| The present work examines semiquantum chaos in vibrating quantum billiards, which may be used to explore nonadiabatic behavior in polyatomic molecules. A d-mode Galërkin expansion of a quantum billiard whose boundary has s mechanical degrees-of-freedom is interpreted physically as a molecule with s excited nuclear modes and a d-fold electronic near-degeneracy.; After introducing the problem, we consider in detail the derivation of semi-quantum physics from the Born-Oppenheimer approximation, its application to molecular systems, and its relation to vibrating quantum billiards. We also review the notions of quantum chaos and quantum billiards to further connect this dissertation with the literature.; We then formulate the infinite-dimensional problem describing vibrating quantum billiards and consider its symmetries. Using Bloch variables for the quantum-mechanical degrees-of-freedom, we derive equations of motion for finite-dimensional truncations. We consider the cases d = 1, d = 2, and d = 3 in detail. We also analyze the radially vibrating spherical quantum billiard and vibrating rectangular quantum billiard as special cases.; Using an adiabatic action-angle formulation, which we prove to be equivalent to the Bloch formulation, we apply Melnikov's method to examine chaos and a priori unstable Arnold diffusion in this system analytically. We also study the relative facility of chaotic onset of the classical and quantum-mechanical degrees-of-freedom when perturbing from an integrable configuration.; Finally, we summarize the present work and conclude with a discussion of future research concerning vibrating quantum billiards, other semiquantum systems, and other areas of quantum chaos and Hamiltonian dynamics. |
