Feynman’s Derivation of the Schrödinger Equation

This is goosebump material!

It is fascinating how concepts from classical mechanics port over into quantum mechanics. If we'd stuck with F=ma, perhaps we may not have gone far. However, the Hamiltonian and Lagrangian approaches are sweet ports.

Noether's theorem [1] connects symmetries of a lagrangian with corresponding conservation laws. Time translation symmetry implies energy conservation. Space translation symmetry implies momentum conservation in the direction of translation. This ports directly to QM. Feynman, in his lecture series, mentions another such symmetry - phase translation symmetry - that exists for QM actions and asks what conservation law does that correspond to. It is the conservation of charge. Mind blown!

[1]: https://en.wikipedia.org/wiki/Noether%27s_theorem

> It's actually not true to say that Schrodinger's equation, or F=ma, cannot be derived.

I don't think I said it couldn't be derived. I said there wasn't a "fundamental reason why the Schrödinger equation is X". Using Lagrangians to derive it then begs the question "but why must electrons obey the Lagrangian?".

> This is because in order to derive something, you need to first start from a set of assumptions. You can pick any assumptions you want, that is your privilege, as it is mine. If you want F=ma to be an assumption, you can go on and derive things from that.

Well, F = ma was a bad example, I grant you. It's actually a definition of what a force is, nothing more. But it's relevant, because it has a similar purpose (except in a different field of physics). A better example of a fundamental law would be action-reaction or something.

But yes, you're free to pick any assumption you like. But if you assume X, which came about because of assumption Y, it shouldn't be a shock that X can be used to derive Y. At that moment, they are just different notations for the same assumption. You might argue (and hell, I might even agree) that Lagrangians are so much more mathematically pleasing, so make a better assumption. But that doesn't change the fact that you're dressing up assumption Y as assumption X.

> But truly, F=ma is not the most general assumption in theoretical physics.

> If you look in any decent book on classical mechanics, you will find that F=ma in fact is derived from a much more general principle, the Lagrangian.

The Lagrangian is great for solving many problems. But it is definitely not more general than Newtonian mechanics. It can't deal with friction, or quite a few other non-conservative forces. On the plus side, solving oscillating systems is much easier. And it's so much nicer when not using Cartesian coordinates.

> That is the same Lagrangian that is the subject of the OP, which is used to derive the Schrodinger equation; and it's not just a coincidence.

While this is correct, Lagrangian mechanics refers to concepts such as energy which are defined from forces. In particular, Lagrangian's deal with systems with only conservative forces (but total energy may change with time). So it's really a circular argument to say "this concept can be derived from this even more abstract concept, which was actually defined from the first concept and at the end of the day is a guess."

There's nothing wrong with guessing fundamental laws. That's how the scientific method works, after all. You ask a question, guess the answer, predict what your guess would imply and test your predictions.

Extra ) in your link.

> The Lagrangian is great for solving many problems. But it is definitely not more general than Newtonian mechanics

Sure it is. It's true that for certain systems, you can derive one from the other so in that sense they might be "just different notations" for the same physics (I'm disregarding non-conservative forces, which aren't really considered in fundamental physics). But the Lagrangian formalism really consists of two parts: Hamilton's principle and a choice (postulate/guess) of Lagrangian (or Lagrangian density). When we say that Lagrangian formalism is more general than Newtonian mechanics, it means we can describe physics using the Lagrangian which we can't get to via Newton's laws. For example, you would be hard-pressed to derive general relativity from Newton's laws, but if you start with the Einstein-Hilbert action, you can derive Einstein's field equations.

> A better example of a fundamental law would be action-reaction or something.

This is another example where the Lagrangian formalism is more general. In this case, a translation-invariant Lagrangian implies conservation of (canonical) momentum. But in more complex systems, Newton's third law might may fail when canonical momentum is still conserved (the prototypical example is the Lorentz force law).

> Using Lagrangians to derive it then begs the question "but why must electrons obey the Lagrangian?".

This isn't circular as much it is one less level of indirection. It's like if you say the reason a dropped object accelerates to the Earth is gravity, you have just shifted the question to "why must gravity behave as an inverse squared law". You can go further and say that Newtonian gravity is not fundamental, but is an approximation of from general relativity, where the dropped object isn't really accelerating. Again, the question is shifted to "why is general relativity described by the Einstein field equations". At each level of reduction you describe one phenomenon by something more fundamental. That you don't have a further explanation doesn't logically preclude that it's more fundamental.

Griffiths is not great but for someone completely unfamiliar with the subject it will hold your hand quite nicely and is not pretentious at all.

For someone a little more knowledgeable it might get a bit boring...