Complementary pictures of the N-boson problem.

Abstract

We consider a system of N identical bosons which interact in one dimension via attractive pair potentials and obey nonrelativistic quantum mechanics. This system is studied from two complementary viewpoints. The equivalent two-body method approximates the system as a collection of (N-1) independent two-particle systems with coupling constants enhanced by the factor N/2, and this yields energy lower bounds. The collective-field method approaches the problem from the standpoint of the limit as N\ensuremath{\rightarrow}\ensuremath{\infty} and leads to energy upper bounds. We find that Gaussian trial functions for the N-particle Hamiltonian and Gaussian trial densities in the collective-field theory lead to precisely the same energy upper bounds. The upper bounds provided by field theory allow for systematic improvement via a variational principle. Details are worked out for two exactly soluble problems, namely, the harmonic oscillator and the attractive \ensuremath{\delta} potential.