Derivation of the SE

Thrice

A visiting professor today made the comment that you can't really derive the SE. Curious what you guys think. I've seen handwaving arguments using wave packets & I suppose more formal arguments using some time evolution operator. It should be possible to get it from first principles, right?

Schrodinger's Dog

<unverified links deleted>

I was sure I'd seen a derivation before, here's two, I am by no means savy enough to determine if these are correct.

jtbell

The first link sets off a couple of warning bells in my head. First, it's published by the Fondation Louis de Broglie. Somebody here was asking about it not too long ago, and the link they gave as an example of what they published looked pretty "far out" to me. I suspect that the FLdB has an "open minded" policy that encourages submission of non-mainstream material. Second, although I haven't read very far yet, the author starts out by claiming that his derivation renders unnecessary the probabilistic interpretation of the wave function. Definitely non-mainstream!

The second link shows a derivation not of the SE itself, but a "quaternion analog" to the SE. At the end author Doug Sweetser notes,

So this is also a non-mainstream approach, and even the author doesn't know yet whether it's going to work out.

masudr

Lots of analogies can be made etc. but ultimately the Schrodinger equation in its most general form must be assumed to hold true.

jostpuur

As well we could discuss about derivation of classical theory. You cannot derive quantum mechanics, similarly as you cannot derive Newton's mechanics either. You just have to accept Newton's laws.

Or on the other hand, if you consider some heuristic arguments being derivation of Newton's laws, in the same spirit you can also derive quantum mechanics somehow.

The question is ultimately about what we mean by "deriving" something.

Thrice

Well you two know how to "derive" SR from invariance of c & the relativity principle. I assumed that's the only thing he could have meant because otherwise (as jostpuur said) the statement is trivially true.

I guess I was thinking something like this.

jtbell

But which first principles? That's the key question. In order to derive something, you have to start with something else that's given as true.

I have not studied this in depth myself, but I'm sure there are others here who have done so. I would not be surprised to find that there are a variety of ways to "axiomatize" QM, and that some of them assume the SE as a postulate and some of them don't.

With relativity, people usually start from the postulates of the Principle of Relativity and the invariance of [itex]c[/itex], because that's how Einstein did it in his original 1905 paper, which is the starting point for modern relativity theory.

But there is no similar single "starting point" for QM, historically speaking. Schrödinger himself came up with his equation by making an analogy between mechanics and optics, but this is more of an "inspiration" or "motivation" for the SE than a proper "derivation" of it. For some more details, see

http://www.physicsforums.com/showthread.php?p=418069

Thrice

Any, right? Any (reasonable) set leading to the SE should be enough to reject the general premise that "The SE cannot be derived," because this premise implies that there are no such axioms.

dextercioby

There's no derivation in an axiomatical approach. It's part of the axioms.

However, another approach could provide a derivation.

cartuz

Your Visiting Professor is right!
Quantum Mechanic is a theory with its own axiomatic. As any serios theory has its own axiomatic. There is different interpretations of QM. Any interpretation has own axioms.
Copengagen intepretation. It is almost the official famous interpretation. In this interpretation the Schro"dinger equation is one of postulates. In other words in this interpretation SE is one of axioms Quantum Mechanics. And it is the most of axioms. In addition the source of Planck constant cann't find nobody. And Planck constant is in SE and it is very important part of SE.
Other interpretations of Quantum Mechanic, De-Brougle-Bohm for example, as the target to derive SE. In this interpretation SE is follow from classical mechanics equation named Jacobi-Hamilton equation. But from classical physics we cann't to derive SE. We are need others postulates (axioms) for this. And Planck constant we cann't derive in this case too.

quetzalcoatl9

the schrodinger equation can be derived by assuming nothing more than (a) that energy is quantized and (b) a complex wavefunction describes everything about the system. The derivation follows from applying classical dynamics to these assumptions.

Thrice

But how is that different from any other derivation? Results are always implied by the axioms preceeding them. I'd appreciate an example of some other approach.

dextercioby

Is that so ? Can you prove your statement ?

quetzalcoatl9

yes, please see:

http://arxiv.org/abs/physics/0610121

quetzalcoatl9

ok, i guess we also need either the de Broglie relation or the relativistic energy-momentum equation (so it is not purely classical)

cartuz

In http://arxiv.org/abs/physics/0610121 authors change the electric field E to Psi-function. It is formal operation.
In Schrödinger paper
<<An undulatory theory of the mechanics of atoms and molecules
E. Schrödinger
Phys. Rev. 28, 1049-1070 (1926)>>
he derive his equation from wave equation for the wave-function but this equation he postulated. Here Schrödinger appear that SE similar to wave equation.
There is another way. But it is alternative to official Copenhaven Interpretation. You can read here
<<Derivation of the Schrödinger equation from Newtonian mechanics
E. Nelson
Phys. Rev. 150, 1079-1085 (1966)>> It is derivation is named de-Broile-Bohm Interpretation. But in this case it is used another axioms not official Copenhagen’s and it is not Quantum Mechanic in the ordinary sense. More right it is the stochastic interpretation because in this axioms used unknown random fields postulate.
I'm like very much the derivation SE from Jacobi-Hamilton equation. But it is not ideal too.

quetzalcoatl9

yes, one of the assumptions that i stated is the assumption that the system can be completely described by a complex wavefunction. i guess we also need to assume that it is normalized, but this is also done in classical electrodynamics.

the "big leap" needed is quantization of energy appled to SR.