EXACT ANALYTICAL EXPRESSIONS AND NUMERICAL VALUES OF DIAGONAL MATRIX ELEMENTS <rk>nlj, <g|rk|g>, <g|rk|f> AND <f|rk|f> WITH DIRAC'S RADIAL FUNCTIONS g(r) and f(r) OF H-LIKE ATOMS AND THE SYMMETRY OF APPELL'S FUNCTION F2(1, 1)

Abstract

This article deals with the connection between a.m. the diagonal matrix elements with the P. A. M. Dirac radial functions g(r) and f(r) and the diagonal matrix elements <rk>nl with the E. R. Schrödinger radial functions Rnl(r) of H-like atoms (as Z=1, n≤8 and -7≤k≤4) and new properties of the P. E. Appell function F2(x,y) to the vicinity of the singular point (1, 1). In Tables 1–4, exact numerical values of these integrals are first given here, i.e., actually for all the D. I. Mendeleev table. (The author has all values of these integrals and graphs of Dirac's radial functions for the arbitrary nlj-states of H-like atoms and Z ≤ 137).