A High-Energy Scattering Approximation

Abstract

A high·energy scattering approximation which expresses the wave function as a sum of the second Born form and correction terms is developed. It is shown for example of a Gaussian potential that the approximation well gives the scattering amplitude for the coupling parameter larger as well as smaller than unity.In previous work,l),2) an approximation method was developed to describe the scattering of a high-energy Schrodinger particle from a central potential of finite range. Under the conditions that the typical strength of the potential is much smaller than the incident energy and its range is greater than the reduced wavelength, the method can well give the scattering amplitude in the weak and intermediate coupling cases far beyond the conventional angular range of validity of Glauber's formula. 3 ) It . is applicable to the scattering from an optical potential by analytic continuation,I) andits relativistic extension is possible within the frawework of the non-relativistic Schrodinger equation by modifying the wave number and the potentia1. 2 ) Its advantage is that the scattering amplitude as well as the wave function is expressed in terms' of integrals evaluated along a straight line parallel to the incident direction independent of a scattering angle. In this respect, it contrasts with the eikonal approach 4H ) which has a mathematical problem of changing the integration path after having calculated the scattering amplitude from the wave function and ambiguity on selecting an integration path in evaluating the off-shell amplitude. On the other hand, however, the method has a physically important problem of extending an allowed range of the coupling parameter. In the present paper, an attempt is reported to improve the method by introducing manageable correction. The notations in what follows are the same as those in Refs. 1) and 2). The scattering of a particle with an incident wave number ko from a potential V(r) is described by the integral equation W(r)=exp{iko(so' r)}+ j G(r, r') U(r') W(r')dr'(1)References 1) and 2) adopted its approximate solution W(r)=exp{iko(so'r)}+ jG(r, r')U(r') 1fft(r') dr' ,where 1fft(r) is the internal wave function for the scattering from the square well