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,

Any attempt to shift the meaning of an equation as central to modern physics had first be able to regenerate all of its results. I believe that the quaternion analog to Schrödinger equation under the listed constraints will do the task. These is an immense amount of work needed to see as the constraints are relaxed, whether the quaternion differential equations will behave better.

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]

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

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

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 http://en.wikipedia.org/wiki/Schr%C3%B6dinger_picture" [Broken].

 

[jtbell]

It should be possible to get it from first principles, right?

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 $c$, 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

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

 

[Thrice]

But which first principles?

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]

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?

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]

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

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]

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.

Is that so ? Can you prove your statement ?

 

[quetzalcoatl9]

Is that so ? Can you prove your statement ?

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]

yes, please see:

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

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]

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.

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 applied to SR.

 

[reilly]

Physics is, after all, about natural phenomena, and test for correctness is almost always based on experiments, on empirical evidence. That is, QM, Newton, Einstein, and most other physics describes the world outside of us. If we neglect the variability of necessarily subjective observations, we find common agreement on the basic phenomena of classical physics. Unless I'm badly mistaken, there is absolutely no way for us to deduce the necessity of what we perceive from any formal argument. That is, the phenomena of Nature are given. The physicist's job is to make sense of the observed and predicted phenomena that are part of the common human perceptual experience. Can this be done axiomatically? I very seriously doubt it.

With a slight of hand here, and finesse there, some folks can use a sophisticated scam approach to derive the Schrodinger Eq.. But why bother, unless you can before the fact demonstrate that atomic spectra, in detail, can be derived from first principles, or show that electron diffraction is necessary, and so on. And so it's the crazy phenomena of spectra and particle diffraction, spin, Bose and Fermi statistics, radioactive decay, and so on, that drive QM.

The proof of the pudding, is the extraordinary success of the SE, or of Newton, or of Einstein when used to describe and predict natural phenomena.

Regards,
Reilly Atkinson

Note that energy is generally not quantized for free particles, as in scattering theory for example.

 

[quetzalcoatl9]

With a slight of hand here, and finesse there, some folks can use a sophisticated scam approach to derive the Schrodinger Eq.. But why bother, unless you can before the fact demonstrate that atomic spectra, in detail, can be derived from first principles, or show that electron diffraction is necessary, and so on. And so it's the crazy phenomena of spectra and particle diffraction, spin, Bose and Fermi statistics, radioactive decay, and so on, that drive QM.

I disagree. No one is saying you can derive experimental results, only that the Schrodinger equation follows naturally and does not itself need to be taken as a postulate as many quantum chemistry books erroneously claim. (As an aside, many books erroneously claim that the Pauli principle is also a postulate, when in fact it follows naturally from deriving the Dirac equation).

The axioms:

(a) E=hv (accepted from experimental evidence)
(b) Maxwell's equations (assumes the concept of a field in euclidean space, the remaining details follow from exact vector analysis)
(c) Special Relativity (exactly true if spacetime metric is euclidean and F=ma is true), you get de Broglie equation also for free here by accepting (a) above
(d) that the system can be completely described by a complex wavefunction (assumption)

then you can exactly derive the Klein-Gordon and Schrodinger equations, it is not a scam. Our assumption are completely known before hand, as are the experimental considerations. This is axiomatic.

unfortunately, few people have the time to go derive things like the SE for themselves when taking courses and so many people aren't aware of how these equations come about! Those who don't bother taking courses at all just assume that physicists are talking out of their...well, you get it. Aka the "what the bleep do we know" attitude - when in fact we know quite a bit. Give humanity some credit!

For that matter, if you believe that the SE equation is true, the HUP can be exactly derived, it is not a postulate either!

 

[masudr]

quetzalcoatl9:

Does this work for systems of particles too? What about for things like the electromagnetic field?

 

[cartuz]

In my opinion there is no unification of quantum axioms(postulates) because there is no suitable and rightable axioms in QM now. We have many variants of axioms. Everyone takes that he think the most. In other words Quantum Mechanics haven't the suitable foundation now! It is in the process of standing about a 100 years.

 

[Thrice]

Physics is, after all, about natural phenomena, and test for correctness is almost always based on experiments, on empirical evidence.

I agree with that, but it doesn't help the professor's point. It means he needs to decide what he meant by "derive."

I think there's a place for derivations. A theory doesn't need to be correct in order to work.

 

[dextercioby]

(As an aside, many books erroneously claim that the Pauli principle is also a postulate, when in fact it follows naturally from deriving the Dirac equation).

? Say what ? To me, this is nonsense, so please explain your claim.

 

[quetzalcoatl9]

? Say what ? To me, this is nonsense, so please explain your claim.

have you derived the dirac equation? the pauli matrices are a direct result of making the SE conform with SR.

this wikipedia article does it nicely:

http://en.wikipedia.org/wiki/Dirac_equation

are you satisfied?

 

[dextercioby]

Part of that article i wrote it myself, however you seem to have eluded my question. So, please, reread the quoted part of your post and try to see what part of it i claimed to be nonsensical.

 

[quetzalcoatl9]

Part of that article i wrote it myself, however you seem to have eluded my question. So, please, reread the quoted part of your post and try to see what part of it i claimed to be nonsensical.

which part? that books erroneously claim it a postulate? i can't read your mind, so why don't you just tell me

 

[dextercioby]

Pauli principle is not a postulate, it is a consequence. Here we agree. However, you clearly state that "the Pauli principle (...) follows naturally from deriving the Dirac equation", which to me is rubbish.

 

[quetzalcoatl9]

Pauli principle is not a postulate, it is a consequence. Here we agree. However, you clearly state that "the Pauli principle (...) follows naturally from deriving the Dirac equation", which to me is rubbish.

why? do you not agree that it's mathematical statement lies in the anti-commutation property of the spin matrices?

 

[dextercioby]

How does spin matrices anticommutation imply that the two identical 1/2 spin particles can't have the same quantum numbers ?

 

[quetzalcoatl9]

if 2 things are anticommutative then if thing 1 is the same as thing 2 the anticommutator will be zero (which in this case could be the wavefunction). I am sure you know all of this, so why the quiz?!

e.g.

$$\psi_a , \psi_b> = N(\psi_a \psi_b - \psi_b \psi_a)$$

which is obviously 0 if $$\psi_a = \psi_b$$