From that, in the low momentum limit, you will get the Schroedinger Equation with spin-orbit interaction term as well. I remember the derivation in Lewis Ryder's Quantum Field Theory for relativistic bispinor field. Already by the last paper he references Born's interpretation of the wave function, which he does not seem to like. (The third introduces "Rayleigh-Schrödinger" perturbation theory and has applications.) Actually, Schrödinger gives what we would recognize as the square of the time-dependent equation we know - basically he avoids complex-valued equations/solutions which he didn't think could be physical. The time-independent equation came first, Schrödinger introduced the time-dependent equation in his fourth paper. I remember the reasoning being pretty hard to follow the last time I read it, which was a while ago, but in general these sorts of big breakthroughs are a combination of inspired guesswork and even a bit of luck. It looks like you can see the papers here. Schrödinger's second QM paper is basically him trying to explain his thought process behind coming up with his equation (the first paper was just presenting the equation, solving it for Hydrogen, and showing that it reproduces the famous Rydberg formula).
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