时间传播多体波函数的显式格式
Explicit schemes for time propagating many-body wavefunctions
摘要：
在原子和分子物理学以及化学中，许多含时过程的精确理论数据需要含时薛定谔方程的直接数值解，从而促进了非常有效的时间传播子的发展。这些通常涉及非常大的一阶微分方程系统的解决方案，其特征在于高度的刚度。我们分析和比较Fatunla和Arnoldi的显式一步算法的性能。这两种算法具有完全相同的稳定性函数，因此共享相同的稳定性属性，结果证明是最佳的。然而，它们各自的准确性差异很大，并取决于所涉及的物理情况。为了测试这种准确性，我们使用的预测校正计划，其中的预测是Fatunla的或Arnoldi的算法和校正，一个完全隐式的四阶段Radau IIA方法的顺序7。我们考虑两个物理过程。第一种是通过短而强的电磁脉冲电离原子系统;原子系统包括一维高斯模型势以及原子氢和氦，两者都是全维的。第二个过程是当对平面双电子量子点施加与时间无关的微扰时，双电子量子态的退相干。尽管该系统的哈密顿量与时间无关，但相应的微分方程却表现出惊人的刚性。对于一维高斯势，我们详细讨论了监测的时间步长的显式算法的可能性。在其他的物理情况下，更苛刻的计算方面，我们表明，这两种算法的准确性强烈依赖于刚度的问题。
英文摘要：
Accurate theoretical data on many time-dependent processes in atomic and molecular physics and in chemistry require the direct numerical solution of the time-dependent Schrödinger equation, thereby motivating the development of very efficient time propagators. These usually involve the solution of very large systems of first order differential equations that are characterized by a high degree of stiffness. We analyze and compare the performance of the explicit one-step algorithms of Fatunla and Arnoldi. Both algorithms have exactly the same stability function, therefore sharing the same stability properties that turn out to be optimum. Their respective accuracy however differs significantly and depends on the physical situation involved. In order to test this accuracy, we use a predictor-corrector scheme in which the predictor is either Fatunla's or Arnoldi's algorithm and the corrector, a fully implicit four-stage Radau IIA method of order 7. We consider two physical processes. The first one is the ionization of an atomic system by a short and intense electromagnetic pulse; the atomic systems include a one-dimensional Gaussian model potential as well as atomic hydrogen and helium, both in full dimensionality. The second process is the decoherence of two-electron quantum states when a time independent perturbation is applied to a planar two-electron quantum dot where both electrons are confined in an anharmonic potential. Even though the Hamiltonian of this system is time independent the corresponding differential equation shows a striking stiffness. For the one-dimensional Gaussian potential we discuss in detail the possibility of monitoring the time step for both explicit algorithms. In the other physical situations that are much more demanding in term of computations, we show that the accuracy of both algorithms depends strongly on the degree of stiffness of the problem.
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