  Without our ability to model and manipulate the band structure of
semiconducting materials, the modern digital computer would be impractically
large, hot, and expensive. In the undergraduate QM curriculum, we studied the
effect of spatially periodic potentials on the spectrum of a charged particle
in one dimension. We would like to understand how to extend these methods to
model actual crystalline materials. Along the way, we will explore the
construction of periodic potentials in three dimensions, and we use this
framework to relate the single-particle Hamiltonian to the potential
contribution from each atom. We then construct a crude model system analogous
to the semiconductor silicon, and demonstrate the appearance of level splitting
and band gaps as the strength of the potential is varied, in accordance with
our intuition from the one-dimensional case. We discuss refinements of the
model to include many-particle effects, and finally we show how a careful
choice of the potential function leads to good agreement with the correct band
diagram for silicon.
