An efficient quantum mechanical method for the electronic dynamics of the three-dimensional hydrogen atom interacting with a linearly polarized strong laser pulse

We present a method whereby the 3D dynamics of the electronic wave packet in a hydrogen atom can be calculated efficiently. The method is constructed so as to satisfy the following two requirements: the wave function is zero at the Coulomb singular point so that the numerical difficulties concerning the singularity are avoided; the coordinate system is chosen so that the differential operators in the Hamiltonian can be well evaluated by the 3-point finite difference formula even near the singular point. The generalized cylindrical coordinate system (ξ^λ, z, φ) is introduced to satisfy the above conditions, and the value of λ is determined to be 3/2. The Schrödinger equation is discretized in time and space and solved by the Peaceman-Rachford method. The λ = 3/2 coodinate system helps the reduction in the number of grid points. To examine the numerical stability and accuracy of our method, we first apply it to cases where no laser field is turned on. The errors for the ordinary cylindrical coordinate system (λ = 1) are more than ten times as large as those for λ = 3/2. We then apply the method to the case where the atom interacts with a linearly polarized strong laser pulse. Our method is highly reliable and is a powerful tool for analyzing ionization processes of the hydrogen atom.