If an electron-hole pair is formed, where does the electron "go"? I've seen the explanation before that holes are basically electron deficiencies in an atom and that the hole "moves around" by electrons from surrounding atoms shuffling to fill that spot which is a very spatial way of viewing what a hole is. However, when I think about the band structure picture, the excitation of an electron from the valence band to the conduction band leaves behind an empty spot in the valence band hence being the "hole" but that's just a momentum space picture.
What's the connection and which one is more correct? How can I reconcile the bandstructure picture and the emergence of holes in a semiconductor. Is the "spatial picture" just a useful but incorrect tool for explaining the origin of holes?
 A: It is also not very clear to me. I try below an explanation.
For a single piece of doped semiconductor, I agree that there is no meaning in talking about holes "moving" from place to place in the k-space.
But in a junction, the band structure is not constant due to the diffusion of the dopants atoms. It can be thought as several thin slices of different materials held together.
When a hole is created in one of that slices, it is in this case localized. Because it is related to the local band structure.
As the slices are in contact, holes, as available energy states, can diffuse to the neighbours.
A: In energy/momentum space the solutions of the wavefunctions give specific energy levels, with a width due to quantum mechanical uncertainties due to higher order exchanges.Because of energy and momentum conservation, there is no "probable energy" for a given electron (except within the width). There are no independent conservation laws in space-time.  The energy momentum  conservation laws (energy momentum) constrain the space location ,orbitals, of electrons around the atoms and molecules to be in a "probable location".
See the possible  orbitals of an electron around an atom.

If the electron is in a 2p state, i.e.higher than ground energy, there is a probable hole in the 1s state. In space time it means that part of the time there is no electron where there would have been if the electron were in the ground state.
Extending this to a solid is not easy , the way the band structure model fits the solid state data, because again energy and momentum conservation affect the space-time probability loci in a probability fuzzy way, but this does not mean that the concept of a "hole" is not correct.  It is a probable hole in a complicated quantum mechanical orbitals solution for the solid.  There is no contradiction with the band theory, imo.
To answer the title :

If an electron-hole pair is formed, where does the electron “go”?

To an appropriate to its energy level orbital of the solid, with the probability given by the solution of the space-time wavefunction of the solid.
