The continuous spectra of hydrogen-like atoms

Abstract

This investigation consists of a theoretical study, by the Wave Mechanics, of the intensities of the affinity spectra of hydrogen like atoms. The main properties of the eigenfunctions for the continuous range of eigenvalues are investigated. They are shown to be real, and their asymptotic expansions are derived. The theory of the normalization of continuous eigenfunctions is applied, and their normalization factors are obtained. The integrals for the coordinate matrices corresponding to transitions from the continuous states to the discrete levels are then evaluated. It, then, is shown that the squares of the complete Schrodinger matrices for the three coordinates x:, y, and z, are equal, their common value being derived. The general formulae are applied to the special cases of the continuous spectra associated with the Lyman and Balmer discrete levels. The absorption coefficients of these spectra, as a function of the frequency, are deduced and plotted. It is found that for equivalent ratios of the absorbed frequency to the critical ionization frequency of the discrete level, the probability of absorption from the Balmer level is approximately nine times that from the Lyman level. The values of the matrices, for any discrete state, are given for the long wave length limit of the continuous spectra. Finally, the variation with wave length and atomic number of the absorption coefficients for both the long and short wave length limits of the Lyman and Balmer continuous spectra are briefly discussed.