Time dependent Schrِdinger equation on arbitrary structured grids: Application to photoionization

A systematic technique for conservatively discretizing the time dependent Schrِdinger equation on an arbitrary structured grid is given. Spatial differencing is carried out by finite volumes, and temporal differencing is carried out semi-implicitly. It is shown that the resulting algorithm conserves probability to within a round-off error regardless of the grid geometry. The algorithm is efficient for both serial and parallel computation. The conservative nature of the algorithm, and its phase accuracy, are demonstrated for a bound state, and for a free state in an electromagnetic field. The ionization rate for a hydrogen atom in a strong electromagnetic field is computed, and compared with the rate from tunneling theory. The regime of validity of tunneling theory is clarified.