Visualization of the Exact Solution of Dirac Equation

Abstract

The exact solution of Dirac equation is schematically shown for the hydrogen atom. It is important to have images of the spinors to study the relativistic effects in molecules containing heavy metals such as lanthanoids. The s, p, d, and f spinors have characteristic shapes. The main purpose of this paper is to make it possible to view them all at once by presenting an illustrated book of their 3D density plots. We have also given the 3D density plots of the Schrödinger orbitals. One can easily observe that most of the nodes observed in the nonrelativistic orbitals vanish by comparing the spinors and the corresponding Schrödinger orbitals. We have calculated the mean value of r(< r > ) for Z = 100. It is well known that <r > is reduced by the relativistic effects, however, we have found that their effects are not uniform. Their effects are largest for spinors of j = 1/2, i.e., s_1/2 and p_1/2. With this paper, we present Mathematica notebooks for plotting and computation, which would be useful also as teaching materials.