Analytic Harmonic Approach to the N-body problem

TL;DR

This paper introduces an analytic harmonic oscillator approach to simplify and solve the quantum N-body problem, enabling calculation of energies, eigenmodes, and correlation functions for various particle systems.

Contribution

The authors develop a novel analytic method replacing two-body interactions with harmonic oscillators, allowing exact solutions for complex N-body quantum systems.

Findings

Energy, radius, and condensate fraction increase with scattering length and particle number.

Radii decrease as the number of bosons increases.

Method applicable to fermions, mixtures, and exotic geometries.

Abstract

We consider an analytic way to make the interacting N-body problem tractable by using harmonic oscillators in place of the relevant two-body interactions. The two body terms of the N-body Hamiltonian are approximated by considering the energy spectrum and radius of the relevant two-body problem which gives frequency, center position, and zero point energy of the corresponding harmonic oscillator. Adding external harmonic one-body terms, we proceed to solve the full quantum mechanical N-body problem analytically for arbitrary masses. Energy eigenvalues, eigenmodes, and correlation functions like density matrices can then be computed analytically. As a first application of our formalism, we consider the N-boson problem in two- and three dimensions where we fit the two-body interactions to agree with the well-known zero-range model for two particles in a harmonic trap. Subsequently, condensate fractions, spectra, radii, and eigenmodes are discussed as function of dimension, boson number N, and scattering length obtained in the zero-range model. We find that energies, radii, and condensate fraction increase with scattering length as well as boson number, while radii decrease with increasing boson number. Our formalism is completely general and can also be applied to fermions, Bose-Fermi mixtures, and to more exotic geometries.