Variational Approach to the Many-Body Problem

Abstract

Certain problems connected with the choice of trial variational functions for the quantum many-body problem are discussed. It is convenient to make use of the reduced-density-matrix formulation developed by Löwdin and Mayer, since a system whose Hamiltonian contains only 1- and 2-body interactions is characterized by the two simplest reduced density matrices ${\Gamma }^{\mathrm{}\left(1\right)\mathrm{}}$ and ${\Gamma }^{\mathrm{}\left(2\right)\mathrm{}}$. It is known that symmetry, Hermiticity, and normalizability conditions are not sufficient to ensure a physically realizable choice of the functions ${\Gamma }^{\mathrm{}\left(1\right)\mathrm{}}$ and ${\Gamma }^{\mathrm{}\left(2\right)\mathrm{}}$. However an additional nontrivial restriction is imposed by the fact that such functions must be possible variational extrema. The implications of this condition are investigated in some detail and it turns out that all the nonphysical choices of ${\Gamma }^{\mathrm{}\left(1\right)\mathrm{}}$ and ${\Gamma }^{\mathrm{}\left(2\right)\mathrm{}}$ which have been exhibited by various authors are eliminated. The condition is essentially nonquantum inasmuch as an analogous argument should be possible for a classical system. Moreover one of the consequences of this condition is that the system (classical or quantum) may exhibit macroscopic behavior, i.e., order-disorder transitions. The analysis leads to apparently reasonable choices of trial forms for ${\Gamma }^{\mathrm{}\left(1\right)\mathrm{}}$ and ${\Gamma }^{\mathrm{}\left(2\right)\mathrm{}}$, although it is still not known whether any further restriction must be imposed to complete the sufficiency argument.