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.