Split-operator spectral method for solving the time-dependent Schrödinger equation in spherical coordinates

Abstract

A spectral method previously developed for solving the time-dependent Schrödinger equation in Cartesian coordinates is generalized to spherical polar coordinates. The solution is implemented by repeated application of a unitary evolution operator in symmetrically split form. The wave function is expanded as a Fourier series in the radial coordinate and in terms of Legendre functions in the polar angle. The use of appropriate quadrature sets makes the expansion exact for band-limited functions. The method is appropriate for solving explicitly time-dependent problems, or for determining stationary states by a spectral method. The accuracy of the method is established by computing the Stark shift and lifetime of the 1s state in hydrogen, the low-lying energy levels for hydrogen in a uniform magnetic field, and the 2p-nd dipole transition spectrum for hydrogen.