Absorbing Boundary Conditions for the Schrr Odinger Equation

A large number of di erential equation problems which admit traveling waves have very large (typically in nite) naturally de ned domains, with boundary conditions de ned at the domain boundary. To be able to numerically solve these problems in smaller subdomains of the original domain, arti cial boundary conditions must be de ned for these subdomains. One such arti cal boundaryconditionswhich can minimize the size of such subdomainsare absorbingboundary conditions. A techniqueused to reduce the necessary spatial domainwhen numerically solving partial di erential equations that admit traveling waves is the imposition of absorbing boundary conditions. Such absorbing boundary conditions have been extensively studied in the context of hyperbolic wave equations. A general absorbing boundary condition will be developed for the Schr odinger equation with one spatial dimension, using group velocity considerations. Previously published absorbing boundary conditions will be shown to reduce to special cases of this absorbing boundary condition. The well-posedness of the Initial Boundary Value Problem of the absorbing boundary condition, coupled to the interior Schr odinger equation, will also be discussed. Extension of the general absorbing boundary condition to higher spatial dimensions is demonstrated. Numerical simulations using initial single Gaussian, double Gaussian and Pseudo-delta function distributions will be given, with comparision to exact solutions, to demonstrate the re ectivityproperties of various orders of the absorbing boundary condition.