Solution of the Coupled Dirac Radial Equations

The matrix Eq. (6.68) may also be written explicitly for each composite index i = , Ki, tttj(i) as a set of two coupled ordinary differential equations of first 

As ordinary differential equations, they can be solved analytically. We proceed to solve the two coupled radial equations and present the analytic expression for the spinor of the ground state. For this purpose, we introduce the substitutions. 

The definition of z depends on the explicit choice of the electron-nucleus attraction potential energy. If it is of the form of point particle interactions as in Eq. (6.3), V r) = —Ze lr, the choice for z is as given above. In general, one may write z = — -i (compare section 6.9). These substitutions help to rewrite the system of coupled differential equations as. 

we study the short-range behavior, r 0, for which we assume that the radial functions may be expanded in a Taylor series around the origin. 

5 Radial Dirac Equation for Hydrogen-Like Atoms 209 

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