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

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