Similarities between the radial Schrödinger-Coulomb wave function and the radial Dirac-Coulomb wave function and an application to photoionization of hydrogen

Abstract

Both the radial Schr\"odinger-Coulomb wave function and the radial Dirac-Coulomb wave function can be presented in a similar way using the regular solution ${f}^{0}(r)$ and the quantum-defect theory parameters $A$ and $B.$ The radial wave function normalized per unit energy for the continuum state is ${B}^{1/2}{f}^{0}(r)$ and the radial wave function normalized to unity for a bound state is ${\mathrm{NA}}^{1/2}{f}^{0}(r),$ where $N$ is the normalization factor. The photoionization cross section of the hydrogen atom can be expressed analytically in terms of algebraic forms using the ground state and continuum states.