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

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 $\mathcal{H}$ ($p_{\alpha }, \alpha$) $\left(\alpha \mathrm{}\right|n)={\omega }_n(\alpha \mathrm{}\left|n\right)\mathrm{}$. We suppose the set of many-body states $|n〉$ to extend indefinitely, and we take the transform $|\alpha 〉=\Sigma |n〉$ ($n|\alpha$) to define a many-body generating state of the band which is precisely localized in $\alpha$ space. The basic assumption of collectivity, that changing the state of at most a few particles cannot much alter the value of $\alpha$, 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.