Cluster Expansions and the Theory of Many-Boson Systems

Abstract

The previous configuration-space cluster integral treatment of the properties of the ground state of a many-boson system is modified by including those higher order diagrams consistent with the pair excitation approximation, which were p~eviously omitted. The cluster expansions involve a parameter, analogous to the fugacity in classical expansions, whose definition automatically accounts for the depletion of the free particle ground state. The resulting expectation value for the Hamiltonian is .in agreement with that obtained by field-theoretic methods for states of pair-excitation type. § I. This paper is concerned with the connection between two different methods for describing the ground state and low-lying excited states of a many particle boson system under the assumption of strong short range repulsive forces between pairs of particles. One of these methods uses perturbation theory in the op­ erator formalism of second quantization, as developed by Lee, Huang, and Yang1', by Brueckner and Sawada,2> and extended by others. 3>-5> These are well­ ordered calculations, in the sense that results given for the energy of the ground state and low-lying excited states are exact to a certain order in the relevant expansion parameter. These calculations are valid at low densities, for example, the region of validity of the ground state energy for hard spheres is defined by (pa3)11 2 -15> Here the ·ground state wave function is expressed as a product of pair functions f(ri 1), and thus takes account of correlations in pairs of particles brought about by the repulsive interactions. The expectation value for the ground state energy is then expressed in terms of cluster integrals, similar to the Ursell-Mayer de­ velopment in classical statistical mechanics.16 > Finally, the variational principle is used to derive a Schrodinger-like equation for the pair function, or is used to vary a parameter in an assumed form for the pair function. 7>,B),lO) In this procedure, only certain subsets of the complete set of cluster integrals can in