Center-of-Mass Motion in Many-Particle Systems

Abstract

An explicit construction is found for a unitary operator which insures the free motion of the center of mass of any many-particle wave function on which it is allowed to act. The transformation is used to calculate recoil correction terms for the internal energy and external interactions of nuclei, and some numerical evaluations are given for cases of interest. The many-body harmonic-oscillator problem is exactly soluble when one uses the transformation, and one is thus enabled to give a more general discussion of the spurious states.

Unequal freezing and melting temperatures for clusters

“…The conclusions, presented rersely here, show that clusters of as few as five particles can be expected to exhibit sharp freezing temperatures Tr-below which no hquid-me form may exist, and sharp melting temperatures T,,, above which no solid-like form may exist; moreover Tf and Tm , which both depend on I'?, are not the same for small Iv, probably for all finite N. "Exist" here refers to existence in thermodynamic equilibrium, not to metastable or transient forms. A fuller account is being submitted for publication We briefly describe the model and then show in stages how the conclusions are derived: the existence of on!y a solid-like form (possibly including isomers) at low temperatures, the coexistence of solid-like and liquid-like forms above a temperature T&V), the The model supposes that the energy levels Ei(~) of the IV-body cluster are continuous functions of a parameter y measuring the non-rigidity of the cluster [6,7] _ At one extreme, y = 0 and the cluster can be described by an effective Ham~tonian like that for a conventional, nearly-rigid molecule, with small-amplitude, harmonic oscillations and rigid rotations_ At the other extreme, r = 1 and the cluster can be described by an effective Hamiltonian for a highly non-rigid but cohesive, cluster of low density: here we use the Gartenhaus-Schwartz model [8] of pairwise harmonic attractions. The parameter 7 can be defied by extension of the parameter of nonrigidity for a diatomic moIecule.…”

Correlation diagram for rigid and nonrigid three-body systems

“…In the center of mass frame this hamiltonian can be rearranged to give (3N -3) degenerate normal modes [4 ] . The symmetry group is known to be SU (3N--3) [S 1, and the eigenfunctions are bases for irreducible representations of SU(3N -3) which are symmetric under permutations of oscillator quanta.…”

“…We idealize the non-rigid limit to be a system of N identical particles with pairwise attractive harmonic interactions, so that (1) In the center of mass frame this hamiltonian can be rearranged to give (3N -3) degenerate normal modes [4 ] . The symmetry group is known to be SU (3N--3 One may correlate states of this highly symmetric oscillator model to systems with lower symmetry subject to the kinematic requirement of definite orbit2 angular momentum J, parity X, and permutational syntmetry l$_ F or enumerating states of given L and I'& in a particular level, the chain…”

“…Deviations from rigid moiecute behavior may be treated by the method of Dalton [3 J . This method and its natural extensions assume that different permutational and chiral isomers or different geometries transform among themselves by large-amplitude motions, which have frequencies low compared to vibrational and rotational frequencies of the molecule in its center of mass.We idealize the non-rigid limit to be a system of N identical particles with pairwise attractive harmonic interactions, so that (1)In the center of mass frame this hamiltonian can be rearranged to give (3N -3) degenerate normal modes [4 ] . The symmetry group is known to be SU (3N--3) [S 1, and the eigenfunctions are bases for irreducible representations of SU(3N -3) which are symmetric under permutations of oscillator quanta.…”

High-energy approximations to nuclear scattering

“…Moreover, it allows to take the c.m. motion of the nucleus correctly into account by employing the Gartenhaus-Schwartz transformation [57].…”