# Origin of the schroedinger's wave equation

## Main Question or Discussion Point

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.

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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.
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.

vanhees71
Gold Member
2019 Award
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.

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.

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jtbell
Mentor
How did schroedinger arrive at the wave equation?
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.

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?

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.

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.
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.

bhobba
Mentor
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.
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).
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

dextercioby
Homework Helper
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.

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bhobba
Mentor
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 take 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.
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

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vanhees71
Gold Member
2019 Award
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.

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.
Thanks

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
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.

bhobba
Mentor
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.
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

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

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.

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
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.

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bhobba
Mentor
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.
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

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.
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.

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.
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...

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bhobba
Mentor
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.
Those postulates (ie the operator eigenvalue and expectation postulates) are assumed in both approaches - the difference is you assume Galilean invariance instead of the Schrodinger's equation. To be specific the axioms of the two approaches are exactly the same except you have Galilean invariance in one and the Schrodinger equation in the other. Its pretty obvious to most people exposed to it which is the more fundamental approach - but to each his own I suppose. The assumption of Schrodenger's Equation looks like its pulled out of a hat - that the laws of physics are the same in inertial reference frames - well its part of many areas of physics and pretty intuitive. The assumption of absolute time is usually not stated explicitly.

Thanks
Bill

Those postulates (ie the operator eigenvalue and expectation postulates) are assumed in both approaches
no. starting with schrödinger equation + definition of <ψ|ψ> as probability distribution of location + fourier transform to obtain the probability of impulse is all you need. this is enough to derive operators of impulse and location. futhermore one finds that solving the stationary problem (eigentvalue problem) helps greatly in solving the schrödinger equation. so it's not a postulate unlike in the other approach. and finally as a matter of fact solving the hydrogen atom was more important during the development of quantum mechanics then which transformation behavior a soultion would have and as a matter of fact the lorentz invariant klein gordon equation does not solve the hydrogen atom (galileian invariance is not the general assumption one would prefer to start with if it was possible).

so assuming nature is describable by the eigenvalues of operators instead of schrödinger eq. doesn't look any less pulled out of a hat.

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bhobba
Mentor
no. starting with schrödinger equation + definition of <ψ|ψ> as probability distribution of location + fourier transform to obtain the probability of impulse is all you need. this is enough to derive operators of impulse and location. futhermore one finds that solving the stationary problem (eigentvalue problem) helps greatly in solving the schrödinger equation. so it's not a postulate unlike in the other approach. and finally as a matter of fact solving the hydrogen atom was more important during the development of quantum mechanics then which transformation behavior a soultion would have and as a matter of fact the lorentz invariant klein gordon equation does not solve the hydrogen atom (galileian invariance is not the general assumption one would prefer to start with if it was possible).

so assuming nature is describable by the eigenvalues of operators instead of schrödinger eq. doesn't look any less pulled out of a hat.
That does not allow you to derive that the eigenvalues of a Hermitian operator are the possible outcome of an observation in for example spin. The probability distribution of location is an example of the trace formula E(R) = Tr (pR) but can not be used to derive it generally. You need to assume those things in either approach.

If you want to see exactly what assumptions go into an axiomatic treatment using Schrodinger's equation - see for example chapter 8 Quantum Mechanics Demystified by David McMahon:
https://www.amazon.com/dp/0071455469/?tag=pfamazon01-20
Postulate 1: States of physical systems are represented by vectors.
Postulate 2: Physical observables are represented by operators.
Postulate 3 : The possible results of a measurement are the eigenvalues of that operator.
Postulate 4 : The probability of obtaining a given measurement result is given by the Born rule.
Postulate 5 : The state of a system after measurement is the eigenvector of the associated eigenvalue.
Postulate 6: The time evolution of a quantum system is governed by the Schrödinger equation.

The above were taken directly from Davids book but are virtually identical from what you see in other books.

Ballentine basically assumes axiom 1 to 5 but by being sneaky and assuming physical continuity (he uses it to derive the other stuff in Chapter 9) after an observation gets it down to two. He also assumed the trace and density operator form of the Born Rule which is its most general expression.

The difference is Ballentine, using those axioms, derives it from Galilean invarience which IMHO its true basis - not as a separate axiom. Galilean invarience is used in many areas of physics not just QM.

Thanks
Bill

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there are many ways if you want to start with schrödinger, McMahon presenting only one one of those many.

his postulates 2 can be left out and derived if you define how the probability densities of observables arise from the wave function which is an implicit definition of borns rule. these definitions can be seen as canonical formulation of the results of some experiments. the operator formalism itself arises from well known mathematical techniques of functional analysis to solve PDEs so it really isn't required to postulate it.

postulate 3 is somewhat void because it does not give you specific definitions for any observable e.g. of location. so actually you need one definition for each observable in any case although this is not explicitly stated to be such.

postulate 1 is unnecessary due to postulate 6 and the interpretation of the wave function through other postulates (born rule).

if you try to construct QM from experimental results the historic schödinger approach is just the intuitive way to go because the experiments hardly reveal that observables can be represented by operators.

the usual formulation via schrödinger is also analogous to most other physical theories that start with the equations of motion as a postulate (e.g. maxwell, newton, gerneral relativity, ...). these equations define what kind of objects you are dealing with (i.e. wave, particle) which i personally find more fundamental then how these transform. and all these equations of motion are pulled out of a hat - or derived from experiments. and in case of maxwell it was actually a surprise that their transformation behavior was not galilean but lorentzian. and on the other hand starting QM with a lorentz invariant theory leads you to klein-gordon equation which doesn't yield the right solutions for hydrongen.

anyway, what are we arguing about? both approaches are just the same true, mathematically equivalent, both are useful to outline different aspects of QM and none of their set of axioms can be derived from any classical theory. since i see no purpose debating personal preferences i end this discussion at this point.

vanhees71
Gold Member
2019 Award
If you want to see exactly what assumptions go into an axiomatic treatment using Schrodinger's equation - see for example chapter 8 Quantum Mechanics Demystified by David McMahon:
https://www.amazon.com/dp/0071455469/?tag=pfamazon01-20
Postulate 1: States of physical systems are represented by vectors.
Postulate 2: Physical observables are represented by operators.
Postulate 3 : The possible results of a measurement are the eigenvalues of that operator.
Postulate 4 : The probability of obtaining a given measurement result is given by the Born rule.
Postulate 5 : The state of a system after measurement is the eigenvector of the associated eigenvalue.
Postulate 6: The time evolution of a quantum system is governed by the Schrödinger equation.

Bill
I don't know this book, but from the "axioms" given above I'd conclude that one has to read it with a grain of salt and with already knowing the subject since they are only very sloppily formulated, not to say they are wrong.

Ad 1: This is not true. If this were true there would not exist a physically successful non-relativistic quantum theory since there is no unitary representation of the Galilei group that gives physically correct results. Correct is that pure states are represented by rays in Hilbert space, and this opens the possibility to introduce unitary ray representations, and this leads to the well known non-relativistic quantum theory supposed to be presented in this textbook.

Ad 2: The operators must be restricted sufficiently to represent operators. In the standard formulation they are essentialy self-adjoint operators on Hilbert space.

Ad 3: If this were true, position and momentum couldnt' be measured, because they have an entirely continuous spectrum. You must thus postulate that possible outcomes of measurements are in the spectrum of the essentially self-adjoint operators of the (corrected) postulate 2.

Ad 5: is definitely wrong. It dependends on the measurement apparatus! This postulate is not needed to formulate quantum mechanics at all. Only if you restrict yourself to ideal filter measurements, this postulate may be reformulated in some way to make sense to begin with, and then it might be derivable from the very definition of such kinds of measurements.

Postulate 6 is somewhat unrelated to the others. It depends somewhat on what you mean when speaking about "the Schroedinger equation". Within an operator-Hilbert-space based system of postulates the dynamics should be introduced by postulating the existence of the Hamilton operator of the system and defining a covariant time derivative of operators, representing observables (only in the special choice of the Heisenberg picture this covariant time derivative coincides with the usual time derivative of the operators!).

Ballentine is definitely more careful in his book, and I'd rather recommend this book than this one, which seems more to mystify than demystify quantum mechanics already with its postulates!

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