Pictorial Representations of the Electron Cloud for Hydrogen-Like Atoms

Abstract

It is well known that the solutions of the wave-equation for hydrogen-like atoms may be represented graphically by interpreting $\Psi {\Psi }^{*}$ as a probability density. The probability density factors ${\Phi }_m{{\Phi }_m}^{*}·{\left[{\Theta }_{m,l}\right]}^2·{\left[R_{n,l}\right]}^2=\Psi {\Psi }^{*}$ are represented graphically and briefly discussed and compared with the electron orbits of four classical models. Graphs for $s$, $p$, $d$, $f$, $g$, and $h$ electrons are given. An attempt to combine the probability density factors and form some graphical representation of $\Psi {\Psi }^{*}$ has resulted in the construction of a mechanical device or model, see Fig. 5, which when photographed, gives very closely the desired result. Photographs for the magnetic states $m=0,\pm 1,\pm 2,\pm 3,\cdots$ are given for $1s$, $2p$, $3d$, $4f$, $2s$, $3p$, $4d$, $5f$, $3s$, $4p$, and $5d$ electrons, see Fig. 6.