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

Abstract

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.

Numerical solutions of the time-dependent Schrödinger equation in two dimensions

“…Thus it seems that with virtually the same effort the calculation gives much more accurate results. A comparison of the two expansions [5] shows that for smaller values of r the calculation is indeed much more efficient, however for larger r the accuracy decreases. The kinetic energy operator resulting from Eq.…”

“…If exact analytic solutions are available the error can be calculated by comparison. If that is not the case, a good estimate of the error can be made by comparing the solution for particular M and r to the one obtained when one or both of the M and r are increased by unity [5].…”

“…(4.4) Since it turns out that e 2 and η 2 are very similar, evaluating η 2 provides a method to estimate the accuracy in the absence of an analytic solution [5,7].…”

“…The "method of choice" for some years is the Chebyshev polynomial expansion of the timeevolution operator with (inverse) Fourier transformations to deal with the spatial development as time progresses [1,2]. More recently the Padé approximant representation of the time-evolution operator is exploited [3][4][5][6][7]. This approach is unitary, stable, and allows for systematic estimate of errors in terms of powers of the temporal and spatial step sizes.…”

Scattering of quantum wave packets by shallow potential islands: A quantum lens

“…The reader is referred to Refs. [11][12][13] for a comprehensive description of the method and its implementations.…”