ORDER 1 N CORRECTIONS TO THE TIME-DEPENDENT HARTREE APPROXIMATION FORA SYSTEM OF N+1 OSCILLATORS/

We solve numerically to order 1/N the time evolution of a quantum dyna mical system of N oscillators of mass rn coupled quadratically to a ma ssless dynamic variable. We use Schwinger's closed time path formalism to derive the equations. We compare two methods which differ by terms of order 1/N-2. The first method is a direct perturbation theory in 1 /N using the path integral. The second solves exactly the theory defin ed by the effective action to order 1/N. We compare the results of bot h methods as a function of N. At N=1, when we expect the expansion to be quite innacurate, we compare our results to an exact numerical solu tion of the Schrodinger equation. In this case we find that when the t wo methods disagree they also diverge from the exact answer. We also f ind at N=1 that the 1/N corrected solutions track the exact answer for the expectation values much longer than the mean field (N=infinity) r esult.