Quantitative Derivation and Scattering of the 3D Cubic NLS in the Energy Space

We consider the derivation of the defocusing cubic nonlinear Schr"{o}dinger equation (NLS) on ${\mathbb{R}}^3$ from quantum $N$-body dynamics. We reformat the hierarchy approach with Klainerman-Machedon theory and prove a bi-scattering theorem for the NLS to obtain convergence rate estimates under $H^1$ regularity. The $H^1$ convergence rate estimate we obtain is almost optimal for $H^1$ datum, and immediately improves if we have any extra regularity on the limiting initial one-particle state.