Efficiency and accuracy of numerical solutions to the time-dependent Schrödinger equation

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.