A theorem on the single particle energy in a Fermi gas with interaction

Abstract

Synopsis This paper investigates single particle properties in a Fermi gas with interaction at the absolute zero of temperature. In such a system a single particle energy has only a meaning for particles of momentum ‖k‖ close to the Fermi momentum kF. These single particle states are metastable with a life-time approaching infinity in the limit ‖k‖ → kF. The limiting value of the energy is called the Fermi energy EF. As a special case of a more general theorem, it is shown that for a system with zero pressure (i.e. a Fermi liquid at absolute zero) the Fermi energy EF is equal to the average energy per particle E0/N of the system. This result should apply both to liquid He3 and to nuclear matter. The theorem is used as a test on the internal consistency of the theory of Brueckner 1) for the structure of nuclear matter. It is seen that the large discrepancy between the values of EF and E0/N, as calculated by Brueckner and Gammel 2), arises from the fact that Brueckner neglects important cluster terms contributing to the single particle energy. This neglection strongly affects the calculation of the optical potential.