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The double-slit experiment shows the fundamentally probabilistic nature of quantum mechanical phenomena. On Wikipedia one can read:

This type of experiment was first performed, using light, by Thomas Young in 1801, as a demonstration of the wave behavior of light.

Einstein proposed interpreting the square of amplitude as probability density (Born's rule). However, who was the scientist in the history of quantum mechanics, who was the first to present a closed formula for the probability density that a photon passing through the slits will hit the screen at a certain point say $x$? I know this function is the square of the underlying wave function or the mod square, when the underlying wave function is given as a complex function. What does the wave function look like according to this scientist and what does the probability density function look like? Actual formulas of the wave function and the probability density function and literature references would be welcome.

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    $\begingroup$ When you researched this, how far back were you able to trace the existence of such formula(s)? It is considered good form here to state one's prior research as part of the question to avoid unnecessary duplication of work. $\endgroup$
    – njuffa
    Jan 5, 2022 at 8:11
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    $\begingroup$ By that point, after a century of diffraction experiments on light, what you call "formulas" were standard background for any practicing physicist: so a mere word, e.g. Einstein's, would instantly evoke such formulas. Which aspect of the "formulas" are you referring to? $\endgroup$
    – Cosmas Zachos
    Jan 5, 2022 at 16:16
  • $\begingroup$ My guess is that this was A-G Fresnel. (Of course he did not speak of "probability density of photons" but rather of "light intensity" but the formula is certainly due to him. $\endgroup$
    – Alexandre Eremenko
    Jan 6, 2022 at 14:48

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