Characteristics of the scattering states of an n-dimensional hydrogen atom have been studied. The normalized wave functions of scattering states on “k/2π scale” and the calculation formula of phase shift are presented.The analytical properties of phase shift are discussed, and analytical formulas for calculating bound-continuous transition matrix elements are also given. The relevant results of scattering states for the usual hydrogen atom (n=3) reported in the literature are contained in more general conclusions of this paper as special cases.