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Invariant Manifolds and Collective Coordinates

T. Papenbrock, T. H. Seligman

Published Jan 1, 2000• 3 citations
OpenAlex

Abstract

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.

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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.
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.
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 present a novel extension of the concept of scars for the wave functions of classically chaotic few-body systems of identical particles with rotation and permutation symmetry. Generically there exist manifolds in classical phase space which are invariant under the action of a common subgroup of these two symmetries. Such manifolds are associated with highly symmetric configurations. If sufficiently stable, the quantum motion on such manifolds displays a notable enhancement of the revival in the autocorrelation function which is not directly associated with individual periodic orbits. Rather, it indicates some degree of localization around an invariant manifold which has collective characteristics that should be experimentally observable.
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.
General properties of many-body systems Definition of individual system So far we have studied the properties of single-body systems in prescribed external potentials. An individual physical system comprises a wave and a particle, both evolving in three-dimensional Euclidean space. Of course, isolating a system and representing the influence of other matter by an ‘external potential’ is an approximation to the real state of affairs in that we neglect the source of the potential. As a fundamental theory of matter quantum mechanics should apply to a closed many-body system (and ultimately to the universe as a whole) and reduce to a theory of systems of a few degrees of freedom as a special case under conditions where it is legitimate to neglect the ‘rest of the universe’. The extension of the quantum theory of motion to many-body systems is straightforward, although it displays some striking features not evident in the one-body case. To begin with, we define an individual n-body system as comprising: ( a ) A wavefunction ψ = ψ( x 1 , …, x n , t ) defined in a 3 n -dimensional configuration space in which x l , …, x n provide a set of rectangular Cartesian coordinates. ( b ) A set of n point particles pursuing trajectories x i ( t ) i = 1, …, n , in three-dimensional Euclidean space. A single configuration space trajectory is equivalent to n particle trajectories in Euclidean space. When we speak of a ‘many-body system’ we mean then a single wavefunction together with a set of particles.