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International Journal of Quantum Chemistry

Solution of the time-dependent Schr�dinger equation using a continuous Fourier transform

Wilfredo Rodríguez

Published Jan 1, 1991• 0 citations
Publisher

Abstract

We discuss the solution of the time-dependent Schrödinger equation. The split operator method is used to propagate an initial wave function and the continuous Fourier transform is applied to obtain the energy eigenvalues. High resolution in the energy is obtained without line shaping techniques.s
DOI
10.1002/qua.560400813

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We discuss the extension of the Lewis-Riesenfeld technique of solving the time-dependent Schr\"odinger equation to cases where the invariant has continuous eigenvalues. The extension is carried out for a general Ermakov system. We also discuss extensions to NNN-dimensional systems and the calculation of the propagator for a general Ermakov system.
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.