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I'm an undergraduate mathematics student trying to understand some quantum mechanics, but I'm having a hard time understanding what is the status of the Schrödinger equation.

In some places I've read that it's just a postulate. At least, that's how I interpret e.g. the following quote:

Where did we get that (equation) from? Nowhere. It is not possible to derive it from anything you know. It came out of the mind of Schrödinger. -- Richard Feynman

(from the Wikipedia entry on the Schrödinger equation)

However, some places seem to derive the Schrödinger equation: just search for "derivation of Schrödinger equation" in google.

This motivates the question in the title: Is the Schrödinger equation derived or postulated? If it is derived, then just how is it derived, and from what principles? If it is postulated, then it surely came out of somewhere. Something like "in these special cases it can be derived, and then we postulate it works in general". Or maybe not?

Thanks in advance, and please bear with my physical ignorance.

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The issue is that the assumptions are fluid, so there aren't axioms that are agreed upon. Of course Schrodinger didn't just wake up with the Schrodinger equation in his head, he had a reasoning, but the assumptions in that reasoning were the old quantum theory and the deBroglie relation, along with Hamiltonian idea that mechanics is the limit of wave-motion.

These ideas are now best thought of as derived from postulating quantum mechanics underneath, and taking the classical limit with leading semi-classical corrections. So while it is historically correct that the semi-classical knowledge essentially uniquely determined the Schrodinger equation, it is not strictly logically correct, since the thing that is derived is more fundamental than the things used to derive it.

This is a common thing in physics--- you use approximate laws to arrive at new laws that are more fundamental. It is also the reason that one must have a sketch of the historical development in mind to arrive at the most fundamental theory, otherwise you will have no clue how the fundamental theory was arrived at or why it is true.

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In a nutshell, Schrödinger equation is an educated guess. The "derivation" is just the process of guessing. – C.R. Jun 22 '12 at 4:12
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@KarsusRen: Absolutely not! This is completely wrong. There was no guesswork involved in the thing. Have you read Schrodinger's paper? Einstein got the same equation independently when he heard of the result. It follows from deBroglie's relation, the semiclassical limit, and this is enough to uniquely specify the equation. It is only that later it is seen to be more fundamental than the thing it is derived from, so the derivation ends up logically going the other way. But the historical derivation is correct, and is a mathematically justified argument, like any other in physics. – Ron Maimon Jun 22 '12 at 7:02
The semiclassical limit, which in first order is nothing other than the good old Hamilton-Jacobi equation, does not (!) completely specify the quantum theory. So a mathematically derivation of the Schrödinger equation is not possible. But one can motivate it, which then is called "derivation". Often this is enough - if not, one has to build abstract quantization theories, which try to construct quantum theory. But up to now, none of these quantization schemes works perfectly. A good overview can be found in arxiv.org/abs/math-ph/0405065. – altertoby Jun 22 '12 at 21:51
@altertoby: This is totally false. Einstein indeed showed that the Hamilton Jacobi equation is the semiclassical wave equation, but this is not a sensible wave equation, since it only gives the phase of the wave. HJ equation uniquely determines the Schrodinger equation, since you need the magnitude part to work together with the phase part. This is what Schrodinger shows in his paper. This is a rigorous derivation, based on semiclassical ideas. It is not necessary to read the paper you linked, it is useless for this discussion. I know how to quantize. Please read Schrodinger instead. – Ron Maimon Jun 23 '12 at 2:16
The other path to quantum mechanics was through the semiclassical operator construction of Kramers and Heisenberg. In this path, which is also nearly rigorous (although not as rigorous as Schrodinger's, due to the fact that the analysis is fundamentally perturbative, which is why Heisenberg didn't discover tunneling). In this path, Heisenberg calculates the semiclassical commutator of p and q and shows that it is $i\hbar$, and then postulates that this is true for all n. This postulate is justified from Einstein's A and B coefficients, which give the Harmonic oscillator matrix elements. – Ron Maimon Jun 23 '12 at 2:21
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The Schrödinger equation is postulated. Any source that claims to "derive" it is actually motivating it. The best discussion of this that I'm aware of this is in Shankar, Chapter 4 ("The Postulates -- a General Discussion"). Shankar presents a table of four postulates of Quantum Mechanics, which each given as a parallel to classical postulates from Hamiltonian dynamics.

Postulate II says that the dynamical variables x and p of Hamiltonian dynamics are replaced by Hermitian operators $\hat X$ and $\hat P$. In the X-basis, these have the action $\hat X\psi = \psi (x)$ and $\hat P\psi = -i\hbar\frac{d\psi}{dx}$. Any composite variable in Hamiltonian dynamics can be built out of x and p as $\omega(x,p)$. This is replaced by a Hermitian operator $\hat \Omega(\hat X,\hat P)$ with the exact same functional form.

Postulate IV says that Hamilton's equations are replaced by the Schrödinger equation. The classical Hamiltonian retains its functional form, with x replaced by $\hat X$ and p replaced by $\hat P$.

NB: Shankar doesn't discuss this, but Dirac does. The particular form of $\hat X$ and $\hat P$ can be derived from their commutation relation. In classical dynamics, x and p have the Poisson Bracket {x,p} = 1. In Quantum Mechanics, you can replace this with the commutation relation $[\hat X, \hat P] = i\hbar$. What Shankar calls Postulate II can be derived from this. So you could use that as your fundamental postulate if you prefer.

Summary: the Schrödinger equation didn't just come from nowhere historically. It's a relatively obvious thing to try. Mathematically, there isn't anything more fundamental in the theory that you could use to derive it.

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I had an electrical engineering course in university which consisted entirely of teaching us how to "derive" the Schrödinger equation. I don't remember much but it was using the rules of calculus to manipulate some other assumed laws of physics. But I guess our teacher might have been wrong in stating that we were "deriving" the equation. – Kmeixner Jan 29 at 23:11

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