Fourth order real space solver for the timedependent Schrödinger equation with singular Coulomb potential
Abstract
We present a novel numerical method and algorithm for the solution of the 3D axially symmetric timedependent Schrödinger equation in cylindrical coordinates, involving singular Coulomb potential terms besides a smooth timedependent potential. We use fourth order finite difference real space discretization, with special formulae for the arising Neumann and Robin boundary conditions along the symmetry axis. Our propagation algorithm is based on merging the method of the splitoperator approximation of the exponential operator with the implicit equations of second order cylindrical 2D CrankNicolson scheme. We call this method hybrid splitting scheme because it inherits both the speed of the split step finite difference schemes and the robustness of the full CrankNicolson scheme. Based on a thorough error analysis, we verified both the fourth order accuracy of the spatial discretization in the optimal spatial step size range, and the fourth order scaling with the time step in the case of proper high order expressions of the splitoperator. We demonstrate the performance and high accuracy of our hybrid splitting scheme by simulating optical tunneling from a hydrogen atom due to a fewcycle laser pulse with linear polarization.
 Publication:

Computer Physics Communications
 Pub Date:
 November 2016
 DOI:
 10.1016/j.cpc.2016.07.006
 arXiv:
 arXiv:1604.00947
 Bibcode:
 2016CoPhC.208....9M
 Keywords:

 Timedependent Schrödinger equation;
 Singular Coulomb potential;
 High order methods;
 Operator splitting;
 CrankNicolson scheme;
 Strong field physics;
 Optical tunneling;
 Physics  Atomic Physics;
 Physics  Computational Physics;
 Quantum Physics
 EPrint:
 doi:10.1016/j.cpc.2016.07.006