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Physical Review

Many-Body Problem with Strong Forces

Robert Jastrow

Published Jun 1, 1955• 1,148 citations
Publisher

Abstract

A trial solution constructed from two-particle functions is applied to the problem of the $N$-particle system with strong interactions. In a variational treatment based on this wave function the expectation values, which cannot be factored into single-particle integrals, are evaluated by a cluster development in powers of the particle density. The procedure is illustrated by a calculation of the pair distribution function and zero-point energy of the hard sphere gas, for Bose and Fermi statistics.

DOI
10.1103/physrev.98.1479

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A trial solution constructed from two-particle functions is applied to the problem of the NNN-particle system with strong interactions. In a variational treatment based on this wave function the expectation values, which cannot be factored into single-particle integrals, are evaluated by a cluster development in powers of the particle density. The procedure is illustrated by a calculation of the pair distribution function and zero-point energy of the hard sphere gas, for Bose and Fermi statistics.
A new variational approach is developed for studying the properties of systems of particles interacting through singular short-range repulsions that give rise to strong two-particle correlations. The correlated trial function ${\ensuremath{\Psi}}_{\ensuremath{\gamma}}={e}^{S}{\ensuremath{\Phi}}_{\ensuremath{\gamma}}$ (state $\ensuremath{\gamma}$) results, with proper choice of SSS, in a simple form for the energy expectation value 〈H〉〈H〉〈H〉---as well as for other matrix elements of interest---which is devoid of all reference to the strong repulsions except through e2S{e}^{2S}e2S factors and hence is particularly suited to calculation. In many cases an independent-particle type ${\ensuremath{\Phi}}_{\ensuremath{\gamma}}$ seems appropriate. The cluster evaluation of this form for 〈H〉〈H〉〈H〉 is discussed, both in the few-particle and many-particle cases. Using the techniques of Iwamoto and Yamada, simplified convergent cluster expansions for the energy expectation value are derived for many-fermion and many-boson systems. A program for application of this method to nuclear problems is being initiated.
A discussion is given of the wave function of a many-particle system. The wave function itself is quite complex, containing all combinations of "clusters" of interacting particles. Expectation values, however may be calculated in a straightforward manner, as is shown in detail. In particular, the expectation value of an operator may be expanded in "linked clusters," each "linked cluster" being weighted with its probability of occurrence in the wave function. The theory is applied to the calculation of the momentum spectrum and pair correlation functin for a dilute gas, a degenerate Fermi-Dirac system, and a degenerate Bose-Einstein system of hard spheres.
A new variational approach is proposed for the study of the many-body problem with the purpose of combining the advantages of perturbation-theoretic and variational approaches, while avoiding their disadvantages. A trial wave function is constructed so that a general expansion for the expectation value of an arbitrary physical quantity can be derived. Each term in the expansion can be calculated with the use of the techniques of Feynman diagrams, but in contrast with usual perturbation-theoretic approaches, every term in the expansion always converges. The procedure for applying the method to real systems is illustrated by a calculation of the ground-state energy of the electron gas at high densities.
The pair distribution function of a hard-sphere Bose gas is evaluated using a modified cluster expansion formula developed in the preceding papers of this series [H. Ichimura, Prog. Theor. Phys. 31 (1964), 538; ibid. 37 (1967), 484]. The contributions for the pair distribution function are expressed in terms of diagrams, and a partial summation of the chain-type diagrams is carried out. The temperature dependence of the resultant expression is investigated for low temperatures, at short and long distances. The result is compared with the expression for the pair distribution function which has been obtained for a weakly interacting Bose gas. The two- and three-particle distribution functions for a degenerate ideal Bose system are derived.
The properties of a Bose system of particles with repulsive interactions have previously been treated using perturbation theory in the formalism of second quantization. Others have also considered this problem by dealing with the wave function in configuration space, using the theory of cluster expansions. In these latter papers, variation with respect to a parameter in a trial function for the ground state has been shown to yield a ground-state energy close to the exact asymptotic expressions obtained from perturbation theory. The connection between the two methods is not immediately obvious from these cluster expansion treatments. It is shown here that, as one might expect, the cluster integral method can be handled so that it is completely equivalent to the pair approximation in perturbation theory.
A formulation is given whereby the grand partition function of a many-body system satisfying Bose-Einstein or Fermi-Dirac statistics is expressed in terms of certain UUU functions defined for the same system with Boltzmann statistics. It is then shown that these UUU functions can be evaluated in successive approximations in terms of a binary kernel BBB which can be computed from a solution of the two-body problem. The approach to the limit of infinite volume is studied. The example of a hard sphere interaction is discussed in some detail.