  I present numerical study of an elastic scattering by solving second order
differential equations of Schroedinger Equation for some types of central
potential (eg. square well, Yukawa, and Woods-Saxon) to find the wave function
inside the potential, and extending the solution with the region outside the
range of potential. The wave function outside the range of potential can be
expanded in terms of spherical bessel and neumann function. At the boundary,
logarithmic derivative is found which becomes a base to compute the phase
shift. Once we have a phase shift, the scattering amplitude can then be
expanded in terms of polynomial legendre, and the differential cross section
can be deduced. Furthermore, the scattering at low energies is studied and the
connection to the effective range expansion is discussed to determine the
scattering length and effective radius. The signature of the virtual state from
the scattering of $1$ neutron from $11$ nucleons are found for each potential
and the energies are best described by the inclusion of the
$\mathcal{O}(k^{10})$ in the effective range expansion.
