How did Schrödinger find his equation?

[Tumorsito]

So the question is pretty simple, how did he came up with the wave function and why does the Schrödinger eq model / predict the change of the wave function throw time?

 

[Isaac0427]

I have a little commentary I can give a little later, but for right now I would suggest reading the reasoning from Schrödinger himself:
http://web.archive.org/web/20081217...toricPaper/Schroedinger/Schroedinger1926c.pdf

 

[fresh_42]

Here is what Wikipedia (and Google translate) has about it:

The equation named after him was postulated by Schrödinger in 1926 . The starting point was the idea of matter waves, going back to Louis de Broglie , and the Hamilton-Jacobi theory of classical mechanics. The effect $S$ classical mechanics is identified with the phase of a matter wave. As soon as typical distances are smaller than the wavelength, diffraction phenomena play a role, and the classic mechanics must be replaced by a wave mechanics.

The Schrödinger equation cannot be derived from classical physics, but is a postulate. Formally, however, the Schrödinger equation can be derived from the Hamilton function (expression for the energy) of the problem under consideration
$$
E=\dfrac{\mathbf{p^2}}{2m} +V(\mathbf{r},t)
$$
can be derived by replacing the classic quantities energy, momentum and location according to the correspondence principle with the corresponding quantum mechanical operators :
\begin{align*}
E&\longrightarrow \hat{E}=i \hbar\dfrac{\partial}{\partial t}\\[6pt]
\mathbf{p}&\longrightarrow \hat{\mathbf{p}}=-i \hbar \nabla\\[6pt]
\mathbf{r} &\longrightarrow \hat{\mathbf{r}} = \mathbf{r}
\end{align*}
Subsequent application to the unknown wave function $\psi = \psi (\mathbf{r}, t)$ gives the Schrödinger equation
$$
i \hbar \dfrac{\partial \psi}{\partial t}=-\dfrac{\hbar^2}{2m}+\Delta \psi + V\psi
$$

In the same way, the Hamilton function can be converted into a Hamilton operator.

Historically, Schrödinger started from Louis de Broglie's description of free particles and introduced analogies between atomic physics and electromagnetic waves in the form of de Broglie waves (matter waves):
$$
\psi(\mathbf{r},t) = A \,\exp\left(-\dfrac{ i }{\hbar}(Et- \mathbf{p}\cdot\mathbf{r})\right)
$$
in which $A$ is a constant. This wave function is a solution to the Schrödinger equation just mentioned
$V (\mathbf{r}, t) = 0$. In the statistical interpretation of quantum mechanics (founded by Max Born ) there is the square of the amount $|\psi|^{2}$ the probability function density of the particle to the wave function.

 

[George Jones]

By going on a winter skiing vacation in the Alps with his mistress instead of his wife.

In the late fall or early winter of 1925, Schrödinger gave a colloquium on de Broglie's work. Peter Debye attended this colloquium and commented that if Schrödinger wanted to talk about waves, he should have a wave equation.

Schrödinger first looked for a relativistic wave equation, and he "came up with" the Klein-Gordon equation (before Klein and Gordon). This gave the wrong energy spectrum for hydrogen, as it does take into account spin. He then went on his vacation, made a non-relativistic approximation, and "came up with" the Schrödinger equation for hydrogen.

Arriving home, he separated variables, and found that he could not solve the radial equation. He went to his wife's lover, Hermann Weyl, for help with solving this.

 

[vanhees71]

All this can be nicely read in the biography

W. Moore, Schrödinger - Live and Thought

 

[sysprog]

Michael Raymer uploaded this paper, which he co-authored: https://www.researchgate.net/profile/Michael_Raymer/publication/260262749_The_Maxwell_wave_function_of_the_photon_Invited_Paper/links/0fcfd511571719af6d000000/The-Maxwell-wave-function-of-the-photon-Invited-Paper.pdf (available for pdf download here: https://www.researchgate.net/publication/260262749_The_Maxwell_wave_function_of_the_photon_Invited_Paper)

and which contains this illustration:

Fig. 1 Flow chart for derivations of electron and photon wave equations, m = rest mass, s = spin, v = velocity.

 

[vanhees71]

Well, are the problems with the "non-local" operator in the 2nd box discussed in this paper? I don't click on research gate links anymore.

 

[hilbert2]

Well, are the problems with the "non-local" operator in the 2nd box discussed in this paper? I don't click on research gate links anymore.

Didn't find this mentioned anywhere when I read it.

 

[vanhees71]

Well, the idea of "wave functions", i.e., a 1st-quantization formalism for (interacting) relativistic QT, is at least problematic. Also phenomenologically QT phenomena at realtivistic energies are characterized by creation and distruction processes of particles, and that's why QFT ("2nd quantization") is anyway the much more convenient tool to describe relativistic QT than any 1st-quantization formalism. The only example I have for the latter is Dirac's formulation of QED as hole theory. As far as I know it's indeed equivalent to the standard formulation as a relativistic local QFT. The latter formulation is way more convenient than the former, and that's why nowadays nobody teaches QED in term's of Dirac's hole theory, as far as I know.

 

[sysprog]

Well, are the problems with the "non-local" operator in the 2nd box discussed in this paper? I don't click on research gate links anymore.

I guess I'm not as 'seasoned' as you are in that regard $-$ the paper doesn't discuss 'Schrödinger steering', but it does make brief reference to the fact that the photon doesn't exhibit the locality that is characterizable with Schrödinger's ('($v<<c$)-bounded' approximation) equation.

Just after Fig. 1 the paper says this:

​

In modern terms, a photon is an elementary excitation of the quantized electromagnetic field. If it is known a priori that only one such excitation exists, it can be treated as a (quasi-) particle, roughly analogous to an electron. It has unique properties, arising from its zero rest mass and its spin-one nature. In particular, there is no position operator for a photon, leading some to conclude that there can be no properly defined wave function, in the Schroedinger sense, which allows localizing the particle to a point. On the other hand, it is known that even electrons, when relativistic, don’t have properly defined wave functions in the Schroedinger sense [2,3], and this opens our minds to broader definitions of wave functions. In relativistic quantum theory, one distinguishes between charge-density amplitudes, mass-density amplitudes, and particle-number density amplitudes. These can have different localization properties. Photons, of course, are inherently relativistic, so it is not surprising that we need to be careful about defining their wave functions.​

And later:

 

[vanhees71]

So it seems to be a good paper. Is there a link to a respectable source like the arXiv. The problem with research gate is that once I wanted to download a paper, for which I had to register, and after that my e-mail inbox was spammed like hell by all kind of information about papers I didn't want to read ;-)).

Indeed, a 1st-quantization formalism ("wave function") for photons is at least problematic. I'd not use it at all. Photons are created and destroyed (emitted and absorbed) all the time. So why should one insist on a wave-function picture.

 

[sysprog]

@vanhees71 Here's an arXiv link for the paper: https://arxiv.org/ftp/quant-ph/papers/0604/0604169.pdf

 

[vanhees71]

I'll have a look at it. It's a theory paper typed in MS Word though

 

[Demystifier]

It's a theory paper typed in MS Word though

In my experience, such papers are either trivial or wrong or both. And they never use units $c=1$ or $\hbar=1$. And usually work with $h$ and $\nu$ rather than $\hbar$ and $\omega$.

 

[hilbert2]

In my experience, such papers are either trivial or wrong or both. And they never use units $c=1$ or $\hbar=1$. And usually work with $h$ and $\nu$ rather than $\hbar$ and $\omega$.

I think conference proceedings contain more MS Word papers than actual journal articles do.

 

[jtbell]

http://web.archive.org/web/20081217...toricPaper/Schroedinger/Schroedinger1926c.pdf

I summarized the basic argument from that paper in a long-ago post:

https://www.physicsforums.com/threads/schrodingers-equation.57867/#post-418069