Consequences of wave function orthogonality for medium energy nuclear reactions

Abstract

In the usual models of high-energy bound-state to continuum transitions no account is taken of the orthogonality of the bound and continuum wave functions. This orthogonality induces considerable cancellations in the overlap integrals expressing the transition amplitudes for reactions such as ($e,e^{\prime }p$), ($\gamma ,p$), and ($\pi ,N$), which are simply not included in the distorted-wave Born-approximation calculations which to date remain the only computationally feasible heirarchy of approximations. The object of this paper is to present a new formulation of the bound-state to continuum transition problem, based upon flux conservation, in which the orthogonality of wave functions is taken into account ab initio. The new formulation, while exact wave functions are used, offers the possibility of using approximate wave functions for the continuum states without doing violence to the cancellations induced by orthogonality. The method is applied to single-particle states obeying the Schrödinger and Dirac equations, as well as to a coupled-channel model in which absorptive processes can be described in a fully consistent manner. Several types of absorption vertex are considered, and in the ($\pi ,N$) case the equivalence of pseudoscalar and pseudovector $\pi \mathrm{NN}$ coupling is seen to follow directly from wave function orthogonality.