Numerical solution of the 3D time dependent Schrödinger equation in spherical coordinates: Spectral basis and effects of split-operator technique

We study a numerical solution of the multi-dimensional time dependent Schrodinger equation using a split-operator technique for time stepping and a spectral approximation in the spatial coordinates. We are particularly interested in systems with near spherical symmetries. One expects these problems to be most efficiently computed in spherical coordinates as a coarse grain discretization should be sufficient in the angular directions. However, in this coordinate system the standard Fourier basis does not provide a good basis set in the radial direction. Here, we suggest an alternative basis set based on Chebyshev polynomials and a variable transformation. Furthermore, it is shown how the use of operator splitting produces a splitting error which introduces high frequency modes in the numerical solution in the case of the singular Coulomb potential. Incorporating the Coulomb potential into the radial Laplacian provides a much better splitting. Fortunately our new basis set allows this in some cases. Numerical experiments are presented which demonstrate the advantages and limitations of our technique. Details are demonstrated by 1D toy examples, while the superior efficiency is demonstrated by a 3D example.