A direct numerical approach to three nucleon bound state problems

Abstract

This paper describes a direct numerical method for solving three nucleon bound state problems. The method has been tried out and developed on a sequence of two and three nucleon problems of increasing difficulty. This work forms part of an attack on the problem of calculating the triton binding energy with the nucleon-nucleon interaction described by the Hamada-Johnston potential. The method uses the analysis of Derrick and Blatt which reduces Schrödinger's equation for the ground state of the triton to an eigenvalue-eigenfunction problem consisting of a set of linked partial differential equations in three variables. In the main calculation described the full three nucleon ground state problem is simplified in two respects. Firstly, the spin orbit forces are omitted from the potential leaving only central and tensor forces. Secondly, in the expansion of the ground state wavefunction in terms of states of definite total angular momentum, iso-spin, and parity, only nine of the 16 components are included resulting in a problem involving nine partial-differential equations. A direct numerical approach is employed in which functions of three variables are represented by three-dimensional tables of numbers and derivatives by finite-difference operators. The numerical problem is thus obtained in the form of an eigenvalue-eigenvector problem. It is solved on a sequence of mesh sizes and an approximation to the analytic binding energy obtained by extrapolating to zero mesh size.