How Schro discovered his equation

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I keep reading that Schro's equation was the best one that fit the data. Well, what data was he looking at? I keep waiting to see the data that he analyzed to come up with his equation and I never run into it.
 
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I'm a little skeptical that no one know the answer to this
 
I think the main thing that first convinced people the Schrodinger equation was right was that it predicted the correct spectrum for the hydrogen atom.
 
Not an expert in QM. AFAIK, Schrödinger's equation is quite different from the classical wave equation. The former is an equation for the dynamics of the state of a (quantum?) system, the latter is an equation for the dynamics of a (classical) degree of freedom. As a matter of fact, Schrödinger's equation is first order in time derivatives, while the classical wave equation is second order. But, AFAIK, Schrödinger's equation is a wave equation; only its interpretation makes it non-classical...
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Is it possible, and fruitful, to use certain conceptual and technical tools from effective field theory (coarse-graining/integrating-out, power-counting, matching, RG) to think about the relationship between the fundamental (quantum) and the emergent (classical), both to account for the quasi-autonomy of the classical level and to quantify residual quantum corrections? By “emergent,” I mean the following: after integrating out fast/irrelevant quantum degrees of freedom (high-energy modes...
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