On a canonical functions approach to the elastic scattering phase‐shift problem

The determination of the phase‐shift δ ρ ( E ) (related to a central potential V ( r ), a total energy E , and an angular momentum ρ) is considered. The “canonical functions” approach already used for the eigenvalue problem is adapted to that of δ. The conventional approach computes the radial wave function y ρ ( E ; r ) starting at r s ∼ 0 (with convenient initial values) and stepping on toward a large value r = R ∼ ∞, where y ρ is matched to its asymptotic value y ρ ( R ) ∼ a sin( kR – ρ π/2 + δ ρ ) and δ is deduced. The present approach starts at any “origin” r 0 , replaces the use of the wave function y by that of the “canonical functions” α and β (well defined for given V , E , and ρ) and defines two functions q ( r ) and Q ( r ) in terms of α and β. When r → O , q ( r ) approaches a constant limit giving Q ( r 0 ), and thus the starting problem is avoided. Using this value Q ( r 0 ), the function Q ( r ) is generated for r > r 0 . The function Q ( r ) reaches a constant limit when r → ∞; this limit is precisely tan δ; thus, the “final” matching problem is avoided. The present method is applied to the Lennard–Jones potential function for low and high E and for low and high ρ. The comparison of the results of the present method with those of confirmed numerical methods show that the present method is competitive.