One-Particle Motions in Many-Particle Systems and the Optical Model in Nuclear Reactions

Abstract

A possible scheme of the systematic one-particle motion III a many-particle system is presented from the first principle as a time-dependent formalism. The theory is formulated and interpreted exclusively for the optical model in nuclear reactions, although the present formalism can be utilized to study various problems in solid state physics. First the one­ particle amplitude is so defined as to describe the processes of elastic scattering. Then it is shown that the systematic part of the amplitude, corresponding to the coarse-grained motion of the system, obeys the one-particle Schrodinger equation with the optical potential, and that the fluctuating part of the amplitude is governed by the Langevin-like equation with the same optical potential and by the fluctuation-dissipation theorem. This is just the scheme assumed in the previous paper from the semi-phenomenological point of view. The optical potential can be calculated from its definition given as the Fourier transform of the so-called self­ part appearing in the equation of the one-particle Green function in the medium of the target nucleus. From the definition it is easily seen that the optical potential is, in general, non-local and slightly energy-dependent. The optical,potential is decomposed into two parts, one being the static (or energy-independent) part to be observed in the target nucleus in the fixed ground state and the other representing reactions of nuclear excitations. It is inferred that the former would not be so different from the corresponding term of the one­ particle potential as expected in the ordinary shell model, and that the latter is small due to the average effect originating in the energy spread of the incident beam. The former is purely real, while the latter has an imaginary part which is responsible for the probability dissipation of elastic scattering. The face of the optical potential may be of the same type irrespective of the question whether the incident beam is a simple short wave-packet or a !nixed beam, so far as the coarse-grained motions are pursued. Finally it is proved that the fluctuation-dissipation theorem holds for the, correlation function of the fluctuating source or amplitude if the system is excited in quasi-equilibrium.