On my way to learn about the very beginning of quantum mechanics and its different formulations, starting with Heisenberg infinite matrices and Schrödinger's wave functions, I can really not find till now a single reference in which it is explained how Heisenberg and Schrödinger were doing quantum mechanics i.e. determining probabilities about measurements for positions and general observables within their own formulation framework of, respectively, infinite matrices and wave functions (i.e. without talking about Heisenberg and Schrödinger's pictures inside a Hilbert space 𝐿2(ℝ) for example). I mean for example for Schrödinger, given a wave function 𝜓(𝑥) of a fixed system, say an electron, what was exactly his interpretation of 𝜓(𝑥)? (Before Born's interpretation came I mean) and how he did to do computations with it in order to predict the probabilities for the position, the momentum the energy and so on?
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$\begingroup$ Others will surely provide several references to look at, but I thought I'd mention a book I've had for a long time (since 1977 I think) and keep wishing I'd take the time to go through it -- The Quantum Physicists and an Introduction to Their Physics by William H. Cropper (1970). Every time I look over this book I'm amazed at how well it appears to cover the type of concerns you bring up. Moreover, at least to me, the book seems (continued) $\endgroup$– Dave L RenfroCommented Nov 18, 2023 at 20:26
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$\begingroup$ pitched at that rare level between a gobbledygook profusion of technical words without really conveying any understanding beyond how the words are used by physicists (what I find a lot of popularizations of physics do, especially recent decades stuff involving Hawking, Penrose, Witten, etc.) and highly technical math jargon that to me can often appear designed to conceal whether an argument is mathematically rigorous or only mathematically suggestive. (moments later) See also A technical and historical introduction to quantum mechanics. $\endgroup$– Dave L RenfroCommented Nov 18, 2023 at 20:26
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$\begingroup$ Thank you so much for the reference @Dave L Renfro $\endgroup$– user19358Commented Nov 18, 2023 at 21:44
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I don't think Heisenberg and Schrödinger calculated probabilities at the beginning, as the role of probabilities in quantum mechanics was not known then. Schrödinger initially interpreted the squared magnitude of the wave function as the charge density.
