Why can we use single electron Schrodinger equations to describe electrons in solids? A solid is clearly a many electron system. Yet we often use single electron Schrodinger equations to calculate the quantities of interest. Probably this is most common in semiconductors. Why is that, or maybe I should ask what lets us do that?
 A: That's a good question. My PhD was in solid state physics. In principle you should calculate the electronic properties of a solid by having a Schrodinger equation that takes into account the interactions between each of the electrons and the nuclei. In practice that is utterly impossible, so you adopt the one-electron model in which you represent the effects of all the other electrons and nuclei as an averaged-out potential. The results you get are pretty accurate. So what allows you to get away with it seems to be that the individual electron-electron interactions are not important to the overall effect. It's a bit like modelling a person's walk down a crowded street- rather than keep track of all the times they have to dodge this way or that to avoid someone in front of them, you can just assume that their walking pace is reduced overall by some factor. Provided you make a sensible choice of the factor you can create realistic estimates of how long the walk takes.
A: I would like to build on @Marco_Ocram's answer.
What we are discussing here is the "independent electron approximation" and more complex elaborations upon it like Hartree Fock - and why they seem to work so well when the system is clearly an interacting many electron system.  Quoting from Ashcroft and Mermin in their textbook "Solid State Physics", in the chapter "Beyond the Independent Electron Approximation" under the heading "Fermi Liquid Theory":

"... even in the Hartree-Fock approximation we continue to describe the stationary electronic states by specifying which one-electron levels are present ..."
  "it is still far from obvious that an independent electron description, with suitably modified energies, will be anywhere near the mark in describing the electrons in an actual metal.  There is, however, reason to expect that this may be the case for electrons with energies near the Fermi energy.  The argument, due to Landau, can be divided into two stages.  The first is fairly straightforward, but the second is very subtle indeed."

Now, when Ashcroft and Mermin describe something as "very subtle indeed" I shudder to summarize it, but here goes:
It is possible to re-map all the individual electron levels into quasi-particles to maximize their independence and minimize interactions.  Then we note that the available energy floating around the system, $k_B T$, leaves the deep occupied states and the high energy unoccupied states unable to play a role.  Thus, when we can warp things like periodic potential energies, effective masses, and what-not in a band-structure-type paradigm we have enough degrees of freedom to describe these quasi-electrons or "collective excitations" by one particle Schrodinger equations.
I encourage reading the original discussion in the text.
