A Test of a New Interacting N-Body Wave Function

TL;DR

This paper applies a novel group-theoretic N-body wave function approach to a three-dimensional system of harmonically-interacting bosons, demonstrating strong convergence and matching with independent exact solutions at first order.

Contribution

It presents the first application of a new invariant perturbation formalism to a 3D N-body bosonic system, enabling analytical solutions and comparison with exact results.

Findings

First-order density profile matches independent exact solutions.

Method shows strong convergence to all orders.

Applicable to fully-interacting N-body quantum systems.

Abstract

The resources required to solve the general interacting quantum N-body problem scale exponentially with N, making the solution of this problem very difficult when N is large. In a previous series of papers we develop an approach for a fully-interacting wave function with a general two-body interaction which tames the N-scaling by developing a perturbation series that is order-by-order invariant under a point group isomorphic with S_N . Group theory and graphical techniques are then used to solve for the wave function exactly and analytically at each order. Recently this formalism has been used to obtain the first-order, fully-interacting wave function for a system of harmonically-confined bosons interacting harmonically. In this paper, we report the first application of this N-body wave function to a system of N fully-interacting bosons in three dimensions. We determine the density profile for a confined system of harmonically-interacting bosons. Choosing this simple interaction is not necessary or even advantageous for our method, however this choice allows a direct comparison of our exact results through first order with exact results obtained in an independent solution. Our density profile through first-order in three dimensions is indistinguishable from the first-order exact result obtained independently and shows strong convergence to the exact result to all orders.