The Hamiltonian and Time Evolution for Classical Statistical Systems and Quantum Systems and a Possible Origin of exp(iEt) in the wavefunction

We consider the possibility that both classical statistical mechanical systems as well as quantum mechanical time independent systems evolve according to the Hamiltonian H. This suggests that given the energy of the system, the Hamiltonian operating on a density or wavefunction will bring back the density or wavefunction with a weight factor of E, energy. For the quantum mechanical case, E represents a frequency. We argue that it is a frequency for both the quantum and statistical mechanical case, related to how quickly the system interacts under the Hamiltonian. We also examine an aspect of interference in the quantum wavefunction which may be related to a 2x2 model of a particle which cycles back and forth as it moves with an average velocity v in a certain direction. We argue that the plane waves stirred up in a potential (or in the Fourier series of the wavefunction) are associated with periodic time motion (due to zitterbewegung). We argue that as these plane waves interfere, they create a new constant average energy throughout a system with a potential which acts as the period of periodic time behaviour, namely exp(iEt)