Variational Principle for Scattering with Tensor Forces

Abstract

A reformulation of the variational principle in differential form for the phase shift ${\delta }_l$ of a central-force scattering problem is presented. This is considered to be the most simple that can be formed in terms of the inside-wave function representing the difference between the wave function and an appropriately chosen asymptotic form. It is then generalized by means of matrix notation so as to provide corresponding variational principles for the three parameters that arise in scattering with tensor forces, the two phase shifts ${\delta }_{\alpha }$, ${\delta }_{\beta }$ and the mixture parameter $\epsilon$. These all develop from the differential formulation of Schrödinger's equation and do not depend on the integral equation formulation as does the one originally presented by Schwinger for the phases ${\delta }_{\alpha }$ and ${\delta }_{\beta }$, and the extension of it by Blatt and Biedenharn to the parameter $\epsilon$.