Applications of Variational Principles to Scattering Problems

Abstract

The Schwinger variational principle is reduced to a convenient form by using trial solutions of the form ${\ensuremath{\Sigma}}_{i}{c}_{i}{u}_{i}(x)$. The reliability of the formalism for polynomial trial solutions is studied for several fields, and wherever possible comparisons are made with phase shifts determined from other variational principles.The forward scattering amplitude is determined variationally for scattering in the static field of the hydrogen atom, and it is demonstrated that the undisturbed wave as trial solution provides more accurate results for all incident energies than second Born approximation. The improved variational estimate of the total cross section for this field approaches that of Born at the higher energies.