Approximate Solution of a Finite Many-Particle System with Translational Invariance. I

Abstract

A formal technique is developed to study internal properties of finite many-particle systems possessing translational invariance. The basic objective is to discover a method of calculating, in a modified shell model, average values of operators referred to the center of mass, particularly those which have to do with the surface of the system, such as the density of particles. The formalism is based on the method of redundant variables, in which the shell-model wave function describes the internal degrees of freedom of the system, while a superfluous variable is introduced to satisfy the conservation law and to handle the collective motion. The spurious degeneracy which results from the introduction of the extra variable is removed by suitable subsidiary conditions. With the aid of these restricted solutions, it is shown that the density with respect to the center of mass can be calculated in terms of a one-body operator. The dynamics of the system is then formulated in terms of density matrices suitably defined from these restricted solutions. Within the context of the modified shell model, the density matrices consist of the Dirac density matrices of the shell model plus correction terms which depend nonlinearly on these Dirac matrices and matrix elements of the coordinate $X$. The single-particle states of the shell model are finally chosen to minimize the expectation of the Hamiltonian. The resulting variational equation contains a homogeneous part which has the structure of the conventional Hartree-Fock equation together with an inhomogeneous term depending nonlinearly on the single-particle states. The solution of this equation is finally expanded about the nontranslationally invariant Hartree-Fock solution to first order in perturbation theory.