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Physical Review

Simplified Derivation of the Binary Collision Expansion and Its Connection with the Scattering Operator Expansion

A. J. F. Siegert, Ei Teramoto

Published Jun 15, 1958• 45 citations
Publisher

Abstract

The binary collision expansion for the density matrix of a system of $N$ particles with pair interaction was derived by Huang, Lee, and Yang by expansion in the interaction and subsequent summation over terms represented by certain classes of diagrams. A simpler derivation has been obtained by the use of the $\frac{N(N\ensuremath{-}1)}{2}$ integral equations which relate the density matrix of the system to the density matrices of systems in which only one pair of particles interacts. Successive substitution of the integral equations into each other yields an expansion which by a trivial additional step becomes the binary collision expansion. Our derivation shows also that the binary collision expansion is the Laplace inverse of the coordinate representation of the expansion in terms of scattering operators.

DOI
10.1103/physrev.110.1232

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Physical Review A•Jan 1, 1970•1 citation
The binary collision expansion for the density matrix of a system of NNN particles with pair interaction was derived by Huang, Lee, and Yang by expansion in the interaction and subsequent summation over terms represented by certain classes of diagrams. A simpler derivation has been obtained by the use of the N(N\ensuremath−1)2\frac{N(N\ensuremath{-}1)}{2}2N(N\ensuremath−1)​ integral equations which relate the density matrix of the system to the density matrices of systems in which only one pair of particles interacts. Successive substitution of the integral equations into each other yields an expansion which by a trivial additional step becomes the binary collision expansion. Our derivation shows also that the binary collision expansion is the Laplace inverse of the coordinate representation of the expansion in terms of scattering operators.
It is pointed out that when there are only two interaction terms in the Hamiltonian, the binary collision expansion (BCE) for the scattering operator can be resummed into a closed form. When there are more than two interaction terms, one can resum the BCE into a continued-fraction-like form by using repeatedly the two interactions resummation. Concise derivations of the BCE are first given, the time-dependent BCE being obtained in a form applicable to both quantum evolution operators and classical frequency modulated oscillators.
A statistical-mechanical theory of evaluating the distribution functions of a many-body system is presented. The theory is a natural extension of the quantum statistical theory of Lee and Yang for the grand partition function and gives a new formalism which is different from that developed recently by Fujita, Isihara, and Montroll. The density matrices are first developed in the Uhlenbeck-de Boer UUU functions. A diagrammatical consideration separates out nonconnected products from connected products of the UUU functions, yielding an expansion formula which is simpler than that reported by de Boer some time ago. Application of the resulting expression to free bosons and fermions is made. Then the distribution functions are developed in the binary kernel introduced by Yang and Lee. This expansion is used for the evaluation of the pair distribution function of a hard-sphere Bose gas at the lowest temperature. The results improve upon those reported previously by Lee, Huang, and Yang and others. The normalization and divergence difficulties encountered by Fujita and Hirota are removed. Actually, their interpretation of Lee, Huang, and Yang's results in terms of the chain diagrams is not satisfactory. Instead the chain diagram results may be compared with the more recent results by Wu. Use of the approximate pseudopotential $\ensuremath{\varphi}=8\ensuremath{\pi}a\ensuremath{\delta}({r}_{1}\ensuremath{-}{r}_{2})$ is reflected in insufficiency of the chain diagram approximation. Satisfactory and consistent results are obtained when a new set of diagrams is taken into consideration.
A many-body pseudopotential for a hard-sphere system which is mathematically equivalent to the boundary conditions imposed on the wave functions is obtained as a generalization of Huang and Yang's binary pseudopotential, and the lowest order term of the ground state energy resulting from the triple collision is calculated for the Bose system. Instead of the direct perturbational treatment of the pseudopotential, the use of the ρk-representation for the Bose system is proposed and the ground state energy is calculated in this representation. It is shown that the energy thus calculated is of the form of an expansion in powers of (ρa3)1/2, where ρ is the average density and a the hard-sphere diameter, which is agreement with Lee and Yang's conclusion drawn from the binary collision expansion method. Some discussions associated with the roton spectrum of liquid He4 are given.
If the lifetime of a collision is defined as the difference between the times in which a particle moves in free space and under an interaction from a scattering center to the outside (or in the reverse direction), this paper shows that exactly the same relation between the lifetime operator Q and the scattering matrix S as was found by Smith (1960) can be derived. This can be done in a natural way provided that the time operator is defined as [Formula: see text] and the manifold of the asymptotic outgoing (or incoming) states is chosen to be the set of corresponding eigenstates. The part of the time needed for the particle moving freely from the scattering center to R is equal to R/v.
An explicit reduction in the number of terms of the expansion of the reduced two-particle density matrix for atomic systems is shown to occur when the expansion functions are eigenfunctions of Lz(1)+Lz(2){L}_{z}(1)+{L}_{z}(2)Lz​(1)+Lz​(2) and Sz(1)+Sz(2){S}_{z}(1)+{S}_{z}(2)Sz​(1)+Sz​(2).