A spectral Phase–Amplitude method for propagating a wave function to large distances

The phase and amplitude (Ph–A) of a wave function vary slowly with distance, in contrast to the wave function that can be highly oscillatory. Hence the Ph–A representation of a wave function requires far fewer computational mesh points than the wave function itself. In 1930 Milne presented an equation for the phase and the amplitude functions (which is different from the one developed by Calogero), and in 1962 Seaton and Peach solved these equations iteratively. The objective of the present study is to implement Seaton and Peach’s iteration procedure with a spectral Chebyshev expansion method, and at the same time present a non-iterative analytic solution to an approximate version of the iterative equations. The iterations converge rapidly for the case of attractive potentials. Two numerical examples are given: (1) for a potential that decreases with distance as 1/r31/r3, and (2) a Coulomb potential ∝1/r∝1/r. In both cases the whole radial range of [0–2000] requires only between 25 and 100 mesh points and the corresponding accuracy is between 10−310−3 and 10−610−6. The 0th iteration (which is the WKB approximation) gives an accuracy of 10−210−2. This spectral method permits one to calculate a wave function out to large distances reliably and economically.