# Origin of the schroedinger's wave equation

1. Sep 26, 2012

### Aniket1

How did schroedinger arrive at the wave equation?
I recently read in a book about the concept of wave packets using fourier analysis and the wave equation was derived by forming a differential equation of the fourier integral.
But some books say that there is no formal proof of the schroedinger's equation.

2. Sep 26, 2012

### ImaLooser

He guessed, actually, along with general knowledge of the form of wave equations. S had no idea that his wave equation had anything to do with probability. When this was discovered, S was displeased.

Nothing in physics can ever be proved. Proof is for mathematics.

3. Sep 26, 2012

### vanhees71

There is indeed no formal proof for Schrödinger's equation, because there is in fact not much more fundamental than that equation in (nonrelativistic) quantum mechanics.

Another question is, how Schrödinger arrived at his equation. As stressed above, it could not have been a formal derivation from more fundamental grounds but heuristic arguments. As far as I know the history, the idea of "matter waves" was around at the time when Schrödinger started to think about "quantum mechanics as an eigenvalue problem". The idea goes back to Louis de Broglie's PhD thesis, which has been considered so strange an idea by his referees that they sent a copy to Einstein for a recommendation of the thesis, and Einstein gave a positive judgement. The idea was that a particle is described by a matter wave, which at that time has been interpreted as a classical field which describes the particle itself as a smeared cloud. Then the stationary states (standing waves) of this waves of an electron around an atomic nucleus were supposed to describe the energy states of this electron within the atom and give a more "natural" explanation for Bohr's quantization condition in his famous Bohr-Sommerfeld model of the atom, which at this time was known to be wrong or at least not applicable to other atoms than the hydrogen atom without very artificial additional rules, which were different for each atom. This is not a very good state of affairs for a fundamental physical theory. So the hope was that with de Broglie's wave description one could find one comprehensive model for all atoms. The problem was that there was no equation of motion for this waves, and Schrödinger wanted to find one.

Now, in electrodynamics, ray optics follows from Maxwell's equations as the socalled Eikonal approximation which is a formal expansion of the wave function describing the electromagnetic fields in powers of the wavelength. It is valid if any structures relevant for the light propagation are large compared to the wavelength. In a way ray optics can be interpreted as point mechanics of "light particles", and now Schrödinger turned this line of argument around and asked which wave equation leads to the equations of classical mechanics of particles.

First he studied the relativistic case, leading him to an equation we know today as Klein-Gordon equation. When he calculated the standing waves of an electron around a proton to get the energy levels of hydrogen, he got a wrong fine structure of spectral lines. This fine structure before had been explained by Sommerfeld's relativistic treatment of the Bohr model. By coincidence this even gave the correct fine-structure splitting. So it was known that it is a relativistic effect, but obviously the relativistic wave equation didn't give the right answer. So Schrödinger first investigated the non-relativistic case, leading to his famous equation, now named after him the Schrödinger equation. It turned out that it gave the correct non-relativistic spectrum (of course without fine-structure splitting), and that's why Schrödinger worked this theory out in a series of brillant papers.

Somewhat earlier Heisenberg had invented another scheme to describe the quantum theory of the atom, which has been quickly worked out by Sommerfeld, Jordan, Pauli, and Heisenberg himself to what's known as "matrix mechanics". Schrödinger could prove that his formulation was equivalent with matrix mechanics. The advantage was that it was much easier for the theoretical physicists to deal with partial differential equations than with matrices to attack real-world problems. Despite this advantage, the first non-relativistic description of the hydrogen atom was given by Pauli using matrix mechanics and a very clever trick using the accidental O(4) symmetry of the problem within matrix mechanics. With Schrödinger's equation one could find the solution without applying fancy tricks.

Then the problem arised, how to interpret Schrödinger's wave function. It was pretty quickly clear that an interpretation as a kind of classical wave, describing particles as extended (or "smeared") objects contradicted the observations since always when an electron is detected it appears as a point particle. On the other hand there were clearly "wave properties". That's why Born came up with his probability interpretation which still is the only interpretation consistent with all observations. Despite this fact the debate on the interpretation is still going on nowadays, but that's another story.

4. Sep 26, 2012

### Killtech

in mathematical terms the Schrödinger equation is to be treated as an anxiom/assumption/definition thus cannot have a proof. it is true by postulate.
the theory based on this postulate corresponds to experimental evidence which renders it a theory describing parts of our world (instead of just a theory without an application).

@vanhees71, thx for providing historical context. makes a nice reading.

Last edited: Sep 26, 2012
5. Sep 26, 2012

### Staff: Mentor

He made an analogy between mechanics and optics:

classical mechanics <--> geometrical optics (ray optics) and Fermat's principle.

quantum mechanics <--> wave optics and Huygens's principle.

6. Sep 26, 2012

### HomogenousCow

I too find it interesting how the equation came to be.
I mean the other fundamental equations all came from empirical evidence (Einstein field equations, Maxwell's equations,Newtons equations etc)
But the Schrodinger equation sort of just came out of nowhere?

7. Sep 26, 2012

### sweet springs

Hi.

Historically de Broglie first declared an idea of matter wave or de Broglie wave. Shrodinger investigated the equation that de Broglie wave satisfy.

Regards.

8. Sep 26, 2012

### JohnRC

Yes, you can in fact "derive" the time independent form of the Schroedinger Wave equation from the condition that the de Broglie wave pattern for a particle should form a standing wave.

9. Sep 26, 2012

### Staff: Mentor

That's not true. You will find the proof of it from its real basis, Galilean invarience, in Ballentine - QM - A Modern Development - Chapter 3.

Schrodenger basically developed it from analogies with wave equations in other areas of physics and the idea of De Broglie. There is an outline here in Chapter 6:
http://www.physics2000.com/PDF/Calculus2000.pdf

Thanks
Bill

10. Sep 27, 2012

### dextercioby

Whether SE can be derived or not depends on the system of axioms you'll be willing to accept. There in this respect 2 different axiomatic formulations, one due to Dirac and von Neumann which takes SE as an axiom and the other due to Weyl and Wigner which takes symmetry groups as fundamental (among them the Galilei group). In this latter formulation, the SE is a theorem. An there's the 3rd due to von Neumann in which the states are described by the density operator and the SE equation can also be seen as a theorem.

Last edited: Sep 27, 2012
11. Sep 27, 2012

### Staff: Mentor

That's it. Interestingly Ballentine defines states by the density operator (its one of his two axioms) and accepts probability invariance under the Gallelli Group which the derivation depends.

That said I think anyone would recognize that the group derivation is more fundamental and in line with the modern view that symmetry is what lies at the foundation of much of physics such as gauge fields. Schrodinger's equation may be logically equivalent but I know which sounds more reasonable as a postulate.

Also I am not sure Dirac had it as an axiom - he derived it from accepting the Poisson Brackets had the same algebraic structure in QM as in Classical Mechanics. Most definitely though many axiomatic treatments have it as an axiom such the treatment in Essential QM by Bowman - nice book but not my preferred approach.

Thanks
Bill

Last edited: Sep 27, 2012
12. Sep 27, 2012

### vanhees71

It's true that the group-theoretical definition of the representing operators of observables is better than hand-waving "correspondence principles" like "canonical quantization", which latter only works by chance for the Heisenberg algebra in Cartesian coordinates. Even a simple thing like transforming this standard quantum mechanics into spherical coordinates doesn't work in the naive way, but one has to use the correct vector calculus of the differential operators $\vec{\nabla}$ and the $\Delta =\vec{\nabla}^2$. The reason behind this is of course the group-representation theory of the rotation group SO(3) (or it's covering group SU(2)).

The group theoretical method thus gives a better founded quantum theory also for more complicated systems like the quantization "spinning top" (as, e.g., a model for rotational modes of molecules). A nice discussion of such issues can be found in Hagen Kleinert's textbook of path integrals.

13. Sep 27, 2012

### Aniket1

Thanks

14. Sep 28, 2012

### Aniket1

I read chapter 6. I did not understand why the equation is derived assuming an exponential solution (replacing ω by d/dt and k by d/dx). The solution could've had any other form.

15. Sep 28, 2012

### Staff: Mentor

Its in Chapter 3 not 6 and is derived from the commutation relations of the generatotors of the Galilei Group. Chapter 6 is about the Harmonic oscillator and has nothing to do with the derivation of the Schrodinger equation.

If you want to discuss that best to start another thread.

Thanks
Bill

16. Sep 29, 2012

### Meselwulf

He somewhat... guessed it if my memory serves me well.

17. Sep 30, 2012

### ycheff

We can get equation of free wave by combination of Maxwell equations.
Schrödinger simply added potential energy to the free wave equation. It was not formal proof.

18. Sep 30, 2012

### Killtech

well, you can of course derive it from another set of equivalent QM axioms/postulates. but starting with a different set of definitions for the theory you won't get any closer of getting a derivation of the theory itself since your new postulates cannot be derived from anything else. so you can just begin as schrödinger did with the equation (+its fourier transform for the impulse state + measurement postulates)

the schrödigner eq. for a QM particle cannot be derived from any set of axioms of classical physics without adding additional postulates.

Last edited: Sep 30, 2012
19. Sep 30, 2012

### Staff: Mentor

If you cant see that Galilean Invariance ie the laws of physics are the same in all intertial reference frames and absolute time is more fundamental than Schrodinger's equation then I think you have a funny view of physics. Its like when people learn of Noethers Theorem and the claim is made it, along with invarience principles, implies conservation laws it is immediately evident that is the true basis of conservation laws even though the two are logically equivalent. The other point is Galilean Invariance, especially the idea of absolute time, is an assumption that is usually understood and not explicitly stated in axiomatic treatments of non relativistic QM that assume Schrodinger's equation - you can probably derive it - although I have never seen it done - so I am not sure how complete the other axiomatic treatments are anyway.

The proof does not assume classical principles only - it is based on the two fundamental postulates of QM you will find in the previously referenced book by Ballentine that other axiomatic treatments also assume.

The issue is this - invariance principles are obviously more fundamental than stuff like conservation laws and Schrodinger's equation that are derived from them even though they are logically equivalent. And besides the number of axioms is smaller than assuming each of those separate things.

Thanks
Bill

20. Sep 30, 2012

### Killtech

this is exactly what i was referring to! the core underlying structure of the later derived schrödinger equation comes from the fundamental QM postulates spread over chapter 2 of that book as far as i could see it browsing through. these postulates are even less intuitive then the schrödinger equation and probably the reason why QM didn't start with this algebraic formulation.

while you may like this approach more it does not make the usual start from schrödinger equation any wrong nor does it yield any more results. what you start from is rather personal preference. after all you can derive under which transformation schrödinger equation is invariant to get to the approach the book chooses.

it is true that bellentines approach is more general because it doesn't require modifications to the postulates to get the relativistic case. but as said before this approach loses any intuition by postulating nature can be described in terms of eigenvalue problems on a infinite dimensional hilbert space...

Last edited: Sep 30, 2012