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.