In this talk we derive and compute the semi-classical limit of the Schrodinger equation with lattice potential. In [Gerard-Markowich-Mauser-Poupaud, Comm. Pure Appl. Math. 1997], the limit is derived under the assumption that energy bands are O(1) separated, namely, the system is adiabatic. However, in reality, this assumption is generically invalid. We remove the assumption, and obtain a general model by performing multi-scale variable separation with the Bloch decomposition and the Wigner transformation. Asymptotically this new full system recovers the old one in the adiabatic region. In the computation, we decompose the domain into regions depending on the distance to the energy band-crossing points, and apply associated schemes in different regions. A nature extension to the diabatic transition beyond the Born-Oppenheimer approximation will also be given at the end of the talk.

Co-authors: Lihui Chai, Shi Jin and Omar Morandi.