Does exciting an electron across the band gap change either its position or its localization? I suspect that exciting an electron from its valence band to conduction band doesn't change its position, since the difference between the two bands are just their energies, but I want to know for sure.
But what about its localization? That is less intuitive to me.
 A: When you are exciting an electron from VB to CB, you normally talk in terms of transition from a state $|k\rangle$ to $|k'\rangle$ and $E(k')>E(k)$. Specifying $|k\rangle$ automatically rules out an precise determination of $|x\rangle$ as the state $|k\rangle$ is spread over the entire real space with specific weight at each point.
A: The conduction and valence band electron are only differed by their Wannier orbital (wave function within each unit cell), so exciting a Bloch electron from one band to another changes the Wannier orbital of the electron, for example, from a bounding orbital to an anti-bounding orbital. In the simplest 1D example of a periodic potential, exciting the electron at the Brillouin zone boundary corresponding to shifting the antinode of the electron wave function from the low potential position to a higher potential position. In this sense, exciting electron across the band can be understood as moving its most probable position within the unit cell from a lower potential region to a higher potential region. However it is wrong to think that the Bloch electron has a definite position or localization center in the lattice. The Bloch electron has the same probability to appear in each unit cell through out the whole lattice. What is changing is only the shape of the electron cloud in the unit cell.
