Extension of the Faddeev equation to Nuclei

Abstract

The Faddeev equation was derived for slving the three body problem. The wave function Ψ is written as the sum of three amplitudes. Each one ${\psi_{ij}}({r_{ij}},{r_k};\alpha )$ is described in one of the three available Jacobi coordinate systems: ${r_{ij}}={x_i}-{x_j},{r_k}= sqrt3({x_k}-X),X=\frac{1}{3}({x_1}+{x_2}+{x_3})$ in terms of the coordinates ${x_i},(i=1,2,3),$ of equal mass particles. The other degrees of freedom a can be spin, isospin etc… Each amplitude is the solution of one Faddeev equation

$$(T - E)\psi ij = - V({r_{ij}})\Psi $$

(1)

for pairwise potentials $V({r_{ij}})$. Then each equation is projected on bipolar harmonics generating coupled two variables integrodifferential equations in coordinate space.