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CronoDAS comments on How do you tell proto-science from pseudo-science? - Less Wrong Discussion
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Huh, I've seen something vaguely similar in a physics textbook: the authors "derive" the Schrodinger equation by describing the properties that an equation has to have in order to describe an object (such as a single photon) that "interferes with itself" in the double slit experiment. Another textbook I've read simply says that "derivations" of the Schrodinger equation are basically bogus; the Schrodinger equation is an empirical formula that is chosen because it matches observations, and doesn't actually have any more justification than that.
The best discussion you are likely to find is in Ballentine. If you accept (empirically) Galilean invariance, the STRUCTURE of the Schroedinger equation falls out of group representation theory quite naturally.
The actual specifics of a problem involved picking a potential to use in the problem, and this is empirical. So if you ask the question: What equation does an electron in an atom obey? That is empirical.
If you ask: Given Galilean invariance and a 1/r potential, what equation does an electron in an atom obey? This doesn't need any more empirics.
And assuming Lorentz invariance gives you the Dirac equation, right?
Sadly, with lorentz invariance things get quite a bit more complicated. Adding in Lorentz invariance forces you to deal more directly with spin (and lets you prove spin-statisics), so you end up with the Klein-Gordon equation for spin 0, the Dirac equation for spin 1/2, and variants of the Maxwell equations for spin 1.
But you also get weird "paradoxical" effects trying to interpret the results of those equations along the lines of non-relativistic quantum, so you are forced to push towards full field theory.