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.
The off energy shell effect on the contribution to the energy from the renormalized three-particle ring diagram is studied for the ground state of a system of hard sphere fermions. The linked cluster formalism supplemented with infinite partial summation method, such as the K-matrix method, fails in giving a finite value to the energy of interest. In place of the K-matrix, we make use of the effective potential which improves the pseudopotential introduced by Fermi and exploited by Lee, Huang and Yang. Our potential prevents two spheres from penetrating each other. It gives a finite energy correction whose dominant part varies as a^4 loga with the small value of the core radius a. Correction term of order a^4 is also presented.
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.
The system under consideration is an
N-particle quantum-mechanical system enclosed in a volume
V, in which the particles interact via two-body hard-sphere potentials, with hard-sphere diameter
a. The two-body hard-sphere problem is first discussed by a generalization of Fermi's pseudopotential by means of which the problem is formulated entirely in terms of the scattering phase shifts. It is then shown that a pseudopotential for the
N-body problem can be introduced, and leads to an expansion of the energy levels of the system in powers of
a. The first order energy levels of a Bose and a Fermi system are calculated. For the Bose system, the first order energy formula exhibits an "energy gap" above the ground state, leading to properties of the system not dissimilar to that of a superfluid. The ground state energy for a Bose system is calculated to order
a3 and that for the Fermi system, to order
a2. The physical interpretation and validity of these results are discussed.
The three-particle correlation energy induced by the polarization effects in the ground state system of hard core fermions is studied with the aid of an effective potential which prevents the cores from penetrating each other. The energy varies as a^4 log a with the small value of the core radius a and is not analytic at a = O. The nonanalyticity is responsible for the failure of the linked cluster expansion formalism, complemented with the partial summation method, in giving a finite value to the three-particle correlation energy in quantum fluid. It shows that t, he expansion of the energy beyond third order terms in t, he coupling const, ant, is not possible. The energy is examined more carefully and components behaving as a^4 are identified. Nonanalyitic component besides the a^4 log a term and dominating the a^4 term, such as a^4 log |log a|, is not found.