agoSo are the only informal discussions done in person, or via email?
- mseri 3y agoBoth I’d say. More recently also online calls/conferences/seminars, and (way more rarely but it happens) on twitter/mastodon/…
- staunton 3y agoMaybe math overflow or physics overflow might work in rare cases... For most papers, I don't think there's really much a layperson can actively do to find out what experts think.
- leephillips 3y agoScientists comment on papers by writing papers. For a paper that just appeared, wait a year or so, and check Google Scholar for papers that cite this paper. Check again every few months. If you know physicists with an interest in this field, you can ask them if they’ve seen the paper and what they think of it. If they have an opinion they’ll probably share it with you freely, but they won’t write it down anywhere.
- magicalhippo 3y agoReminds me of a paper by by Hardy[1] where he introduces five reasonable axioms (his words). Classical and quantum probability theory obeys the first four. However the fifth, which states that there exists continuous transformations between pure states, is only obeyed by the quantum theory. In that sense he argues that quantum theory is in a sense more reasonable than classical theory. There's also an interesting link between this and entanglement[2] which seems to rule out other probability theories, leaving only quantum theory able to exhibit entanglement. Not my field at all though, just find these foundational things interesting to ponder. [1]: https://arxiv.org/abs/quant-ph/0101012 https://arxiv.org/abs/quant-ph/0101012 [2]: https://arxiv.org/abs/0911.0695v1 https://arxiv.org/abs/0911.0695v1
- fsckboy 3y agoisn't it more likely that classical mechanics emerges from the Schrodinger equation?
- monadINtop 3y agoit's not more likely, it just does. If we couldn't re-derive all known laws of classical mechanics and thermodynamics from the large scale limit of quantum mechanics, than we would have rejected quantum mechanics as wrong (or incomplete) decades ago. This paper seeks to show that some of the mathematical framework of quantum mechanics "pops out" of some intuitive (depends on your perspective i guess) machinery from classical mechanics. It doesn't really mean much fundamentlly, and doesn't really reflect the historical derivations of the equations, but it is interesting to look in retrospect how readily some of these equations pop out from seemingly basic frameworks. Its also interesting to consider the actual historical discovery of these concepts, or any scientific concept that generalised existing theories to a far deeper and more unifying result (e.g classical -> quantum mechanics, newtonian mechanics -> general relativity). You are required to somehow develop a theory that not only extends beyond horizons currently seen, but also one that correctly replicates the theory it seeks to supercede. And of course, you are limited to theoretical tools you already know, since no-one has yet figure a way to reach into the future and pluck out a more suitable notation or mathematical framework. Its like a literary character trying to write the story it is embedded in. Of course, there are always hints to the keen observer, especially tucked away at the foundations: much of special relativity unravelled itself directly from the laws of electromagnetics, since in the equation the speed of light is never specified, and the naive galilean assumption that everyone made - that time and space are absolute, and speeds must be specified relative to observers - was the veil obscuring our vision. If you take the courage to abandon the doctrine of absoluteness of time and space, and to declare that the speed of light doesn't need to be specified in terms of some preferred reference frame, since the speed of light is invariant for all observers everywhere throughout the universe, the intractable gulfs seperating what we know from what we don't vanish like a mirage, and meld together naturally into a more fundamental, and unified theory. And we can take the same step again, by noticing the strange coinicdence that in Newton's theory of gravitation and mechanics, the inertial mass happens to exactly equal the gravitational mass, magically cancelling each others contribution. If we declare that these two phenomena are infact exactly the same thing seen from different perspectives, and we realise that the apparant difference between somebody accelerating and somebody falling is an illusion, obscuring the fact that both are simply bodies taking the shortest path through the warped 4-dimensional manifold of spacetime, we once again unify all of our observations into a elegant, geometric theory of immense power and stunning beauty, one that can peer into the hearts of dead stars, and into the birth of all things, despite being first revealed in the brain of an absent minded jewish man, sitting in a cluttered office filled with pipesmoke, in a time where europe was fragmented from the collapse of empires and feudal houses, with wars fought with bayonets and horses still in living memory.