Generalizations of the virial relations for the Dirac equation in a central field and their applications to the Coulomb field

Abstract

The virial relations for the Dirac equation are generalized for the case of the non-diagonal radial matrix elements and for the case when the author integrates over an arbitrary finite interval of the radial variable. The case of the Coulomb field is considered separately. Recurrence relations between the solutions of some class of inhomogeneous equations appearing in the relativistic calculations of atoms according to perturbation theory are derived. Some applications of the obtained relations are discussed. In particular, a possibility of their application to the calculations of hyperfine structure of multiply charged ions in the first and higher orders of perturbation theory is demonstrated. The corresponding non-relativistic analogues of the relations obtained are also considered.