Abstract by Cory Vernon
Celestial Molecular Mechanics: Quantum KAM Theory
Celestial Mechanics and particularly KAM Theory utilize much of pure mathematics and applications in physics. Hamiltonian matrices (or operators) form systems of differential equations derived from position and momentum to describe energy. Hamiltonian Mechanics appears in Quantum Mechanics where the eigenvalues correspond to energy minima that are quantum energy states. Some common approaches to better understand dynamical systems (planetary, molecular, etc.…) include Linear Variational Theory and Perturbation Theory. A classic example of the quantum particle in a slanted box is shown using Linear Variational Theory, and Perturbation Theory takes zero order known solutions and adds a perturbation to approximate experimental results. Perturbation Theory is expanded to infinite dimensions with KAM Theory. While KAM Theory originated in Celestial Mechanics, it organizes higher dimensional energy and applies to the behavior of molecular energy and wavefunctions. The application of KAM Theory in the quantum context is explored.