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 $\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}={\ensuremath{\psi}}_{I}{{\ensuremath{\psi}}_{I}}^{*}+{\ensuremath{\psi}}_{2}{{\ensuremath{\psi}}_{2}}^{*}+{\ensuremath{\psi}}_{3}{{\ensuremath{\psi}}_{3}}^{*}+{\ensuremath{\psi}}_{4}{{\ensuremath{\psi}}_{4}}^{*}$ is treated by considering, as on the Schrodinger theory, the independent angular and radial factors separately. Tables and graphs are given for $\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}$ as a function of $\ensuremath{\theta}$ alone. The graphs are compared with and shown to correspond closely to the space quantization of the classical electron orbits. Graphs representative of $\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}$ 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 $\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}$. 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 $m=\ifmmode\pm\else\textpm\fi{}\frac{1}{2},\ifmmode\pm\else\textpm\fi{}\frac{3}{2},\ensuremath{\cdots}\ifmmode\pm\else\textpm\fi{}j$