Single-Particle Energy and Effective Mass and the Binding Energy of Many-Body Systems

Abstract

The relationship between the single-particle energy and effective mass and the binding energy of the many-particle nuclear system is discussed. It is shown that only in the case of first order perturbation theory is it possible to define a physically meaningful single-particle energy $E(p)$ so that both relationships, $E({p}_{F})=(\frac{{p}^{2}}{2M})+V({p}_{F})={E}_{\mathrm{average}}$ and $N{E}_{\mathrm{average}}={(\frac{{p}^{2}}{2M})}_{\mathrm{average}}+\frac{1}{2}{[V(p)]}_{\mathrm{average}}$, are satisfied. More generally a correction term appears, as a result of important many-body contributions to the single-particle energy which arise from the effects of the exclusion principle and from the variation of the self-consistent excitation spectrum with density. The principal effect of the correction is to alter the relationship between ${E}_{\mathrm{average}}$ and the average value of the single-particle energy. Analysis of the optical potential which determines the momentum of a nucleon interacting with the nucleus shows that the same correction term again appears, changing the usual definition of the optical potential.An additional consequence is that it is not possible to fix the effective mass for particle motion from knowledge of the average binding energy and kinetic energy alone, the first order theory underestimating the effective mass by 38% in nuclear matter and by 77% in liquid ${\mathrm{He}}^{3}$.