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Physical Review A

Point Transforms and the Hard-Core Fermi System

Norman M. Witriol

Published Jun 1, 1970• 7 citations
Publisher

Abstract

The method of point transforms, assuming pairwise additivity, as developed by Eger and Gross, is applied to the many-body hard-core Fermi system. The ground-state energy and Landau parameters are found to second order in the hard-core radius. The dilute-gas-limit results are in agreement with those of Huang and Yang and of Abrikosov and Khalatnikov. Methods of extending the calculation to denser systems via the determination of the cutoff function of the transformation are discussed.

DOI
10.1103/physreva.1.1775

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Progress of Theoretical Physics•Jan 1, 2006•1 citation
The method of point transforms, assuming pairwise additivity, as developed by Eger and Gross, is applied to the many-body hard-core Fermi system. The ground-state energy and Landau parameters are found to second order in the hard-core radius. The dilute-gas-limit results are in agreement with those of Huang and Yang and of Abrikosov and Khalatnikov. Methods of extending the calculation to denser systems via the determination of the cutoff function of the transformation are discussed.
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 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 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.
A new method is proposed for treating the hard core potential with a small core radius a. This is an alternative to the method based on Fermi's pseudopotential. Various shortcomings of the latter method are removed with the present method. These shortcomings include the penetrability of hard spheres and incompatibility with the completeness of the unperturbed states, and they result in the invalidity of the pseudopotential method even in applications to effects that are of low order in powers of a. The present method is developed taking account of the three-particle correlation, which is a higher-effect plagued by problems of divergence and the nonanalyticity of the relevant energy as a function of the potential strength. Treatment of the three-particle correlation requires a careful preliminary study of the two-particle interaction in many-body systems, particularly when a hard core potential is involved. In this report, we present basic ideas and obtain improvement over the methods appearing in the literature.