The below shows some excerpt from Feynman's lecture notes.

21–4 The meaning of the wave function

When Schrödinger first discovered his equation he discovered the conservation law of Eq. (21.8) as a consequence of his equation. But he imagined incorrectly that P was the electric charge density of the electron and that J was the electric current density, so he thought that the electrons interacted with the electromagnetic field through these charges and currents. When he solved his equations for the hydrogen atom and calculated ψ, he wasn’t calculating the probability of anything—there were no amplitudes at that time—the interpretation was completely different. The atomic nucleus was stationary but there were currents moving around; the charges P and currents J would generate electromagnetic fields and the thing would radiate light. He soon found on doing a number of problems that it didn’t work out quite right. It was at this point that Born made an essential contribution to our ideas regarding quantum mechanics. It was Born who correctly (as far as we know) interpreted the ψ of the Schrödinger equation in terms of a probability amplitude—that very difficult idea that the square of the amplitude is not the charge density but is only the probability per unit volume of finding an electron there, and that when you do find the electron some place the entire charge is there. That whole idea is due to Born.

Can anyone elaborate on this paragraph?

What was wrong with Schrödinger's understanding? What was his understanding? Which problem did he encounter?

  • $\begingroup$ Why take Feynman for granted here? You can find Schroedinger's papers and Born's papers and read them for yourself. $\endgroup$ – CuriousOne Jun 2 '16 at 0:35
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    $\begingroup$ Possible duplicate of physics.stackexchange.com/q/16387 $\endgroup$ – user108787 Jun 2 '16 at 0:43
  • $\begingroup$ The more accurate phrase describing Schrödinger's interpretation than "wrong" is Heisenberg's term, "trash" - www-history.mcs.st-andrews.ac.uk/PrintHT/Wave_matrix.html - sometimes translated as crap - math.ucr.edu/home/baez/photon/anschaulichkeit.htm Schrödinger literally said and thought that the electric charge of the electron gets dissolved, a part gets here, a part gets there. It's easy to experimentally verify that it's not the case, the electric charge in any volume is always an integer multiple of $e$. $\endgroup$ – Luboš Motl Jun 2 '16 at 4:22
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    $\begingroup$ Schroedinger interpreted $e|\psi|^2$ as charge density, just as we do now. So 1) no, he did not take $|\psi|^2$ to mean "charge density", but yes, he did use the term "amplitude equation"; 2) he definitely did not talk about "charges P and currents J" flying around the atom, but 3) he did point out that |\psi|^2 satisfies a continuity eq. which suggested to him that it likely had the significance of a "weight function" and should be normalized to 1. Now why would he think of normalizing a charge to an adimensional unit if he really had in mind the electron charge? $\endgroup$ – udrv Jun 2 '16 at 5:51
  • $\begingroup$ Not to mention that he also discussed said normalization in the context of a multi-particle wavefunction. On the other hand, Heisenberg worked for Born, who in turn worked hard on quantizing complex orbits, but with so little success that Heisenberg himself eventually decided to tackle the problem in a completely different way. Only his solution was unintuitive, whereas Schroedinger worked in configuration space, just like Born, and could calculate matrix elements, just like Heisenberg. $\endgroup$ – udrv Jun 2 '16 at 5:52

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