Radiation boundary conditions for the numerical solution of the three-dimensional time-dependent Schrödinger equation with a localized interaction

Abstract

Exact radiation boundary conditions on the surface of a sphere are presented for the single-particle time-dependent Schrödinger equation with a localized interaction. With these boundary conditions, numerical computations of spatially unbounded outgoing wave solutions can be restricted to the finite volume of a sphere. The boundary conditions are expressed in terms of the free-particle Green's function for the outside region. The Green's function is analytically calculated by an expansion in spherical harmonics and by the method of Laplace transformation. For each harmonic number a discrete boundary condition between the function values at adjacent radial grid points is obtained. The numerical method is applied to quantum tunneling through a spherically symmetric potential barrier with different angular-momentum quantum numbers l . Calculations for l=0 are compared to exact theoretical results.