Fisher information of radial wavefunctions for relativistic hydrogenic atoms

Highlights: • We present Fisher information per binding energy for relativistic hydrogenic atoms. • The relativistic effect makes the information increase for heavy atoms. • The inverse participation ratio decreases monotonically with Z to the contrary. -- Abstract: We calculate Fisher information and the inverse participation ratio normalised by the electric binding energy for the relativistic hydrogenic atoms based on the radial wavefunctions in position space. Fisher information rapidly decreases as the atomic number increases, but it increases again for heavy atoms for $S_{1/2}$ and $P_{1/2}$ states due to the relativistic effect. We consider not only the ground state but also excited states by numerically evaluating the generalised Laguerre polynomials in the exact Dirac's relativistic radial wavefunctions. This study contrasts with the previous studies of the relativistic many-electron systems (i.e., ground-state neutral atoms), where numerical densities were employed.