Using the real stabilization method in coordinate space， the eigen-function of the single particle scattering state and resonant state in a one-dimensional potential are solved.In addition， the corresponding Wigner functions are given from the single -particle eigen -wave function，and the characteristic distribution of these Wigner functions in phase space is analyzed，which is proved to be a useful tool to distinguish the single-particle resonant state from the general scattering state and thus can be used as a criterion for studying the quantum properties in the relevant quantum measurement experiments.