The best guide to Schroedinger's thinking is his 4 lectures, which are
available in English. I read them when I was a grad student end found
them very understandable.
Shroedinger, as I recall, was guided by the Hamilton-Jacobi equation,
which is first order in the time (see Corben and Stehle for an exposition
that is very close to quantume mechanics). In the Hamilton-Jacobi
viewpoint the Hamiltonian is the generator of time translations. There
is a characteristic function that describes the classical path. In
quantume mechanics the path becomes uncertain by an amount h.
Someone who knew him once told me that he was also guided by his interest
in the physics of violin strings.
Feynman was apparently guided by somewhat similar thinking when he
developed his path-integral formulation of Quantum Mechanics. For an
introduction, one can do much worse than to read the chapter on the
principle of least action in Volume 2 of the Feynman lectures.
--
"Don't push the river, it flows by itself"
Frederick Perls