Oh, boy... where to begin? You know what... there are so many false assumptions in this question, starting with the foundations of QM which have absolutely nothing to do with the double-slit experiment, that I am not even going to start sorting this mess out for you. I will let somebody else set you straight on them. Can the Schroedinger equation be "derived" from an equivalent variational calculus principle? Yes, but that has absolutely nothing to do with anything that your question starts out with.

@CuriousOne: double-slit experiment gives an intuition to see electron as a wave, so what is your point?

We have a few posts here every day from folks who have derived the 100% wrong intuition about quantum mechanics from hearing falsehoods about the double slit experiment. Quantum mechanics didn't start out with the double slit experiment and wavefunctions are not even necessary to do quantum mechanics. They weren't even present in QM until rather late in the game. What's worse is that focusing on the Schroedinger formalism for the single particle case robs you of developing any intuition about real physics questions, which are about relativistic field systems.

@CuriousOne: What makes you belive that I'm trying to undestand the theory in a traditional way? Now I just begin with double-slit experiment and then conclude that there is a probability wave for electron. The rest is just math. I think you are arrogant now.

I never thought for a moment that you were trying to understand the theory at all. Even so, most of your statements are false in both a traditional and a non-traditional understanding of the theory (and by the latter I mean non-traditional interpretations of physicists, not of cranks).

@CuriousOne: OK. Now it should be easy for you to point out a one mistake in my post?

We could start with the very first sentence. Then we can move on to the second sentence. How about the third sentence? Would you like me to go on? :-)

@CuriousOne: Please, go on! Explain what is wrong in the first sentence or in the second? Just easy correction is enough :)

One can't infer anything from a single experiment in physics and wavefunctions were certainly not inferred from the double-slit experiment. I am sure you know the history of quantum mechanics and the chain of evidence that led to its development? Hint: the double slit experiment with electrons is not in there. It wasn't even carried out until way after quantum electrodynamics had already been well developed. The first electron version was done in 1961. Do you know when the wavefunction enters the picture? :-)

@CuriousOne: From double-slit experiment we can conclude that there is a light-like properties in a beam of electrons, and hits on screen are probalistically distributed. Therefore for electron there is a wave and square of that corresponds to intensity/probability.

Let me know when you are ready to learn something about reality instead of regurgitating the things that you "know" and that are just not so. See you in about five years, or so. :-)

@CuriousOne: I suppose that the Holy inquisition does not allow anyone to figure QM out of double-slit experiment. But what you find difficult, I find it easy: from double-slit experiment we have all the information that we need. Then with de broglie's and Plack's relations we are done :)

So now you are implying that you have figured out the double slit experiment? You got ego, for sure. :-)

@CuriousOne: No. But from the results of double-slit experiment I can figure out that there is that probability wave.

So now you are implying that YOU have figured out from observations that are not even necessary and that were not used to derive quantum mechanics how to derive quantum mechanics? Good god, you really have a big ego! All of the words you use you have read on some websites and maybe in some books. They are the results of truly smart people who worked on this for about a century now. Seriously, friend, you can not even tie your shoelaces in scientific terms and you think you have figured this out? Wow.

@CuriousOne: Now you are saying the one can do no assumptions from experimental results! That's the end of science, my friend. All I have been saying is that from double-slit experiment we can infer that there is a "wave" that guides those damn electrons, therefore we can associate a wavefunction for electron.

I am saying that you have no idea about QM, the discovery history of QM and the reasoning of Schroedinger that lies behind the Schroedinger equation, which made Feynman say "Where did we get that (equation) from? Nowhere. It is not possible to derive it from anything you know. It came out of the mind of Schrödinger". Did you even read Schroedinger's papers? Can you even name their title? I don't think so. You have read some most likely incorrect and oversimplifying stuff about the double slit experiment and that is as far as your half-knowledge goes.

@CuriousOne: There is that strange slave-mentality in your mind, but I can assure you that we could end up with Schrödinger equation from different premises! All I have been saying that from double-slit experiment we can infer that there is a wave, just like there is a wave for light.

You think five years will be enough to do your homework about QM? Let me know when you are ready.

@Juho "Now I just begin with double-slit experiment and then conclude that there is a probability wave for electron" this is wrong. Firstly, the wave function is defined on configuration space, it isn't a wave in space. And secondly, it isn't using probability the way a mathematics textbook on probability uses the word. So it isn't a wave in space, and it isn't a probability. A wave function is one (of many) possible mathematical descriptions of a physical system.

OK. I thought that there is something wave-like phenomena in the result of double-slit experiment.

What do you mean by "I thought..."? Three people tell you that you are wrong and you are already faltering in your absolute knowledge about quantum mechanics? Tell me that's not the Juho who, just ten minutes ago, was absolutely certain he could derive the wavefunction from the double slit experiment... :-)

@CuriousOne: I said that from double-slit experiment we can infer/assume that there is a wave that can be attached to electron. That wave is only a tool to derive QM dynamics. But if there is nothing that waves, I stop assuming.