Confrontation of macroscopic and microscopic nuclear collective models

Abstract

Since its start nuclear theory has lived with the dichotomy of viewing the nucleus microscopically, as a system of nucleons, or describing it macroscopically in terms of collective coordinates. In the last decade though, a point transformation has been introduced in which single particle coordinates can be expressed in terms of collective ones plus others, opening the possibility of deriving a microscopic collective model. In the present paper we confront the macroscopic and microscopic collective models, first in a space of two dimensions, in which we find explicitly the unitary representation in quantum mechanics of the canonical transformation that relates them. We then show how to extend every step of the analysis to the three-dimensional problem, though there some of the states required are not yet available in analytic form. One of the fundamental problems in collective models of the nucleus is that of shape. We indicate what are the operators whose expectation values give a reasonable description of the shape in the macroscopic and microscopic collective models and confront them critically.