Not many people know that Schrödinger wrote a a paper involving a dog, and to show solidarity with a fellow scientist, opted to model the dog after a subject of Pavlov's experiments. I called the UIUC library to ask if they had a copy of the paper just a few months ago, but I couldn't remember the title of the paper. The researcher that answered told me, "That sure rings a bell; I'm not sure if it's here or not, though."
I find that often reading a scientist's dissertation will often tell you something about how they approach the rest of their research. Somewhere I actually have a paper copy of Feynman's Thesis under Wheeler, which covers exactly this material. He was clearly enamored of Lagrangian formulations quite early (I didn't know that it was due to a High School teacher).
"A few days before Christmas, 1925, Schrodinger, a Viennese-born professor of physics at the University of Zurich, took off for a two-and-a-half-week vacation at a villa in the Swiss Alpine town of Arosa. Leaving his wife in Zurich, he took along de Broglie's thesis, an old Viennese girlfriend (whose identity remains a mystery) and two pearls. Placing a pearl in each ear to screen out any distracting noise, and the woman in bed for inspiration, Schrodinger set to work on wave mechanics. When he and the mystery lady emerged from the rigors of their holiday on Jan. 9, 1926, the great discovery was firmly in hand."
The Schrödinger equation was not derived, a more accurate word would be guessed. It's a fundamental law, like F = ma (although that is actually more of a definition than a law). There is no fundamental explanation why electrons obey the Schrödinger equation, just like there's no fundamental explanation why macroscopic objects obey F = ma.
A lot of things can be derived from more basic or at least more beautiful principles. It makes perfect sense to say that Schroedinger's equation can be derived from Lagrangian Mechanics, even if this is not historically how it happened.
An especially beautiful constraint is "Schroedinger's equation has to be linear because otherwise you can build a (quantum) computer that can solve NP-complete problems in polynomial time". Schroedinger did not know about NP-completeness, but this notion does give us clues why the equation is what it is.
It is always instructive to see how a "fundamental" law can be derived or at least constrained from other laws. Thermodynamics and the notion of entropy would not have been invented if engineers did not "rederive" what was considered fundamental at the time.
The grandparent post is right in the sense that reductionism always reduces, ultimately, to a guess (in this case, Lagrangian mechanics is the guess).
You're right, though, in that a basic framework can be established, using minimal assumptions and axioms, in which most of classical physics and even much of quantum physics can be derived via (mostly) rigorous mathematical arguments. But that framework itself is, at the end of the day, a guess. It's simply the best guess we have at the given moment in time that fits the evidence (e.g., by allowing us to derive from it known rules and patterns that past generations have verified fit the evidence).
It's actually not true to say that Schrodinger's equation, or F=ma, cannot be derived.
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.
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.
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.
> 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.
> 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.
> Well, F = ma was a bad example, I grant you. It's actually a definition of what a force is, nothing more.
Although I would agree that it's fair to say "F=ma is a definition of what a force is", I think of it more as a law of physics than as a definition.
I would also add that Newton's second law is more than merely the statement F=ma.
The original formulation is:
Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd. [1]
So in addition to stating what the word force means, the second law is also saying there are these things called forces which move things in a certain direction.
The way I think of F=ma personally, is that it means the laws of physics are associated with differential equations. So for me, the statement F=ma implies a huge number of other things, since without also having made the assumption that the laws of physics can be described in terms of vector calculus, the definition of force is rather useless and uninteresting.
Where did we get that [Schrödinger's equation] from? It's not possible to derive it from anything you know. It came out of the mind of Schrödinger.
-The Feynman Lectures on Physics
For anyone who wants to study quantum mechanics and has a math background, I strongly recommend Shankar's quantum textbook. It has the most rigorous, clear, and correct derivations I have seen thus far, starting with vector spaces and moving to physical systems only after a hundred or so pages.
The book is just down right atrocious. The fact that it became almost standard in American universities speaks more of the power of networking than anything else. Shankar's book is anything but rigorous, quite on the contrary.
There are so many good QM books:
For an absolute beginner I'd recommend David Griffiths
Dirac's is very readable
Sakurai's is very good though a bit advanced
Whatever you do, it's difficult to go wrong by not using Shankar's
I personally think that Shankar makes a great companion to Sakurai. Shankar is less rigorous but it explains things in a more accessible way which can be extremely helpful if you find Sakurai too terse on some topics.
What in the world are you talking about? Griffiths is substantially worse than shankar.
Griffiths doesn't even get into Hilbert spaces until chapter three, at which point he delivers a broken and incomprehensible explanation. He attempts to teach Hilbert operators by analogy to the multiplication or application by adjacency people use in normal arithmetic. This is a terrible approach, and confuses the hell out of students who haven't already taken abstract algebra.
In what way is Shankar less rigorous than Griffiths?
The QM class I originally took used Liboff. I could barely understand a word of it and ended up dropping the class. Later I found the Shankar book and was thrilled with it by comparison. I'm not a math whiz though, so lack of "rigor" on a first pass is just fine by me. For an intro level text for the average person, I found Shankar was one of the few accessible ones.
Haven't read Griffiths yet so I can't comment on it.
It's been a long time now, but I pretty much agree with you. I really enjoyed Griffiths as an introduction but for reasons that escape me now I remember having a bad time with Shankar.
Have you ever read Lectures on Quantum Mechanics for Mathematics Students or Quantum Mechanics for Mathematicians? I'd like to know how Shankar's book sits in the space of QM textbooks.
Shankar is all round a very practical book to pass quals, very poorly structured, highly pretentious by dealing with say parth integrals with grassman variables but doing an awful job at basic stuff such as adding angular momentum.
If you've got a math background I'd recommended you skip Shankar even more vehemently than I usually do.
Feynman's proof of the Maxwell equations (FJ Dyson - Phys. Rev. A, 1989 (http://signallake.com/innovation/DysonMaxwell041989.pdf)) shows, that it is possible to derive Maxwells equations from Newtons second law of motion and the uncertainty principle.
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!
This article might be one of the best intros:
Space-Time Approach to Non-Relativistic Quantum Mechanics
R. P. Feynman
Rev. Mod. Phys. 20, 367 – Published 1 April 1948
