What is it about?

Symbolic computation provides excellent tools for solving quantum mechanical problems by perturbation theory. The techniques presented herein avoid the use of an infinite basis set and some of the complications of degenerate perturbation theory. The algorithms are expressed in the Maple symbolic computation system and solve for both the eigenfunctions and eigenenergies as power series in the order parameter. Further, each coefficient of the perturbation series is obtained in closed form. In particular, this paper examines the application of these techniques to R. A. Moore's method for solving the radial Dirac equation. One is confident that the techniques presented will also be useful in other applications.

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Why is it important?

Moore's decoupling technique is useful for solving the Dirac equation for all sorts of potentials. It recasts the problem as a re-normalized non-relativistic Schrödinger equation and a small component which can be treated by perturbation theory. This paper uses computer algebra to demonstrate convergence of the solutions for the hydrogen atom case and shows a number of general methods for the hierarchy of differential equations. It bears a resemblance with the Foldy transformation but is far more useful.

Perspectives

Not enough about the successful applications of Moore's decoupling technique has been published. E.g. it has been used for the alkali atom systems but results are now outdated. By now, it is superseded by other programs like GRASP for numerical relativistic atomic fine-structure calculations but it still has some juice left to finding analytic solutions to exotic problems.

Dr Tony Cyril Scott
RWTH-Aachen University

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This page is a summary of: Perturbative solutions of quantum mechanical problems by symbolic computation, Journal of Computational Physics, April 1990, Elsevier,
DOI: 10.1016/0021-9991(90)90258-3.
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