Extension of the mapped Fourier method to time-dependent problems

Area of Science:

• Quantum mechanics
• Computational physics
• Atomic physics

Background:

• Solving the time-dependent Schrödinger equation with Coulomb fields presents challenges due to the singularity at r=0 and the potential's long-range nature.
• Accurate representation of wave packets requires high grid density near the origin and extension to large distances.

Purpose of the Study:

• To develop and validate a numerical method for integrating the time-dependent Schrödinger equation in the presence of a Coulomb field.
• To accurately compute atomic properties such as eigenenergies and ionization rates for systems like the hydrogen atom.

Main Methods:

• A numerical method employing unequally spaced grid points, mapped to an equally spaced grid, is utilized.
• Fast Fourier Transform (FFT) propagation methods, scaling as N ln N, are applied for efficient computation.
• Classical phase space criteria are used for selecting sampling points.

Main Results:

• The method accurately extracts hydrogen atom eigenenergies from wave-packet autocorrelation functions using filter-diagonalization.
• Calculated ionization rates for hydrogen atoms subjected to half-cycle pulses show excellent agreement with previous results.

Conclusions:

• The developed numerical method provides a highly accurate and efficient approach for studying quantum systems with Coulomb interactions.
• This technique is suitable for calculating various properties of atoms and molecules governed by the Schrödinger equation.