The Optimal Stopping Problems based on Markov processes have aroused much more attentions due to their applications in a wider range of areas. Here, we will begin with a concrete diffusion process, and a new strategy is adopted for solving the problems. First we will explicitly compute some typical hitting times of the diffusion process. Then the Markov processes potential analysis theories could help us to further get the optimal stopping problem solution in a close form, for the infinite horizon case. Also this method can be developed for the finite horizon case, in some sense.