RADIAL MATRIX-ELEMENTS WITH DIRACS FUNCTIONS IN THE THEORY OF FINE AND HYPERFINE STRUCTURES OF HYDROGEN-LIKE SYSTEMS

Investigating the effects of fine and hyperfine structures of hydrogen spectrum there arises practical necessity to evaluate the different m atrix elements with Dirac's radial functions f and g; usually such mat rix elements being evaluted approximately. The article deals with the study of relativistic matrix elements by a new method, i.e. with the h elp of Appell's hypergeometrical functions F-2(x, y) and their new pro perties. Such approach allows: (1) to get exact analytical expressions for these matrix elements by means of Appell's F-2 (and Clausen's F-3 (2)) functions; (2) to reveal ''latent'' symmetry of diagonal matrix e lements of such types as [r(k)](njl) and [g\r(k)\f](njl) with respect to points k(0) = -3/2 and k(0) = -1/2, respectively, the above symmetr y is connected with the property of Appell's function F-2(1, 1) mirror -like symmetry; (3) to prove that the diagonal and off-diagonal matrix elements in a ''pole-domain'' of Euler's gamma-function are given exa ctly analytically by means of ''nonorientable'' series of type F-2(x, y). The examples for the evaluation of diagonal matrix elements for th e K- and L-shells and also of dipole matrix elements for Lyman's serie s are obtained.