Reduction of the N-Particle Variational Problem

Abstract

A variational method is presented which is applicable to N‐particle boson or fermion systems with two‐body interactions. For these systems the energy may be expressed in terms of the two‐particle density matrix: Γ(1, 2 | 1′, 2′)=(Ψ |a 2 + a 1 + a 1′ a 2′ | Ψ) . In order to have the variational equation: δE/δΓ = 0 yield the correct ground‐state density matrix one must restrict Γ to the set of density matrices which are actually derivable from N‐particle boson (or fermion) systems. Subsidiary conditions are presented which are necessary and sufficient to insure that Γ is so derivable. These conditions are of a form which render them unsuited for practical application. However the following necessary (but not sufficient) conditions are shown by some applications to yield good results: It is proven that if Γ(1, 2 | 1′, 2′) and γ(1 | 1′) are the two‐particle and one‐particle density matrices of an N‐particle system [normalized by tr Γ = N(N − 1) and trγ = N] then the associated operator: G(1,2 | 1′,2′)=δ(1−1′)γ(2 | 2′)+σ Γ(1′,2 | 1,2′)−γ(2 | 1)γ(1′ | 2′) is a nonnegative operator. [Here σ is + 1 or − 1 for bosons or fermions respectively.]