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福井工業大学研究紀要. 第一部

Effective Potential for the Quantum Mechanical Hard Sphere Interaction

S. Yamasaki, T. Yamanishi

Published Mar 20, 2003• 0 citations
Publisher

Abstract

The virtue of an effective potential V for the quantum mechanical hard sphere interaction is illustrated. The potential makes the wave function vanish at the surface of the sphere, and has the following advantages over Fermi's pseudopotential V_F: It prevents the wave function from penetrating the sphere; it is Hermitian and can be applied to eigenvalue problems. The potential is used to determine the three-body correlation energy induced by the polarization effects in a fermion system of hard spheres up to terms of order (κ_0α)^4 log(κ_0α) and (κ_0α)^4.

DOI
10.57375/00000243

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•Mar 20, 2004•0 citations
The virtue of an effective potential V for the quantum mechanical hard sphere interaction is illustrated. The potential makes the wave function vanish at the surface of the sphere, and has the following advantages over Fermi's pseudopotential V_F: It prevents the wave function from penetrating the sphere; it is Hermitian and can be applied to eigenvalue problems. The potential is used to determine the three-body correlation energy induced by the polarization effects in a fermion system of hard spheres up to terms of order (κ_0α)^4 log(κ_0α) and (κ_0α)^4.
A new pseudopotential is proposed for the description of the hard core repulsive interaction. The pseudopotential prevents long wavelength spheres from penetrating each other in contrast to Fermi's pseudopotential. The proposed pseudopotential is applied to the ground state of a dilute system of hard sphere fermions in the calculation of the three-body correlation energy, which has been reported to have divergent terms requiring cutoff in an earlier work. The corresponding terms obtained here are found to converge and depend as a4log a on the sphere radius a.
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.
The system under consideration is an NNN-particle quantum-mechanical system enclosed in a volume VVV, in which the particles interact via two-body hard-sphere potentials, with hard-sphere diameter aaa. 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 NNN-body problem can be introduced, and leads to an expansion of the energy levels of the system in powers of aaa. 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{a}^{3}a3 and that for the Fermi system, to order a2{a}^{2}a2. The physical interpretation and validity of these results are discussed.
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.