The fact that the wavefunction - the very quantity whose evolution schrod eqn describes - is not an observable gives it a very different flavor than the electromagnetic field. From Wordnik.com. [Everything You Ever Wanted to Know About Quantum Mechanics, But Were Afraid to Ask] Reference
Thinking more about the question of “deriving” the schrod eqn, maybe an analogy would be to the “derivation” of maxwell’s eqns in, e.g. From Wordnik.com. [Everything You Ever Wanted to Know About Quantum Mechanics, But Were Afraid to Ask] Reference
8.6 in particular shows how path integral approach is equivalent to schrod eqn, and maybe you’ll find path integral a more intuitive starting point?. From Wordnik.com. [Everything You Ever Wanted to Know About Quantum Mechanics, But Were Afraid to Ask] Reference
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