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Physical Review

Structure of a Many-Particle Quantum-Mechanical Medium

Robert Karplus, Kenneth Watson

Published Sep 1, 1957• 5 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.107.1205

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Many-Body Problem in Quantum Statistical Mechanics. I. General Formulation

T. D. Lee, Chongming Yang

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 $U$ functions defined for the same system with Boltzmann statistics. It is then shown that these $U$ functions can be evaluated in successive approximations in terms of a binary kernel $B$ 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.

Physical Review•Mar 1, 1959•210 citations
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 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.
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.
A strongly coupled system---the limiting case of a highly degenerate many-fermion system for which the variation of the kinetic energy is neglected, and the interaction restricted to a region of momentum space neighboring the Fermi surface---has been analyzed in a manner not dependent upon assumptions about the convergence of power series expansions or on partial summations of infinite series. The vacuum expectation value of the resolvent operator, ${〈\frac{1}{(H\ensuremath{-}z)}〉}_{0}$, is expressed as the Laplace transform of the exponential of a function linearly dependent on the volume of the system. It is shown that the linked-cluster expansion of the vacuum expectation value of the resolvent operator has a zero radius of convergence as a power series in the coupling constant. The most serious physical consequence of this is that a nontrivial interaction never results in a "normal" system.
A new cluster expansion technique is developed for the calculation of the energy expectation value of the many-fermion systems with singular potentials.The wave function, which is studied in this paper, is of the formwhich, in a special case, coincides with Jastrow wave function.The formula for the energy expectation value is written in terms of the complex cluster integrals.The method introduced in the derivation of this formula has an analogy to the classical cluster expansion method in the theory 0f an imperfect classical gas of multicomponents.
We develop an approach of calculating the many-body path integral based on the linked cluster expansion method. First, we derive a linked cluster expansion and we give the diagrammatic rules for calculating the free energy and the pair distribution function g(r)g(r)g(r) as a systematic power-series expansion in the particle density. We also present a structured Pad\'e approximation scheme in the momentum space to determine g(r)g(r)g(r). The calculated g(r)g(r)g(r) for distinguishable particles interacting with Lennard-Jones and hard-sphere potential in various attempted schemes of approximation of the diagrammatic series compares very well with the results of path-integral Monte Carlo simulation. Our method is applicable to a wide range of problems of current general interest and may be extended to the case of identical particles and, in particular, to the case of the many-fermion problem.
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.