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Schrodinger's reasoning was remarkable. The high-level description of classical mechanics was formulated by Hamilton, who was starting from optics. He saw a ma

by alecst 3y ago

Schrodinger'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 ago
Yeah, I've been taught that Schrödinger basically made a guess as well... Very interesting how it really happened.
- djmips 3y ago
  Now all I understand is that he made an educated guess. ;-)
- kergonath 3y ago
  The 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 ago
For 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 ago
  I 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.
- alecst 3y ago
  Honestly it's a great question. Even the typical physics major isn't going to be able to give you a great answer because learning how this was derived isn't a part of any curriculum that I know of. But if you can accept the principle of least time (that a light ray will travel along the path that takes the shortest time) then you kind of already accept that (ordinary, classical) light somehow knows the time it takes to go along all possible paths, then chooses the minimum. The action is a kind of thing, discovered by Hamilton, by analogy, that plays the same role in mechanics as time does in optics. I image he just stared at the equations of optics for a while and had an "ah-ha" moment. He had been working on this stuff for decades, on top of being a pretty smart guy already. It's extremely unintuitive that classical systems should minimize anything like a time or an action. Think about it: they travel along the minimum path, but how do they know that that path is the minimum? Do they sample the other paths to know? Well interestingly, Feynman's thesis was about exactly this idea. What happens if you start from the assumption that particles just sample all possible paths (weighted by something having to do with the action/time)? It turns out you can get the Schrodinger equation (and optics equations) from that too. It partially explains how paths "find the minimum." It turns out they don't, but a nice cancellation happens that makes the minimizing path the most probable one.
andypea 3y ago
Thanks! This is really intersesting and insightful. I'd definitely read the blog-post if you get round to writing it!
kergonath 3y ago
> 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. I think you are attributing to Hamilton what was developed by others some time before. The jump from optics to mechanics was about 100 years before Hamilton published his principle. Maupertuis was the one who said that physical objects should follow shortest paths in their phase space in a way analogous to light according to Fermat’s principle. This was developed and generalised by Euler and Lagrange (with the Euler-Lagrange equations to derive equations of motions), and then Hamilton’s principle is another generalisation. It does not affect the point you are making about Schrödinger, though.
- screye 3y ago
  It's always Euler. Can't imagine a more prolific and simultaneously impact full scientist in the modern era
  - leeoniya 3y ago
    John von Neumann?
- adrian_b 3y ago
  Before Hamilton, "action" meant "accumulated living force", i.e. the integral of the kinetic energy. The principle of the minimum action of Maupertuis was true only in some restricted cases, and it was false in most mechanics problems. Hamilton has introduced a new physical quantity, for which he has used only the name "function S". He formulated a variational principle for the "function S", analogous to the principle of the minimum action, but from which it is possible to deduce the system of equations of Lagrange, so it has general applicability. Hamilton's "function S" is relativistically invariant and nowadays it is usually called Hamilton's action. It is the integral of the Lagrangian, not of the kinetic energy, like the traditional action. It is proportional with the phase of the wave function in quantum mechanics. In relativistic mechanics, the Lagrangian is the component of the momentum-energy that is tangent to the trajectory in space-time, so "function S" is the line integral of the momentum-energy over the trajectory in space-time. So Hamilton's variational principle is quite different in meaning and applicability from the principle of the minimum action that existed before him. It remains true in all forms of physics that have been discovered after Hamilton.

by alecst 3y ago

ahartmetz 3y ago

Yeah, I've been taught that Schrödinger basically made a guess as well... Very interesting how it really happened.

djmips 3y ago
Now all I understand is that he made an educated guess. ;-)
kergonath 3y ago
The 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 ago

For 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 ago
I 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.
alecst 3y ago
Honestly it's a great question. Even the typical physics major isn't going to be able to give you a great answer because learning how this was derived isn't a part of any curriculum that I know of. But if you can accept the principle of least time (that a light ray will travel along the path that takes the shortest time) then you kind of already accept that (ordinary, classical) light somehow knows the time it takes to go along all possible paths, then chooses the minimum. The action is a kind of thing, discovered by Hamilton, by analogy, that plays the same role in mechanics as time does in optics. I image he just stared at the equations of optics for a while and had an "ah-ha" moment. He had been working on this stuff for decades, on top of being a pretty smart guy already. It's extremely unintuitive that classical systems should minimize anything like a time or an action. Think about it: they travel along the minimum path, but how do they know that that path is the minimum? Do they sample the other paths to know? Well interestingly, Feynman's thesis was about exactly this idea. What happens if you start from the assumption that particles just sample all possible paths (weighted by something having to do with the action/time)? It turns out you can get the Schrodinger equation (and optics equations) from that too. It partially explains how paths "find the minimum." It turns out they don't, but a nice cancellation happens that makes the minimizing path the most probable one.

andypea 3y ago

Thanks! This is really intersesting and insightful. I'd definitely read the blog-post if you get round to writing it!

kergonath 3y ago

> 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. I think you are attributing to Hamilton what was developed by others some time before. The jump from optics to mechanics was about 100 years before Hamilton published his principle. Maupertuis was the one who said that physical objects should follow shortest paths in their phase space in a way analogous to light according to Fermat’s principle. This was developed and generalised by Euler and Lagrange (with the Euler-Lagrange equations to derive equations of motions), and then Hamilton’s principle is another generalisation. It does not affect the point you are making about Schrödinger, though.

screye 3y ago
It's always Euler. Can't imagine a more prolific and simultaneously impact full scientist in the modern era
- leeoniya 3y ago
  John von Neumann?
adrian_b 3y ago
Before Hamilton, "action" meant "accumulated living force", i.e. the integral of the kinetic energy. The principle of the minimum action of Maupertuis was true only in some restricted cases, and it was false in most mechanics problems. Hamilton has introduced a new physical quantity, for which he has used only the name "function S". He formulated a variational principle for the "function S", analogous to the principle of the minimum action, but from which it is possible to deduce the system of equations of Lagrange, so it has general applicability. Hamilton's "function S" is relativistically invariant and nowadays it is usually called Hamilton's action. It is the integral of the Lagrangian, not of the kinetic energy, like the traditional action. It is proportional with the phase of the wave function in quantum mechanics. In relativistic mechanics, the Lagrangian is the component of the momentum-energy that is tangent to the trajectory in space-time, so "function S" is the line integral of the momentum-energy over the trajectory in space-time. So Hamilton's variational principle is quite different in meaning and applicability from the principle of the minimum action that existed before him. It remains true in all forms of physics that have been discovered after Hamilton.