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Progress of Theoretical Physics

Canonical Coordinate System Suitable for Adiabatic Treatment of Collective Motion: General Case

A. Kuriyama, M. Yamamura

Published May 1, 1984• 9 citations
Publisher

Abstract

For the general case of particle-pair correlation or particle-hole correlation, we define a canonical coordinate system of TDHF symplectic manifold which may be suitable for the adiabatic description of collective motion. Following the prescription of Baranger and Veneroni, we adopt a Slater determinantal state generated by two-step unitary transformation. In this case, the canonicity condition plays an important role.

DOI
10.1143/ptp.71.973

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Progress of Theoretical Physics•Jan 1, 1984•4 citations
For the general case of particle-pair correlation or particle-hole correlation, we define a canonical coordinate system of TDHF symplectic manifold which may be suitable for the adiabatic description of collective motion. Following the prescription of Baranger and Veneroni, we adopt a Slater determinantal state generated by two-step unitary transformation. In this case, the canonicity condition plays an important role.
The geometry of the self-consistent collective-coordinate (SCC) method formulated within the frame-work of the time-dependent Hartree-Fock (TDHF) theory is investigated by associating the variational parameters with a symplectic manifold (a TDHF manifold). With the use of a canonical-variable parametrization, it is shown that the TDHF equation is equivalent to the canonical equations of motion in classical mechanics in the TDHF manifold. This enables us to investigate geometrical structure of the SCC method in the language of the classical mechanics. The SCC method turns out to give a prescription how to dynamically extract a “maximally-decoupled” collective submanifold (hypersurface) out of the TDHF manifold, in such a way that a certain kind of trajectories corresponding to the large-amplitude collective motion under consideration can be reproduced on the hypersurface as precisely as possible. The stability of the hypersurface at each point on it is investigated, in order to see whether the hypersurface obtained by the SCC method is really an approximate integral surface in the TDHF manifold or not.
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〉|n〉∣n〉 of the system exhibits an energy spectrum and relative transition probabilities following the laws of the (in general anharmonic) oscillator described by H\mathcal{H}H (${p}_{\ensuremath{\alpha}}, \ensuremath{\alpha}$) $(\ensuremath{\alpha}|n)={\ensuremath{\omega}}_{n}(\ensuremath{\alpha}|n)$. We suppose the set of many-body states ∣n〉|n〉∣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.
Collective modes of interacting many-body systems can be related to motion on classically invariant manifolds.We introduce suitable coordinate systems.These coordinates are Cartesian in position and momentum space.They are collective since several components vanish for motion on the invariant manifold.We make a connection to Zickendraht's collective coordinates and also obtain shear modes.The importance of collective configurations depends on the stability of the manifold.We present an example of quantum collective motion on the manifold.
We introduce suitable coordinate systems for interacting many-body systems with invariant manifolds. These are Cartesian in coordinate and momentum space and chosen such that several components are identically zero for motion on the invariant manifold. In this sense these coordinates are collective. We make a connection to Zickendraht's collective coordinates and present certain configurations of few-body systems where rotations and vibrations decouple from single-particle motion. These configurations do not depend on details of the interaction.