Variational Methods and the Nuclear Many-Body Problem

Abstract

The general form of the energy of the ground state of a many-fermion system is shown to be exactly of the form proposed by Brueckner and Bethe, without approximation. In a variational treatment, if the trial wave function is picked containing only pair correlations, together with all possible unlinked pairs, it is described by a two-body excitation matrix $〈m_1{m}_2\mathrm{}\left|A\right|\mathrm{}{p}_1{p}_2〉$. Variation of this matrix in the Ritz-Rayleigh principle yields a set of integral equations of the scattering type for the matrix $A$. Hole-state energies are given self-consistently in terms of the matrix $A$, but particle-state energies are Hartree-Fock energies. This may be corrected for by widely enlarging the class of terms admitted into the wave function. If the approximation is then made of omitting a class of terms, defined as cross-linked clusters in $〈\psi \mathrm{}\left|H\right|\mathrm{}\psi 〉$, the particle-state energies are easily renormalized. Variation then leads to an infinite hierarchy of integral equations.