A Hamiltonian model for a molecular segment or molecular chain with phonon or vibrational coupling is introduced which admits analytic solutions. A time correlation function Q(t) for the average position of an electron inserted at the end of a chain with a thermal average of the phonons is defined. A prominent feature of the dynamics is that the phonons drive the electron density to decay to a steady-state distribution along the chain. We demonstrate that two imaging methods based on the time derivatives of Q(t) at zero time are capable of producing the average velocity of the electron along the chain using a reasonable number of the time derivatives. We further show that this average velocity increases as the coupling to the phonons is increased and as the temperature is increased; that is, the decay to a steady state is enhanced in both cases.