Time-Dependent Formulation of the Semiclassical Approximation in Collision Theory

A semiclassical approximation is presented which uses the wave-pocket formulation of collision theory. First an appropriate semiclassical approximation is derived by assuming that the wave packet is and remains narrow during the passage of the interaction region. After this, a process is devised to decompose a macroscopic wave packet into narrow wave packets. The wave function, after the collision, is obtained by summing up the contributions from all the narrow wave packets into which the macroscopic wave packet was decomposed. Finally, using this function, the $S$ matrix and the scattering cross section are calculated.