A new method for solving the Z>137 problem for energy levels of hydrogen-like atoms

摘要

The "catastrophe" in solving the Dirac equation for an electron in the fieldof a point electric charge, which emerges for the charge numbers Z > 137, isremoved in this work by effective accounting of finite dimensions of nuclei.For this purpose, in numerical solutions of equations for Dirac radial wavefunctions, we introduce a boundary condition at the nucleus boundary such thatthe components of the electron current density is zero. As a result, for allnuclei of the periodic table the calculated energy levels practically coincidewith the energy levels in standard solutions of the Dirac equation in theexternal field of the Coulomb potential of a point charge. Further, for Z >105, the calculated energy level functions E(Z) are monotone and smooth.Thelower energy level reaches the energy E=-mc^{2} (the electron "drop" on anuclei) at Zc = 178. The proposed method of accounting of the finite size ofnuclei can be easily used in numerical calculations of energy levels ofmany-electron atoms.