We
study a numerical
solution of the multi-dimen
sional time dependent Schr
xf6;dinger equation u
sing a
split-operator technique for time
stepping and a
spectral approximation in the
spatial coordinate
s. We are particularly intere
sted in
sy
stem
s with near
spherical
symmetrie
s. One expect
s the
se problem
s to be mo
st efficiently computed in
spherical coordinate
s a
s a coar
se grain di
scretization
should be
sufficient in the angular direction
s. However, in thi
s coordinate
sy
stem the
standard Fourier ba
si
s doe
s not provide a good ba
si
s set in the radial direction. Here, we
sugge
st an alternative ba
si
s set ba
sed on Cheby
shev polynomial
s and a variable tran
sformation.
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.