Accurate numerical solutions of the time-dependent Schrödinger equation

Abstract

We present a generalization of the often-used Crank-Nicolson (CN) method of obtaining numerical solutions of the time-dependent Schrödinger equation. The generalization yields numerical solutions accurate to order (∆x) 2r−1 in space and (∆t) 2M in time for any positive integers r and M , while CN employ r = M = 1. We note dramatic improvement in the attainable precision (circa 10 or greater orders of magnitude) along with several orders of magnitude reduction of computational time. The improved method is shown to lead to feasible studies of coherent-state oscillations with additional short-range interactions, wavepacket scattering, and long-time studies of decaying systems.

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

“…In keeping with the expansion of the time-evolution operator discussed in Ref. [4], the operator is written as…”

“…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.…”

Numerical solutions of the Schrödinger equation with source terms or time-dependent potentials

“…The method, like that for the homogeneous Schrödinger equation [1], proves to be capable of high precision and efficiency.…”

“…[8,9] and references contained in them), but most realistic systems require numerical solutions. In an earlier paper [1] (hereafter referred to as I) we presented an accurate and efficient method for obtaining solutions of the homogeneous Schrödinger equation in one dimension and for uncoupled partial waves in three dimensions.…”

“…Recent interest in accurate numerical solutions of the timedependent Schrödinger equation (TDSE) using a generalized Crank-Nicolson (CN) approach [1,2], suggests further study for cases where the Schrödinger equation has a nonhomogeneous term and where the Hamiltonian is time dependent. Over the years the method of choice for solving the homogeneous Schrödinger equation with time-independent interactions has been the Chebyshev expansion of the propagator introduced in 1984 by Talezer and Kosloff [3].…”