Concept of Ideal Collective Coordinate as the Foundation for a Phenomenological Theory of Nuclear Collective Motion: Basic Ideas and Relation to Other Phenomenological Methods

Abstract

The concept of an ideal collective coordinate is introduced by means of the following example: Consider a one-dimensional vibration of a many-body system in the sense that a large subset of states $|n〉$ of the system exhibits an energy spectrum and relative transition probabilities following the laws of the (in general anharmonic) oscillator described by $\mathcal{H}$ (${p}_{\ensuremath{\alpha}}, \ensuremath{\alpha}$) $(\ensuremath{\alpha}|n)={\ensuremath{\omega}}_{n}(\ensuremath{\alpha}|n)$. We suppose the set of many-body states $|n〉$ to extend indefinitely, and we take the transform $|\ensuremath{\alpha}〉=\ensuremath{\Sigma}|n〉$ ($n|\ensuremath{\alpha}$) to define a many-body generating state of the band which is precisely localized in $\ensuremath{\alpha}$ space. The basic assumption of collectivity, that changing the state of at most a few particles cannot much alter the value of $\ensuremath{\alpha}$, is shown to be sufficient to derive a phenomenological theory from the many-body starting point. The phenomenological aspects of a recent theory of rotations due to Villars is seen to be contained in the above formulation as a special case. A brief review is given of the generator coordinate and similar projection methods in order to exhibit their relationship with the present method.