I would put it this way: Time dependence, therefore dispersion is in direct connection with the model Hamiltonian. The uncertainty principle is independent of t

I would put it this way: Time dependence, therefore dispersion is in direct connection with the model Hamiltonian. The uncertainty principle is independent of time dependence and the model Hamiltonian. I think I made a mistake about massless quantum particles, since in 3D a packet disperses in all directions. But in 1D it works. It can't stay stationary, but it moves with "c" in either direction but it stays in a localized area, still the uncertainty principle still holds.