The Representative-Vector Method for Calculating Operator-Moments

Abstract

The utility of operator-moments or traces in the various applications of spectral-distribution theory is well-documented in the literature^1–6 as well as being the subject of a good many of the papers at this conference. From these references, past and current, it is clear that to take full advantage of the powerful entrée that spectral distribution theory offers in nuclear physics, at least, it may be generally necessary to have many moments beyond the first two Hamiltonian moments ‹H› and ‹H²›. In order to calculate, for one example, level densities reliably in the excitation-energy regions of physical interest it is now known that it may be necessary to have the moments ‹J ²_z Hⁿ› and of course ‹Hⁿ› with n ranging as high as 8 or so.⁶ The subject of this paper is a new method for obtaining these higher moments which is based on the use of random multi-particle vectors, which we call random representative vectors (RRV), in conjunction with an appropriate shell-model space and Hamiltonian. With this method it is possible to calculate average properties of very large spaces with well-defined symmetries by averaging the results over a relatively few RRV’s.