van Dijk W, Brown J, Spyksma K
Physics Department, Redeemer University College, Ancaster, Ontario, Canada.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Nov;84(5 Pt 2):056703. doi: 10.1103/PhysRevE.84.056703. Epub 2011 Nov 17.
Recent significant improvements of the numerical solutions of the time-dependent Schrödinger equation beg the question as to whether these recent methods are comparable in efficacy (in terms of accuracy and computational time) to the current "method of choice," i.e., the Chebyshev expansion of the time-evolution operator and the fast-Fourier-transform method of determining the kinetic energy. In this paper we review the methods in question and, by studying the time development of a coherent wave packet in an oscillator well, we are able to assess the effectiveness of the various methods. It turns out that the new generalizations come close (to within an order of magnitude) to being able to generate solutions as precisely and efficiently as the Chebyshev-fast-Fourier-transform method. The strict unitarity of the generalized methods may be an advantage. We also show that the fast-Fourier-transform approach to calculating the kinetic energy can be replaced by straightforward numerical differentiation to obtain the same precision.
这些最新方法在效能（就准确性和计算时间而言）上是否能与当前的“首选方法”相媲美，即时间演化算符的切比雪夫展开以及确定动能的快速傅里叶变换方法。在本文中，我们回顾了相关方法，并且通过研究振子势阱中一个相干波包的时间演化，我们能够评估各种方法的有效性。结果表明，新的推广方法（在一个数量级范围内）接近能够像切比雪夫 - 快速傅里叶变换方法那样精确且高效地生成解。广义方法严格的幺正性可能是一个优势。我们还表明，计算动能的快速傅里叶变换方法可以被直接的数值微分所取代，以获得相同的精度。
