  The phase and amplitude (Ph-A) of a wave function vary slowly and
monotonically with distance, in contrast to the wave function that can be
highly oscillatory. Hence an attractive feature of the Ph-A representation is
that it requires far fewer meshpoints than for the wave function itself. In
1930 Milne developed an equation for the phase and the amplitude functions (W.
E. Milne, Phys. Rev. 35, 863 (1930)), and in 1962 Seaton andPeach (M. J. Seaton
and G. Peach, Proc. Phys. Soc. 79 1296 (1962)) developed an iterative method
for solving Milne's Ph-A equations. Since the zero'th order term of the
iteration is identical to the WKB approximation, there is a close relationship
between the Ph-A and the WKB representations of a wave function. The objective
of the present study is to show that a spectral Chebyshev expansion method to
solve Seaton and Peach's iteration scheme is feasible, and requires very few
meshpoints for the whole radial interval. Hence this method provides an
economical and accurate way to calculate wave functions out to large distances.
In a numerical example for which the potential decreased slowly with distance
as 1/r^3, the whole radial range of [0-2000] covered with 301 mesh points (and
Chbyshev basis functions). The first order iteration of the Ph-A wave function
was found to have an accuracy better than 1%, and was always more accurate than
the WKB wave function.
