Monte Carlo Ground State Energy for Trapped Boson Systems

Diffusion Monte Carlo (DMC) and Green's Function Monte Carlo (GFMC) algorithms were implemented to obtain numerical approximations for the ground state energies of systems of bosons in a harmonic trap potential. Gaussian pairwise particle interactions of the form $V_0 e^{-{|r_i-r_j|^2}/{{r_0}^2}}$ were implemented in the DMC code. These results were verified for small values of $V_0$ via a first-order perturbation theory approximation for which the N-particle matrix element evaluated to ${N\choose2} \frac{V_0}{(1 + 1/{r_0}^2)^{3/2}}$. By obtaining the scattering length from the 2-body potential in the perturbative regime ($\frac{V_0}{\hbar\omega} \ll 1$), ground state energy results were compared to modern renormalized models by P.R. Johnson \emph{et. al}, New J. Phys. {\bf 11}, 093022 (2009).