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Journal of Mathematical Physics

Many-Body Point Transforms. II. An Exact Noncluster Approach to the Hard-Core Many-Body Problem

Norman M. Witriol

Published Dec 1, 1971• 3 citations
Publisher

Abstract

The method of many-body point transforms previously developed by the author is modified to eliminate the interacting particle cluster formalism. The cutoff terms in the transformation, and thus in the Hamiltonian, are therefore removed. For the hard-core many-body problem, as in the previous work, a Hamiltonian is obtained which is Fourier analyzable, Hermitian and amenable to ordinary perturbation and variational techniques while still being equivalent to the original Hamiltonian. No approximations are made. It is demonstrated that the use of the standard zeroth-order approximation of the ground state Bose system, i.e., a constant, for the transformed wavefunction gives a negative expectation value of the energy for the hard core system. This discrepancy arises from the extended range of the new potential generated by the transformation, which necessitates explicit consideration of boundary conditions not satisfied by the above wavefunction.

DOI
10.1063/1.1665559

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The method of many-body point transforms previously developed by the author is modified to eliminate the interacting particle cluster formalism. The cutoff terms in the transformation, and thus in the Hamiltonian, are therefore removed. For the hard-core many-body problem, as in the previous work, a Hamiltonian is obtained which is Fourier analyzable, Hermitian and amenable to ordinary perturbation and variational techniques while still being equivalent to the original Hamiltonian. No approximations are made. It is demonstrated that the use of the standard zeroth-order approximation of the ground state Bose system, i.e., a constant, for the transformed wavefunction gives a negative expectation value of the energy for the hard core system. This discrepancy arises from the extended range of the new potential generated by the transformation, which necessitates explicit consideration of boundary conditions not satisfied by the above wavefunction.
The method of point transforms is extended to the consideration of the truly many-body terms of many-body systems. The point transformation is taken to be regional (one body isolated, two bodies interacting, three bodies interacting, …, N bodies interacting) and continuous in the 3N-dimensional transformed space. The transformed Hamiltonian and wavefunction of an N-body system can then be easily approximated by one-body, two-body, three-body, …, or N-body additivity. As an example of the use of this method, a set of regional point transforms is developed for the short range, strongly repulsive, or hard-core many-body problem. Applying these transforms to the many-body Hamiltonian, we obtain a set of equivalent Hermitian Hamiltonians which are Fourier analyzable and amenable to ordinary perturbation and variational techniques. Assuming two-body additivity, this set is shown to be the set of pairwise point transformations developed by Eger and Gross.
The method of point transformations is used to provide a Hamiltonian for strongly interacting particles that is equivalent to the usual Hamiltonian. The new Hamiltonian is Hermitian, Fourier analyzable, velocity dependent, and free of the difficulties of pseudopotentials. The present work is restricted to the case that two-body encounters are dominant. The theory is expressed in the quantized wave formalism so that number-violating approximations can be applied. The well known results for the spectrum and ground-state energy of the dilute hard-sphere gas are then derived as an elementary application.
The left-unitary operators that introduce the correlations of the hard core in an uncorrelated bound-state wave function of a finite number AAA of particles are explicitly determined up to a unitary transformation. Left unitarity is required in order to keep orthonormality. Using a shell-model basis, transformed according to the correlation operator, it is possible to diagonalize the Hamiltonian in a subspace of the full Hilbert space with the technique of Bloch and Horowitz. Ground as well as excited states are treated on the same footing. In practice, the diagonalization is accomplished using the original single-particle basis and transforming back the Hamiltonian. This results in an AAA-body operator for which a cluster expansion is possible. In this cluster expansion, there are three- and four-body terms easy to evaluate, because they factorize. A particularly interesting application can be made to few-body systems, evaluating the integrals numerically. The unique approximation is then connected with the cut of the basis. The method developed seems comparatively simple and is free of the mathematical problems of the hard core.
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.
The earlier proposed method for exact determination of the energy spectra of Hamiltonians H and the quantities , where W is a finite-dimensional perturbation, is extended to treat a still larger class of Hamiltonians. In particular, some two- and three-dimensional problems became amenable to this approach. In addition, the expressions for the density and local densities of states are deduced. As an example, an impurity problem is considered in a simple cubic lattice with short-range and in a linear chain with long-range interactions.