Rigorous Studies in Quantum Mechanics

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This project concentrates primarily on the investigation of mathematical problems that arise in molecular quantum mechanics. Because nuclei have large masses, their motion is well approximated by classical mechanics. Electrons have much smaller mass, and their motion is truly quantum mechanical. The electronic motion is influenced by the more slowly moving nuclei, and it is well described by the quantum adiabatic approximation. This intuition is the basis for Born-Oppenheimer approximations. Professor Hagedorn, his graduate students, and his collaborators will study situations where Born-Oppenheimer approximations are not sufficiently accurate to describe all the phenomena of interest. For example, they plan to develop accurate "surface hopping models" to include the possibility of electrons making non-adiabatic transitions. Models of this type have been proposed in the chemistry and physics literature, but they have not been derived from first principles. The main goal of this project is rigorously to derive and implement such models. In addition, several other closely related questions involving semiclassical and adiabatic approximations to quantum mechanics will be studied.The motion of electrons and nuclei in molecules is extremely complicated, and all theoretical knowledge of chemistry and molecular physics has come from approximations. The primary goal of this project is to provide greater understanding of these approximations and to develop new and improved approximations. In addition to providing greater theoretical understanding, the thrust of this research is to derive new techniques that will be useful for chemists and molecular physicists.