Semiclassical Models For Quantum Systems And Band Crossings

Multibandquantumsystemsplayfundamentalrolesinthestudyofsolidstatephysicsand quantum chemistry. They are usually described by some Schr(o|¨)dinger equations, suchastheSchr(o|¨)dingerequationwithaperiodicpotentialtodescribethemotionofelectronsinacrystallattice,andthemultilevelSchr(o|¨)dingersystemtodescribethemoleculardynamicsin chemical reaction. These equations can be diagonalized through specific basis repre-sentation, in which the techniques such as Bloch decomposition and Born-Oppenheimerapproximation may be used. Under the adiabatic assumption, the energy bands are wellseparated and the Schr(o|¨)dinger equation has a semi-classical limit governed by the Liou-ville equation. However, when the bands get close to each other, the inter-band transitionphenomena can no longer be ignored in the band crossing zone. The adiabatic approxi-mation is no longer valid, and new semi-classical models need to be developed.In this thesis, we derive the coupled inhomogeneous semi-classical Liouville sys-tem for both the periodic Schr(o|¨)dinger equation and the two-level Schr(o|¨)dinger system.The main idea is using the Wigner transform and the basis representation, retaining theoff-diagonal terms in the Wigner matrix which cannot be ignored near the point of bandcrossing. Our coupled Liouville systems have the following properties:These systems are able to describe the inter-band transition phenomena very well;Thesesystemsareconsistenttotheadiabaticapproximationiftheadiabaticassump-tion holds,;Themultiscaleasymptoticpropertiesofthesesystemsallowonetodevelopefficientnumerical methods.A domain decomposition method that couples these non-adiabatic models with theadiabatic Liouville equations is also presented for a multiscale computation. Solutions ofthese models are numerically compared with those of the Schr(o|¨)dinger equations to justifythe validity of these new models for band-crossings.