Double Giant Quadrupole Resonance in Time-Dependent Density-Matrix Theory

state of the isoscalar giant quadrupole resonance in 16 0. It is found that the peak energy of the double phonon state is nearly twice the energy of the single phonon state calculated in the random phase approximation and that its width is slightly larger than that of the single-phonon state. The double phonon states of giant resonances have been the subject of a num ber of recent experimental 1) - 3) and theoretical 4) - 10) investigations, and some mi croscopic calculations have been performed based on various time-dependent the ories. 8) - 10) In this note we propose a new time-dependent approach based on the time-dependent density-matrix theory (TDDM).l1) TDDM is an extended version of the time-dependent Hartree-Fock theory (TDHF), and its equations of motion determine the time evolution of both one-body and two-body density matrices. We demonstrate that, using appropriate initial conditions for the two-body density ma trix, we can calculate the strength functions of the double-phonon states in a manner similar to that used to calculate the strength functions of single phonon states in TDHF and TDDM.ll) The new method is applied to the double phonon state of the isoscalar quadrupole resonance in 16 0. We first briefly discuss the equations of motion in TDDM and then present the method for calculating the strength function of the double giant quadrupole resonance (DGQR). The formulation of TDDM is based on a truncation of the BBGKY hierarchy 12) of reduced density matrices, in which genuine correlated parts of a three-body density matrix and higher reduced density matrices are neglected. 13) TDDM thus determines the time evolution of a one-body density matrix p and a two body density matrix P2. The equations of motion in TDDM consist of the following three coupled equations for the single-particle wavefunctions 'ljJQ, the occupation matrix nQQI (the expansion coefficients of p) and the correlation matrix CQ(3QI(31 (the expansion coefficients of C2, where C2 is a correlated part of P2): 11)