Test of a general symmetry-derived $N$-body wave function

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 for a confined system of identical bosons with a general two-body interaction. This method 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, yielding a solution for the general $N$-body problem which scales as $N^0$ at any given 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 application of this $N$-body wave function to a system of $N$ fully interacting bosons in three dimensions. We derive an expression for 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 for the wave function 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.

Figure 1

Scaled density profile at $D=3$ for $N$ harmonically interacting particles under harmonic confinement in oscillator units of the confining potential. The dotted curve is the lowest-order density profile, while the dashed curve is the density profile for the first-order wave function. The solid curve is the exact result. The scaling factor, $\sqrt{{\lambda }_{\text{eff}}}$, is explained in the Appendix.

Figure 2

Unscaled density profile at $D=3$ for $N=10000$ particles under harmonic confinement with strong attractive harmonic interactions $\left({\lambda }_p^2=100\right)$ in oscillator units of the confining potential. The dotted curve is the lowest-order density profile, while the dashed curve is the density profile for the first-order wave function. The solid curve is the exact result. The parameter ${\lambda }_p^2$, as explained in the Appendix, is the interaction frequency squared in oscillator units of the confining potential.

Figure 3

Unscaled density profile at $D=3$ for $N=10000$ particles under harmonic confinement with repulsive harmonic interactions $\left({\lambda }_p^2=-1/10000+{10}^{-10}\right)$ in oscillator units of the confining potential. The system is just below the dissociation threshold and very extended. The dotted curve is the lowest-order density profile, while the dashed curve is the density profile for the first-order wave function. The solid curve is the exact result. The parameter ${\lambda }_p^2$, as explained in the Appendix, is the interaction frequency squared in oscillator units of the confining potential.