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

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