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This picture shows the output based on this code and the plot is made in matlab by these lines. The code is a typical "10 liner" - written in fortran - and should run fairly easy on any single processor machine. The physical situation is a free gaussian in the presence of a "gravitational potential". The wavefunction cannot penetrate into the ground at x=-25. With time the wavefunction therefore behaves as a bouncing ball.

Two dimensional split-operator method

• Documentation (*.ps) 
Reprint from paper presented at EuroPVM/MPI 2003 (to be published at Springer Verlag)
• Code (Unix *.tar file)
This file contains code, input files, readme (instructions) as well as an output example.
• Related articles:
  M. R. Hermann and J. A. Fleck, "Split-operator spectral method for solving the time dependent Schrödinger equation in spherical coordinates" Phys. Rev. A, 38, 6000, (1988).
  J. P. Hansen, T. Sørevik and L. B. Madsen, "Electron emission during combined attosecond pulses". This article describes the generalization of the Fleck/Hermann method to 3 dimensions Phys Rev A (in print, 2003).

The present code is developed parallelized and tested in Fortran90/MPIon the IBM Regatta machine at the University of Bergen. It also contains compile options (see Makefile) for Linux clusters. and is based on certain internal fft/legendre routines. Test calculations with the above example results in a near linear speed-up for increasing number of processors up to nprocs=16:

Results from downloadble example: 
 
  A hydrogen atom exposed to a linearly polarized laser pulse is reduced to a two dimensional problem when using spherical coordinates. In the present example the laser pulse is modeled as a sine square envelope and shown below. 

In this example the electric field is varying with time as: As an example we use this electric field: 
 
 

 The probability density in the x-z plane is shown at 5 instants of time indicated by the arrows af the above figure. Initially at t=0 a spherical 1s distribution is displayed. The three following pictures shows the wavefunction at three extrema of the electric field and the tendency of the probability density to follow the field is clearly observed. 

t=0:	t=20:	t=25:
t=31:	t=50=T:

At t=50 the expansion amplitudes |f(r)|^2 for l=0 (upper left), l=1 (upper right), ... l=5 (lower right) has the r-distribution as shown below. The components of l>5 vanish.