Three-Dimensional Quantum Mechanics: Scattering by a Potential

Abstract

In Section 10.1 we found the solutions R _t (k, r) of the radial stationary Schrödinger equation for spherical square-well potentials. Since the radial Schrödinger equation is linear, its solutions are determined up to an arbitrary complex normalization constant, which has to be inferred from the boundary conditions of the three-dimensional problem we want to solve. As we have found in Section 5.5, a harmonic plane wave is an appropriately chosen idealization of an incoming wave packet representing a particle with sharp momentum. We want to apply this finding to the three-dimensional case, that is, the scattering or diffraction of a three-dimensional harmonic plane wave which represents a particle of sharp momentum. Then the normalization of the radial wave function has to be chosen in such a way that, for great distances from the region of the potential, the three-dimensional wave function consists of an incoming plane wave exp(i k · r) and an outgoing wave.