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Say you have an electron at the Fermi level of an insulator. Then, you apply an electric field. No current flows, because all of the states of higher momentum (i.e. kinetic energy) require giving the electron a bunch of energy to get to, due to the band gap. What happens to the energy that should be given to every electron, qV? It seems like applying this field needs to be giving the electrons energy. Is it the case that they are essentially just given potential energy? In that case, is it also accurate to say that there is no classical analog for the "force" that would need to be holding them in place to keep them from moving in an applied potential, given that it is just something that falls out of the SWE and not a classical potential?

Thank you!