Numerov extension of transparent boundary conditions for the Schrödinger equation in one dimension

Abstract

We describe an algorithm for animating time-dependent quantum wave functions in one dimension with very high accuracy. The algorithm employs the Crank–Nicholson approximation for the time dependence along with a Numerov extension of the discrete transparent boundary conditions described recently by Ehrhardt. We illustrate the power of this approach by simulating the decay of alpha particles from radioactive nuclei and the resonance scattering of electrons in a three-layer GaAs–GaAlAs sandwich.

Accurate numerical solutions of the time-dependent Schrödinger equation

“…Preliminary studies with the Numerov spatial integration scheme [18] and the generalized time evolution as described in this work indicate that significant improvements occur if one incorporates appropriate changes in the spatial step size for different regions of space. This is important in the case of discontinuous potentials and potentials that have great variation in some region and little or no variation in other regions.…”

“…[18], since the Numerov method has an error O((∆x) 6 ), and it is difficult to see how it can be generalized systematically to higher order spatial errors. The generalized time evolution algorithm can be applied to Moyer's [18] method.…”

“…[8] and consider the main features of that analysis with a view of determining the improvement brought about by the generalizations of this paper. This problem was revisited by Moyer [18] to illustrate the efficacy of the Numerov method and the use of transparent boundary conditions for the propagation of free-particle wavepackets. The authors of Ref.…”

“…They claim to obtain two orders of magnitude improvement in the time advance. Moyer [18] uses a Numerov scheme for the spatial integration method with error of O((∆x) 6 ) but has a onestage time evolution giving the CN precision in time, i.e., with an error of O((∆t)…”

General recursive solution for one-dimensional quantum potentials: a simple tool for applied physics

“…It is normalized since |Ψ(x, 0)| 2 dx = 1, and its full width at half maximum (FWHM), W , can be chosen by providing σ x = W/ √ 2 ln 4 0.6 W. Numerov-type solutions of the time-dependent SE can handled the time evolution of such wave packet, Eq. (19), through a given potential barrier [17], but in the present recursive method it is not possible to start with such expression of Ψ(x, 0) since it is an artificial expression that happens to have the same shape, at t = 0, of the actual wave packet…”