Understanding conduction as a dynamic process in time In pure metals with perfect periodicity, the Bloch wavefunction representing an electron, is delocalized throughout the crystal for all times and is an eigenstate of the one-electron Hamiltonian. Therefore, there is no change or "propagation" of this wavefunction in time. However, real conduction takes place in time. 
How does one physically or mathematically understand the process of conduction dynamically i.e., in terms of a wavefunction which travels down the crystal from one lead of the battery to another lead connected across the sample in time? 
I want to understand the process of conduction and for simplicity neglecting electron-electron interactions and all deviations from periodicities (such as ionic motion, defects, impurities etc).
 A: Unfortunately, conductivity cannot really be understood in such a simple setting. For free non-interacting electrons, conductivity is infinite, which is just a consequence of momentum conservation. In a perfect lattice, the conductivity is no longer infinite, but it becomes zero instead, or more precisely the time-averaged value becomes zero because the electrons undergo Bloch oscillations. A finite conductivity of the type we actually observe requires some way to dissipate momentum besides the lattice, which requires scattering and means that Bloch wavefunctions are not an exact eigenstate.
In view of this, the simplest case you can consider with finite conductivity is probably that in which the electrons are scattered weakly by impurities, and can be approximated as wavepackets which have a finite spatial extent but still have relatively well-defined quasimomenta, over a sufficiently short time that the spatial extent of the wavefunction doesn't change too much. In this case, the expectation value of the position bounces around as the electron collides off impurities and the dynamics are not too different from a classical Drude model picture. The Bloch oscillations from the lattice could alter this in principle, but in a realistic metal the atoms never build up enough momentum to approach the edge of the band. Unfortunately, I don't really know of a model of conduction that is more sophisticated than this basically classical picture but still simple enough to be intuitive. Quantum transport is a tricky subject.
