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Computer Physics Communications

Numerical solutions of the time-dependent Schrödinger equation in spherical coordinates by Fourier transform methods

Christopher E. Dateo, Volker Engel, Raphael Almeida, Horia I. Metiu

Published Feb 1, 1991• 21 citations
Publisher
DOI
10.1016/0010-4655(91)90268-p

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We present a numerical Fourier-transform method for solving the time-dependent Schrödinger equation in spherical coordinates. The time propagation is accomplished with a novel implementation of a Crank–Nicholson scheme combined with a Feit–Fleck split-operator method. Calculations on model problems give results that are in excellent agreement with those obtained by other methods. The procedure avoids the numerical instabilities associated with the division by the sine of the polar angle. The propagator is unitary, does not require the calculation of any matrix elements, and uses no matrix multiplications other than fast Fourier transforms.
Abstract— The fundamental possibility of a numerical solution of the time-dependent Schrödinger equation for a wave function defined on a non-uniform coordinate grid by the spectral method using the fast Fourier transform algorithm is discussed. The method is based on reducing the non-uniform coordinate grid to a uniform one by a non-linear transformation of coordinates and approximating the obtained evolution operator using the Lie–Trotter–Suzuki product formula. Algorithms for the numerical solution of the first and second orders are developed.