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Quantum Mechanics - Theory of Scattering : Scattering From An Attractive Square Well Potential

As an application of Method of Partial Waves, the scattering from an attractive square well potential has been described. The radial part of the Schrodinger’s equation has been set up for S-wave and solving it by the use of appropriate boundary conditions, the expression for the Phase Shift and the Total Scattering Cross section have been obtained. The formula for the total scattering cross section gives the energy dependence of the cross section at low energies. It shows that low energy scattering by a short range attractive potential can be described in terms of two parameters-the scattering length and the effective range. In the case of resonant scattering, when there is a bound state near zero energy, effective range is related in a simple way to the wave function of the bound state. By exploiting these facts, it is possible to formulate the low energy scattering theory in a way that is largely independent of detailed assumptions concerning the potential function.