Homogenization theory for periodic potentials in the Schr?dinger equation

摘要

We use homogenization theory to derive asymptotic solutions of the Schr?dinger equation for periodic potentials. This approach provides a rigorous framework in which the key concepts in solid-state physics naturally arise (Bloch waves, band gaps, effective mass, and group velocity). We solve the resulting spectral cell problem using numerical spectral methods, and validate our solution in an analytically-solvable case. Finally, we briefly discuss the convergence of our asymptotic approach and we prove that the ground-state k = 0 effective mass is never less than the ordinary inertial mass.