量子李雅普诺夫指数
Quantum Lyapunov Exponents
摘要：
我们表明，它是可能的关联与每个给定的解决方案的时间依赖薛定谔方程的一个特定的相流（“量子流”）的非自治动力系统。这一事实使我们能够引入量子动力学中的混沌（量子混沌）的定义，该定义基于动力系统中混沌的经典理论。这样，我们可以引入可称为“量子李雅普诺夫指数“的量。我们的方法适用于一个非相对论量子力学系统的n个带电粒子，在目前的工作中，数值计算只进行了氢原子。在轨道的计算中，我们首先忽略了自旋对混沌的贡献，然后考虑了量子混沌中的自旋效应。我们展示了如何量子李雅普诺夫指数可以评估，并给出了几个数值结果，描述了在本方法中发现的一些属性。虽然该系统非常简单，经典的对应是正则的，但根据我们的定义，相应的薛定谔方程的大多数非定常解是“混沌”的。
英文摘要：
We show that it is possible to associate univocally with each given solution of the time-dependent Schroedinger equation a particular phase flow ("quantum flow") of a non-autonomous dynamical system. This fact allows us to introduce a definition of chaos in quantum dynamics (quantum chaos), which is based on the classical theory of chaos in dynamical systems. In such a way we can introduce quantities which may be appelled "Quantum Lyapunov Exponents". Our approach applies to a non-relativistic quantum-mechanical system of n charged particles; in the present work numerical calculations are performed only for the hydrogen atom. In the computation of the trajectories we first neglect the spin contribution to chaos, then we consider the spin effects in quantum chaos. We show how the quantum Lyapunov exponents can be evaluated and give several numerical results which describe some properties found in the present approach. Although the system is very simple and the classical counterpart is regular, the most non-stationary solutions of the corresponding Schroeodinger equation are "chaotic" according to our definition.
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