文档介绍：A Critical Study of the Finite Difference
and Finite Element Methods for the
Time Dependent Schrödinger Equation
Simen Kvaal
Thesis Submitted for the Degree
of Candidatus Scientiarum
Department of Physics
University of Oslo
March 2004
Preface
This thesis is perhaps a bit pared to the standards of a cand. scient. degree
(or Master Degree as it will be called in the future in order ply with international
standards). It is however aimed at a broad audience; from my fellow physics students
to mathematicians and other non-physicists that may have interests in the area. I have
tried to cut away material in order to make it a little bit shorter, and some less central
material is moved to appendices for the specially interested readers.
The title of the thesis pretty much describes the aim of this cand. scient. project.
The finite difference and finite element methods are two widely used approaches to
solving partial differential equations. Traditionally the finite element method has been
reserved for engineering projects and not that much in basic research fields such as
atomic physics where the finite difference method is the main method. One reason
may be that finite element methods are plicated to implement and that they
utilize a wide range plicated numerical tools, such as sparse matrices and iterative
solvers for linear equations. For this reason finite element solvers are usually expensive
commercial products (such as FemLab, see Ref. [1]) whose operation hides the numerical
details for the user, an approach that in a way makes scientists feel that they lose control
of the situation.
Some work has been done on the Schr¨odinger equation with finite element methods
early in the eighties and ies, see for example Refs. [2–4], but for some reason
the development has seemed to stagnate. One reason might be the above mentioned
complexity in implementation. There is a huge threshold to climb if one wants to
generalize a simple formulation which