约束量子系统中的几何相位、曲率和外势
Geometric Phase, Curvature, and Extrapotentials in Constrained Quantum Systems
摘要：
我们导出了被无限回复力约束在位形空间（环境空间）的子流形（约束流形）上的量子系统的有效哈密顿量。我们特别注意这个哈密顿量如何依赖于约束流形外部的量，例如约束流形的外部曲率，周围空间的（黎曼）曲率和约束势。特别地，我们发现了一个值得注意的事实，即约束势的扭曲在约束哈密顿量中表现为规范势。这种规范势是几何相位的一个例子，与贝里最初讨论的密切相关。受约束的哈密顿量还包含一个有效势，它取决于约束流形的外曲率、周围空间的曲率和约束势的扭曲。我们的分析的一般性质允许应用到各种各样的问题，如刚性分子，分子系统沿反应路径的演变，和量子条形波导。
英文摘要：
We derive an effective Hamiltonian for a quantum system constrained to a submanifold (the constraint manifold) of configuration space (the ambient space) by an infinite restoring force. We pay special attention to how this Hamiltonian depends on quantities which are external to the constraint manifold, such as the external curvature of the constraint manifold, the (Riemannian) curvature of the ambient space, and the constraining potential. In particular, we find the remarkable fact that the twisting of the constraining potential appears as a gauge potential in the constrained Hamiltonian. This gauge potential is an example of geometric phase, closely related to that originally discussed by Berry. The constrained Hamiltonian also contains an effective potential depending on the external curvature of the constraint manifold, the curvature of the ambient space, and the twisting of the constraining potential. The general nature of our analysis allows applications to a wide variety of problems, such as rigid molecules, the evolution of molecular systems along reaction paths, and quantum strip waveguides.
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