In his lectures on Quantum Physics, Richard Feynmann derives the Hamiltonian matrix as an instantaneous amplitude transition matrix for the operator that does nothing except wait a little while for time to pass.(Chapter 8 book3) The instantaneous rate of change of the amplitude that the wave function is in a specific state is the sum or integral of the Hamiltonian times the wave function. He then says that for a particle in position coordinates (Chapter 16 section 12) the integral of the Hamiltonian matrix times the wave function equals the - h/2m.Laplacian plus the potential. This is Shroedinger's equation and he goes on to say that there is no derivation of this. Shroedinger just came up with it. My question is: starting with the Shrodinger equation how do we find the Hamiltonian matrix? I also wonder whether there is a better motivation than Feynmann's explanation.