Pictorial Representations of the Dirac Electron Cloud for Hydrogen-Like Atoms

Abstract

Previous attempts to represent graphically the Schrodinger electron cloud for hydrogen-like atoms have been so successful that similar graphical representations have been looked for on the Dirac theory. With a single electron specified on Dirac's theory by a set of four wave functions instead of one, the probability density $\Psi {\Psi }^{*}={\psi }_I{{\psi }_I}^{*}+{\psi }_2{{\psi }_2}^{*}+{\psi }_3{{\psi }_3}^{*}+{\psi }_4{{\psi }_4}^{*}$ is treated by considering, as on the Schrodinger theory, the independent angular and radial factors separately. Tables and graphs are given for $\Psi {\Psi }^{*}$ as a function of $\theta$ alone. The graphs are compared with and shown to correspond closely to the space quantization of the classical electron orbits. Graphs representative of $\Psi {\Psi }^{*}$ as a function of $r$ alone are given and compared with the electronnucleus distances of the corresponding electron orbits. The radial and angular factors are brought together by means of a mechanical device described previously, which when photographed represents closely the electron-cloud $\Psi {\Psi }^{*}$. Photographs are given for the states $1{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$, $2{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}, \frac{3}{2}}^{}{}_{}{}^{}$, $3{}_{}{}^2{}_{}{}^{}D_{\frac{3}{2}, \frac{5}{2}}^{}{}_{}{}^{}$ $4{}_{}{}^2{}_{}{}^{}F_{\frac{5}{2}, \frac{7}{2}}^{}{}_{}{}^{}$, and $5{}_{}{}^2{}_{}{}^{}G_{\frac{7}{2}}^{}{}_{}{}^{}$, with $m=\pm \frac{1}{2},\pm \frac{3}{2},\cdots \pm j$