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Physics of Particles and Nuclei Letters

Spectral Method for Solving the Time-Dependent Schrödinger Equation on a Non-Uniform Coordinate Grid

М. А. Захаров

Published Apr 1, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1134/s1547477124702595

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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.
A matrix inverse free method to solve time-dependent Schrodinger equation is presented. The method is not subject to form of Hamiltonian and adopting real space grid system such as structured and unstructured grid, and achieves the order N algorithm even if we adopt unstructured grid systems to adjust complex regions. We have demonstrated some error evaluation problems that have exact solution and application in which it has non-Schrodinger type Hamiltonian, and we have successfully simulated the time-evolution of wave function for all of the case.