@Semiclassical ok.. so I guess the question becomes, how does the discrete dof of the wavefunction interact with the particle then? Because it must affect its velocity. Reading the
Wikipedia page, it says the the probability current is given as $(\psi,D_k\psi)$ with $D_k=\nabla_k -iA$ and $A$ vector potential. But at this point I lost any sort of intuition