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What made scientists believe that we should calculate probability $P$ as the $P = \left|\psi\right|^2$ in quantum mechanics? Was it the double slit experiment? How? Is there anywhere in the statistical mathematics a similar equation where we calculate probability like $P=\left|x\right|^2$ where $x$ is whatever?

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    $\begingroup$ "[F]or a detailed reconstruction of the historical origin of the Born rule within the context of quantum mechanics," see: J. Mehra and H. Rechenberg, The Historical Development of Quantum Theory. Vol. 6: The Completion of Quantum Mechanics 1926–1941. Part 1: The Probabilistic Interpretation and the Empirical and Mathematical Foundation of Quantum Mechanics, 1926-1936, Springer-Verlag, New York, 2000. Source. $\endgroup$ – Keep these mind Feb 8 '13 at 13:46
  • $\begingroup$ I'm not quite sure what the historical origin was, but the Stern-Gerlach experiment and particularly variations with multiple Stern-Gerlach filters can teach us a lot about probabilities in QM. I might write up a more detailed answer in the future but at the moment I don't really have the time. $\endgroup$ – Wouter Feb 8 '13 at 15:12
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    $\begingroup$ Earlier, related question by the same poster: physics.stackexchange.com/q/51962 $\endgroup$ – dmckee Feb 8 '13 at 15:44