Determination of Josephson penetration depth has a crucial importance in calculation of both anisotropy factor and the effective mass of quasi-particles (m(*)) in high temperature copper oxide superconductors. In this study, the effective mass values of the quasi-particles have been determined via third derivative of phase differences of quantum wave function that represents the superconducting state of the system. According to our calculations, the anisotropy factor and m(*) have been affected by temperature varying from 5 to 90 K for over-doped sample. Moreover, the anisotropy factor and m(*) of the superconducting media are found to be changed by temperature, applied magnetic field and the oxygen annealing procedures. Furthermore, the quantum vertex which corresponds to the quantum gravity that has been already determined for optimally oxygen doped mercury cuprate superconductors  has not showed up in over-doped samples investigated.