Based on Mei's embedded atom model molecular dynamics simulations have been performed to investigate the rapidly cooling processes of Cu. The atomic transport property, namely the self-diffusion coefficient, is computed in the liquid state, and the results near the melting point of Cu are in good agreement with experimental data and other computational values. The atom diffusion movements during the long period of relaxation have been also studied around the solidification temperature Tc. To describe the complex microstructural evolutions during the rapidly cooling processes and the long relaxation processes, the pair correlation function and the pair analysis technique are used. It is demonstrated that the crystallization of amorphous Cu is caused by the atomic diffusion.