Deriving the Hamiltonian

1. Apr 15, 2010

lavinia

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.