ITLApplied  Computational Mathematics Division
ACMD Seminar Series
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The Discrete Variable Method for the Time Dependent and Time Independent Schroedinger Equation, Part II

Barry Schneider
National Science Foundation

Wednesday, October 15, 2003 13:00-14:00,
Room B111, NIST Administration Building (101)
Wednesday, October 15, 2003 11:00-12:00,
Room 4550

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. Kearsley

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