Schrodinger's reasoning was remarkable.<p>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.<p>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.<p>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 <i>the same way</i> 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?<p>Guided by that, he was able to add "diffraction" to the equations of matter and come up with the Schrodinger equation.<p>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.
The 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.<p>Hamilton Jacobi theory:
<a href="https://en.m.wikipedia.org/wiki/Hamilton%E2%80%93Jacobi_equation" rel="nofollow noreferrer">https://en.m.wikipedia.org/wiki/Hamilton%E2%80%93Jacobi_equa...</a><p>Optic-mechanical analogy:
<a href="https://en.m.wikipedia.org/wiki/Hamilton%27s_optico-mechanical_analogy" rel="nofollow noreferrer">https://en.m.wikipedia.org/wiki/Hamilton%27s_optico-mechanic...</a><p>Schrödinger equation from HJ theory:
<a href="https://www.reed.edu/physics/faculty/wheeler/documents/Quantum%20Mechanics/Miscellaneous%20Essays/Schrodinger's%20Argument.pdf" rel="nofollow noreferrer">https://www.reed.edu/physics/faculty/wheeler/documents/Quant...</a>
I 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.<p>[<a href="https://iopscience.iop.org/article/10.1088/1742-6596/361/1/012015" rel="nofollow noreferrer">https://iopscience.iop.org/article/10.1088/1742-6596/361/1/0...</a>]
Reminds 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.<p>In that sense he argues that quantum theory is in a sense more reasonable than classical theory.<p>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.<p>Not my field at all though, just find these foundational things interesting to ponder.<p>[1]: <a href="https://arxiv.org/abs/quant-ph/0101012" rel="nofollow noreferrer">https://arxiv.org/abs/quant-ph/0101012</a><p>[2]: <a href="https://arxiv.org/abs/0911.0695v1" rel="nofollow noreferrer">https://arxiv.org/abs/0911.0695v1</a>
If 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.
I 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.<p>[1] <a href="https://pubs.geoscienceworld.org/geophysics/article-abstract/36/3/467/71259/Toward-a-unified-theory-of-reflector-mapping" rel="nofollow noreferrer">https://pubs.geoscienceworld.org/geophysics/article-abstract...</a>
This 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.
This 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: <a href="https://www.quantamagazine.org/famous-experiment-dooms-pilot-wave-alternative-to-quantum-weirdness-20181011/" rel="nofollow noreferrer">https://www.quantamagazine.org/famous-experiment-dooms-pilot...</a><p>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.<p><a href="https://physics.stackexchange.com/questions/397694/what-makes-a-theory-quantum/397974#397974" rel="nofollow noreferrer">https://physics.stackexchange.com/questions/397694/what-make...</a><p>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.
><i>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.</i><p>If they have to "extend," introduce the notice of "phase" and then recover the uncertainty principle from that, the quantum mechanics was <i>not</i> there to begin with. "A bucket of water emerges mathematically from a bucket."
Okay, 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?
A Big missing part is the wave function and superposition principle that Classical Physics cannot emulate.The paper is at best a mathematical curiosity.
A lot of good physics students here!<p>When I was a student of physics and math for physics,
the math was solid but often the physics had to be just <i>swallowed</i> whole. This thread has some explanations missing from the physics sources I had!<p>I'm busy with my startup, but I'd like to see how the physics of Lagrange, Hamilton, Schrödinger, etc., e.g., <i>least action</i>, 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.
That 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.<p>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.<p>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.<p>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.
Is my impression correct — if you introduce fundamental (quantized) randomness, classic physics turns into quantum physics. Or is that an over simplification?
Not 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.<p>[1] <a href="https://physics.stackexchange.com/questions/32112/classical-limit-of-quantum-mechanics" rel="nofollow noreferrer">https://physics.stackexchange.com/questions/32112/classical-...</a>