Deriving the Bloch's theorem

Bloch’s theorem is a general statement about the shape and symmetry of the wavefunction of electrons in a periodic potential, such as in crystalline solids. Bloch’s theorem unravels the subtleties of the electronic behavior in solids, such as why electrons can travel large distances in a crystal lattice, much longer than interatomic spacing, without scattering. In this video, we’re going to go through the basics of Bloch’s theorem, explaining why it is the case and its implications to the behavior of electrons in crystalline materials. We will begin by discussing the topic of spatially periodic potentials and periodic boundary conditions for the wave function of electrons. Next, we’re going to state Bloch’s theorem and prove it by directly solving the time-independent Schrodinger equation in a periodic potential.

Technical Content: Duarte Sousa, Tony Low
Video Production: Katie Low, Tony Low

Erratum
9:25: k is only defined modulo G, thus we can arbitrary shift k by +-G. The step to obtaining the matrix form uses this fact.