Interdiffusion process at interfaces of chemically identical polymers (e.g., deuterated-nondeuterated pairs) with different molecular weights or polymers with similar physical properties, is studied here by varying the diffusion time. Considering the vacancy flux ($J_v$) and adopting the Cahn-Hilliard interracial energy in describing this system, we can see that the variation of the interfacial composition profile with time is asymetric and the interface moves towards the polymer with the lower molecular weight as interdiffusion progresses. Furthermore, interface shift $Deltachi$, which characterizes the interdiffusion between polymers, agrees well with the behaviors of the existing experimental data. We can also obtain the interface shift factor C, which can be converted into values of $D_s$ (self-diffusion coefficient of the smaller molecules), from the slopes of the linear fits to the data of the interface shift.