Spatial Approximation

Abstract

The emphasis here is on the spatial part of the TDSE; it's represented either via its grid or a basis representation, these are used to illustrate the transformation of the TDSE to a matrix algebra problem, either as a system of linear equations or as an eigenvalue problem. Finite differences is applied as a local approximation method of the equation while a trigonometric basis is used as an example of a global approximation method of the solution. Finally, the B-splines basis, a basis standing in between these two extremes, a piecewise polynomial basis with limited spatial support, is introduced and its algebraic properties are discussed.