Cluster Development Method in the Quantum Mechanics of Many Particle System, I

Abstract

In order to investigate the many particle system by the variational principle, the cluster develop­ ment method propcsed by Jastrow is sys~cmatically formulated. It is a dircct generalization of the Hartree method, in which two particle functions are included in the trial function to represent the correlations between particlcs. The cnergyxpectation value is expar..ded in a series, each term of which is expressed in cluster integra's. It is found that the series convergcs rapidly when the cor· relations are marked only fot· small particle separations, so that this method is expected to be useful especially for the system of particles interacting through short range potentials with repulsive cores. § I. Introduction The Hartree method ·for a many particle system, in which each particle is supposed to move independently 2.nd to be influenced by other p2.rticles only through their average field of force, has been an importa.nt approximation a.nd successful in various problems. However, in some cases where the correlations between particles play an important role, this simple approximation is no longer valid. In one case, when the range of the interaction between particles is long compared with the mean distance between them, many particles may be coupled togeth:::r to move in some organized mode of motion as a whole, as is seen in the typical example of the plasma oscillation in- a dense electron gas. The recent study by Tomonaga 1l made the mechanism of this so-called collective motion quite clear_ In :>.nothe: case, when the interaction is strong but of short range, the actual potential determi11cd by the instantaneous configuration of the other . particles fluctuates from the average one considerably, and the image of the independent particle motion becomes far from the truth. Such a situation occurs especially when the interaction potential has a repulsive core at short distance as in liquid helium .or in the atomic nucleus. (In the latter case the existence of the repulsive core seems very likely from the analysis of the nucleon-nucleon scattering/) the binding energy differ~nce between H'l and He3, 3l and so forth.) In fact, if .we apply the Hartree method to these systems, the rc3ulting average potential diverges to infinity. Actually, the wave function of the system must vanish in the region of the configuration space where at least two particles arc fotmd. within the hard core range. In such cases we have to take into· account this strong short range correlation in some way.