Stationary Solution of a Time Dependent Density Matrix Formalism

Abstract

A stationary solution of a time-dependent density-matrix formalism, which is an extension of the time-dependent Hartree-Fock theory to include the effects of two-body correlations, is obtained for the Lipkin model hamiltonian, using an adiabatic treatment of the two-body interaction. It is found that the obtained result is a reasonable approximation for the exact solution of the model. We have recently reported realistic calculations for the damping of giant reso­ nances1>-3> and for low-energy heavy-ion collisions, 4 > based on an extended version of the time-dependent Hartree-Fock theory. The ETDHF theory called the time­ dependent density-matrix theory (TDDM) determines the time evolution of both one-body and two-body density matrices and allows us to consistently treat the effects of the nuclear mean field and two-body correlations. Although the obtained results are quite encouraging, these numerical calculations contain an inconsistency in the treatment of the ground state: the Hartree-Fock (HF) ground state, which clearly is not a stationary solution of TDDM, is used as the initial ground state. In our previous paper 5 > we tried to find a correlated stationary solution of TDDM using the stationary limit of the TDDM equations and the constraints on reduced density matrices derived from general relations among them. The obtained results for the Lipkin model, 6 > however, were not quite satisfactory and the problem of finding the ·reasonable stationary solutions of TDDM has not been settled yet. In this paper we demonstrate that a better stationary solutions of TDDM can be obtained through an adiabatic treatment of two-body interactions. The TDDM equations have been derived from the truncation scheme, proposed by Wang and Cassing,7), 8 > of the well-known BBGKY hierarchy 9 > for reduced density