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.
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机译：本工作研究了振动量子台球中的半量子混沌，它可用于探索多原子分子中的非绝热行为。边界为 机械自由度的量子台球的 d 模式Galërkin扩展在物理上被解释为受 s 激发的分子核模式和 d 倍电子简并性。在介绍了该问题之后，我们将详细考虑由Born-Oppenheimer近似推导半量子物理学，其在分子系统中的应用以及与振动台球的关系。我们还回顾了量子混沌和量子台球的概念，以进一步将本论文与文献联系起来。然后，我们提出描述振动量子台球的无穷维问题，并考虑其对称性。将Bloch变量用于量子机械自由度，我们得出了有限维截断的运动方程。我们详细考虑 d = 1， d = 2和 d = 3的情况。我们还分析了径向振动的球形量子台球和矩形振动的矩形量子台球。使用绝热作用角公式，我们证明它等效于Bloch公式，我们应用梅尔尼科夫方法来分析该系统中的混沌和先验不稳定Arnold扩散。当从可积构型中扰动时，我们还研究了经典自由度和量子力学自由度的混沌发作的相对便利性。最后，我们总结了目前的工作，并就有关振动量子台球，其他半量子系统以及量子混沌和哈密顿动力学其他领域的未来研究进行了讨论。
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