Global integration of the Schr\"odinger equation: a short iterative scheme within the wave operator formalism using discrete Fourier transforms

TL;DR

This paper introduces a fast iterative method using Fourier transforms to solve the time-dependent Schr"odinger equation globally, providing accurate solutions for complex quantum dynamics with potential advantages over traditional methods.

Contribution

A novel iterative scheme employing FFT-based numerical integration for solving the time-dependent Schr"odinger equation globally within the wave operator framework.

Findings

Method achieves high accuracy in quantum dynamics simulations.

Competitive with standard algorithms within convergence radius.

Effective for modeling laser-driven molecular transitions.

Abstract

A global solution of the Schr\"odinger equation for explicitly time-dependent Hamiltonians is derived by integrating the non-linear differential equation associated with the time-dependent wave operator. A fast iterative solution method is proposed in which, however, numerous integrals over time have to be evaluated. This internal work is done using a numerical integrator based on Fast Fourier Transforms (FFT). The case of a transition between two potential wells of a model molecule driven by intense laser pulses is used as an illustrative example. This application reveals some interesting features of the integration technique. Each iteration provides a global approximate solution on grid points regularly distributed over the full time propagation interval. Inside the convergence radius, the complete integration is competitive with standard algorithms, especially when high accuracy is required.