Berry Phase in the Molecular System H3. Master's Thesis of Jialiang Wu.
This thesis presents a comprehensive treatment of the mathematical foundations of the quantum mechanics of molecules. In particular it treats the molecular system H3, which consists of three protons and three electrons. The mathematical formalism of quantum mechanics is developed from first principles, and elementary examples of quantum systems, such as spin systems, the hydrogen atom and molecule are worked out as illustrations of the formalism. Molecular symmetry groups and their representation theory are also treated. The coordinate systems in the shape space of three nuclei are studied. Vector bundles and connections are developed and applied to the Berry-Simon connection in the complex vector bundle over the shape space whose fibres are the energy ground state eigenspaces. ``Berry phase'' is then interpreted as holonomy associated to parallel translation in the usual sense of differential geometry.
Since this thesis was defended (November, 2003) several errors or omissions have been detected in it and if they are not too extensive they will be corrected by Dan Dix. These (planned) amendations with their dates of completion are as follows.
The following topics would have been nice to have been discussed for the sake of completeness, and maybe in the future short accounts will be written by Dan Dix to summarize them.
Discussion of mathematical issues left unresolved in the thesis.