  The decay of $\alpha$ particle from a nucleus is viewed as a quantum
resonance state of a two-body scattering process of the $\alpha$+daughter
nucleus pair governed by a novel nucleus-nucleus potential in squared
Woods-Saxon form. By the application of the rigorous optical model (OM)
potential scattering (S-matrix) theory the genuineness of the potential for the
system is established by giving good explanation of the elastic scattering and
reaction cross sections data of the $\alpha$+nucleus pair. From the pole
position in the complex momentum (k) plane of the S-matrix defined above, the
energy and width of the resonance state akin to the decaying state of emission
of $\alpha$ particle are extracted and from this width, the result of
$\alpha$-decay half-life is derived to account for the experimental result of
half-life in the cases of large number of $\alpha$-emitters including heavy and
super-heavy nuclei. The S-matrix of the full OM calculation above is replaced
by an analytical function expressed in terms of exact Schr\"{o}dinger solutions
of a global potential that closely represents the Coulomb-nuclear interaction
in the interior and the pure Coulomb wave functions outside, and the resonant
poles of this S-matrix in the complex momentum plane are used to give
satisfactory results of decay half-lives of $\alpha$ coming out from varieties
of nuclei.
