What is it about?
The zero-, first-, and second-order differential equations in a previously defined hierarchy of equations giving approximate solutions to the one-particle Dirac equation and the corresponding eigenvalue contributions are each written as power series in α, the fine structure constant, for an arbitrary, spherically symmetric potential. These equations are solved numerically for the hydrogen-atom potential to obtain wave functions to order α² and eigenvalues to order α⁴ for all states with n = 1–4, inclusive. The numerical solutions are then used to evaluate a number of matrix elements to order α². A comparison with the exact expressions shows that the numerical values for the coefficients of the different powers of α have at least six significant figures in the eigenfunctions and eigenvalues and five in the matrix elements. Thus, the procedure is validated and can be applied with confidence to other atomic systems.
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Why is it important?
This is Moore's decoupling technique for the Dirac equation. It extracts something like 3/4 of the 4 x 4 structure of the Dirac equation and is found to be a normalized Schrödinger equation. The remainder can be treated by perturbation theory.
Perspectives
The full potential of Moore's decoupling technique has never been fully realized. Though useful and practical, it has been attacked by people who never understood it. In its time, it could do all the alkalis with fairly good accuracy back in the early 1980s.
Dr Tony Cyril Scott
RWTH-Aachen University
Read the Original
This page is a summary of: Approximate solutions to the one-particle Dirac equation: numerical results, Canadian Journal of Physics, March 1986, Canadian Science Publishing,
DOI: 10.1139/p86-052.
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