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Physical Review

A Note on the Wave Functions of the Relativistic Hydrogenic Atom

Leverett Davis

Published Jul 15, 1939• 42 citations
Publisher

Abstract

The radial functions for the discrete levels of Dirac's relativistic hydrogenic atom are expressed in terms of the generalized Laguerre polynomials which have known properties. This leads to an easy method of evaluating the normalization constants and the average values of ${r}^{q}$.

DOI
10.1103/physrev.56.186

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The radial functions for the discrete levels of Dirac's relativistic hydrogenic atom are expressed in terms of the generalized Laguerre polynomials which have known properties. This leads to an easy method of evaluating the normalization constants and the average values of rq{r}^{q}rq.
will be added. One can explain the nature of the term by examining the relativistic case. Below the formulas for the radial and angular parts of the wave function were evaluated. The term for the energetic levels of the hydrogen atom for specific quantum numbers was given. We calculated the values of several specific radial functions for different quantum numbers, as well as the values of the radial function at the origin of coordinates and wide apart. A number of mean values for different powers of r was calculated. For the continuous spectrum the radial eigenfunctions were calculated.
This chapter begins with the simple solution of the Schrödinger equation for hydrogen atom for which the energy value has been derived. The Bohr radius of H atom has also been computed. After this, this chapter provides the generalised solution of the Schrödinger equation for H atom/hydrogen-like species in which F equation, R equation, and T equations have been derived in a simple and lucid way. This chapter also deals with the Laguerre differential equation, Laguerre polynomials, Rodrigues formula for the Laguerre polynomials, Laguerre associated equation and its solution, associated Laguerre polynomials, Rodrigues formula for the associated Laguerre polynomials, and solution of the radial equation. It also provides normalisation of the radial wave function and complete wave function; for the hydrogen atom, hydrogenic atomic orbital; electron density of 1 s electron; mathematical background of shape of orbitals; the most probable distance of electron from nucleus of hydrogen atom, and average distance of electron from the nucleus of hydrogen atom. The end of this chapter describes the references and adequate computational problems (solved problems and question on concepts).
will be added. One can explain the nature of the term by examining the relativistic case. Below the formulas for the radial and angular parts of the wave function were evaluated. The term for the energetic levels of the hydrogen atom for specific quantum numbers was given. We calculated the values of several specific radial functions for different quantum numbers, as well as the values of the radial function at the origin of coordinates and wide apart. A number of mean values for different powers of r was calculated. For the continuous spectrum the radial eigenfunctions were calculated.