The Discrete Variable Method for the Time Dependent
and Time Independent Schroedinger Equation, Part I
Barry Schneider
National Science Foundation
Tuesday, October 14, 2003 13:00-14:00,
Room B111, Administration Building (101)
Gaithersburg
Tuesday, October 14, 2003 11:00-12:00,
Room 4550
Boulder
Abstract:
The discrete variable representation (DVR) has been found to be a very
effective approach for the numerical solution of the Schroedinger equation.
The advantages of the DVR are that it simultaneously provides an analytic
representation of the kinetic energy operator while preserving the
simplicity of a grid based approach for operators which are local in the
DVR
coordinate. Matrix elements of the (often complicated) local operators of
the potential energy are diagonal and may be evaluated simply at the DVR
grid points, while kinetic energy matrix elements are evaluated
analytically
or by a numerical procedure which yields the analytical result.
The first talk will be devoted to demonstrating the connection between the
classical orthogonal functions Gaussian quadrature and the DVR
representation. In addition, I will provide some illustrations of DVR-type
representations which are not based on the classical orthogonal
polynomials.
The first talk will conclude with a discussion of the manner in which
boundary conditions may be simply incorporated into the DVR approach and
how the finite element element method and the DVR may be combined in an
numerically optimal fashion.
The second talk will be devoted to using the DVR for the solution of the
time-dependent Schroedinger equation. An implict, unconditionally stable
method for propagating the time dependent Schroedinger will be described in
which the DVR is used in both the space and time coordinates. Finally, I
will describe a potentially powerful approach, which uses the Lie-Trotter
product formulae and the finite element DVR to produce numerically stable,
explicit and norm conserving approximations to the time-dependent
Schroedinger equation.
Numerical examples will be used to illustrate the methodology in the course
of the lectures.
Contact: A. J. KearsleyNote: Visitors from outside NIST must contact
Robin Bickel; (301) 975-3668;
at least 24 hours in advance.
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