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If I am right Born published his interpretation of the wave function after Schrodinger published his wave equation. However, according to my QM textbook, all the expected values of quantities (like energy and momentum) are derived from Born's interpretation, i.e. the wave function can only make any sense with the statistical interpretation. So, how and why did Schrödinger derive his equation for a function whose interpretation did not exist at that time?

(Kindly pardon me if the question is stupid, I am just a beginner!)

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    $\begingroup$ I think before Max Born proposed his interpretation, $m|\Psi|^2$ was thought to be the actual mass density of electrons. They probably thought an electron is not a point particle. But later it was understood that scattering experiments can only be explained using Born's interpretation. $\endgroup$ Feb 28 at 8:27
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    $\begingroup$ "How is it possible for Born's interpretation of the wave function to be published after Schrödinger published his equation?" -- History works in mysterious ways. :) Joking aside, this is a great example of how scientific concepts and theories arise from a "blooming and buzzing confusion" where people have partial insights that don't fully make sense and aren't completely coherent until enough pieces of the puzzle come into place. As for the actual physics, I think the answer by Kasi explains it very nicely. :) $\endgroup$
    – ACat
    Feb 28 at 21:04

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What I guessed in the comments was true. Schrödinger mentioned in his $1926$ paper (see below) that "the real continuous partition of the charge is a sort of mean$\dots$".

So he got the right equation but he interpreted it wrongly. He believed that in reality electron has a continuous charge distribution. But he did mention that "no very definite experimental results can be brought forward in favour of his hypothesis".

The following is a relevant excerpt from Schrödinger's $1926$ paper "An Undulatory Theory of Mechanics of Atoms and Molecules" in The Physics Review (Vol. $28$, No. $6$, pp. $1067$):

But this amounts to making the following hypothesis as to the physical meaning of $\psi$ which of course reduces to our former hypothesis in the case of one electron only: the real continuous partition of the charge is a sort of mean of the continuous multitude of all possible configurations of the corresponding point-charge model, the mean being taken with the quantity $\psi\psi$ as a sort of weight-function in the configuration space.

No very definite experimental results can be brought forward at present in favor of this generalized hypothesis. But some very general theoretical results on the quantity $\psi\bar{\psi}$ persuade me that the hypothesis is right. For example, the value of the integral of $\psi\bar{\psi}$, taken over the whole coordinate space proves absolutely constant (as it should, if $\psi\bar{\psi}$ is a reasonable weight function) not only with a conservative but also with a non-conservative system. The treatment of the latter will be roughly sketched in the following section.

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    $\begingroup$ As a general rule posting images of text is very strongly discouraged as images are not searchable. The site has simple methods to allow quoted text to be easily distinguished from the main question body. $\endgroup$ Feb 28 at 15:28
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    $\begingroup$ @StephenG I transcribed it. $\endgroup$
    – benrg
    Feb 28 at 20:34
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    $\begingroup$ @benrg thanks for transcribing. I added the page number as it is no longer visible. $\endgroup$ Feb 28 at 20:38

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