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 withing 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 $L^2(\mathbb{R})$ for example). I mean for example for Schrödinger, given a wave function $\psi(x)$ of a fixed system, say an electron, what was exactly his interpretation of $\psi(x)$? (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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    $\begingroup$ I really don't want to be too pedantic, but please: It is Schrödinger (or, if you don't have "ö", use "oe"). Regarding your question: Have you tried to read (possible translations of) the first papers of Schrödinger? $\endgroup$ Nov 18, 2023 at 14:31
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    $\begingroup$ This looks like a history, HSM project, not one focussed on this site. If you are conversant in the Dirac notation connection between the two formulations, there is no point in asking here, as opposed to HSM... $\endgroup$ Nov 18, 2023 at 15:05
  • $\begingroup$ Heisenberg only managed to do QHO and some anharmonic corrections thereof, but boldly claimed that it is a general method that can solve every system. The reason why he did not get much further than that is because his matrices required both the input and output states to be well-defined, whereas Schrödinger was working for single states (wavefunctions), which directly contradicted Heisenberg's assertion that only measureable quantities should appear in quantum theory. Schrödinger did not try to interpret his wavefunction. $\endgroup$ Nov 19, 2023 at 3:38
  • $\begingroup$ @naturallyInconsistent Well, the history went slightly differently. Indeed, Heisenberg was not the first to solve the problem of the Hydrogen atom with his matrix QM. But Pauli did with matrices slightly before Schrödinger did with his wave equation. Moreover, Pauli did not need input and output states. For one particle, Schrödinger had de Broglie's hypothesis as an interpretation of his wavefunction. $\endgroup$ Nov 20, 2023 at 11:49
  • $\begingroup$ @GiorgioP-DoomsdayClockIsAt-90 yes, I am aware of that. Pauli used LRL vector, which is a tour de force that is considered somewhat too difficult to include in textbooks. $\endgroup$ Nov 20, 2023 at 15:15

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Look here: Quantisierung als Eigenwertproblem E. Schrödinger, 1926 in Annalen der Physik. Link: https://onlinelibrary.wiley.com/doi/10.1002/andp.19263840404

He is solving the eigenvalue problem using an unusual method (Laplace), but gets the correct results (spherical harmonics). Of course, he interprets $E_i$ as energy levels. Regarding the eigenvectors, he isn't so sure (see page 372): He isn't considering what we know as Born's rule. He tends to associate the eigenvectors with the actual orbits ("Phasenwelle"), following de Broglie's ideas.

  • $\begingroup$ I'd have to point out that Laplace's transform is not at all an unusual method. It is part of the wave toolbox both back then and also today. He resorted to that because he did not have the mathematical chops to actually prove the quantisation at the time. It is not a big deal because shortly after publication, Sommerfeld (and many others) quickly managed to give the modern derivation. $\endgroup$ Nov 19, 2023 at 3:40

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