The Optimal Stopping Problems based on Markov processes were and still are interesting topics due to their applications in a variety of subjects. Here, we begin with a concrete diffusion process, and a new strategy is adopted for solving the problems.  First we explicitly  compute some typical hitting times of the diffusion process. Then the Markov processes potential analysis theories  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.