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Physical Review

Pictorial Representations of the Electron Cloud for Hydrogen-Like Atoms

Harvey E. White

Published Jun 1, 1931• 54 citations
Publisher

Abstract

It is well known that the solutions of the wave-equation for hydrogen-like atoms may be represented graphically by interpreting $\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}$ as a probability density. The probability density factors ${\ensuremath{\Phi}}_{m}{{\ensuremath{\Phi}}_{m}}^{*}\ifmmode\cdot\else\textperiodcentered\fi{}{[{\ensuremath{\Theta}}_{m,l}]}^{2}\ifmmode\cdot\else\textperiodcentered\fi{}{[{R}_{n,l}]}^{2}=\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}$ 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 $\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}$ 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 $m=0,\ifmmode\pm\else\textpm\fi{}1,\ifmmode\pm\else\textpm\fi{}2,\ifmmode\pm\else\textpm\fi{}3,\ensuremath{\cdots}$ are given for $1s$, $2p$, $3d$, $4f$, $2s$, $3p$, $4d$, $5f$, $3s$, $4p$, and $5d$ electrons, see Fig. 6.

DOI
10.1103/physrev.37.1416

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The probability density that an electron have certain momenta is given by the square of the absolute magnitude of a momentum eigenfunction ${\ensuremath{\Upsilon}}_{\mathrm{nlm}}(P, \ensuremath{\Theta}, \ensuremath{\Phi})$, in which $P$, $\ensuremath{\Theta}$, and $\ensuremath{\Phi}$ are spatial polar coordinates of the total momentum vector referred to the same axes as the coordinates $r$, $\ensuremath{\theta}$, and $\ensuremath{\varphi}$ of the electron. The following general expression for these functions for a hydrogen-like atom is obtained: ${\ensuremath{\Upsilon}}_{\mathrm{nlm}}(P, \ensuremath{\Theta}, \ensuremath{\Phi})=\left\{\frac{1}{{(2\ensuremath{\pi})}^{\frac{1}{2}}}{e}^{\ifmmode\pm\else\textpm\fi{}\mathrm{im}\ensuremath{\Phi}}\right\} \left\{{\left(\frac{(2l+1)(l\ensuremath{-}m)!}{2(l+m)!}\right)}^{\frac{1}{2}}{{P}_{l}}^{m}(cos\ensuremath{\Theta})\right\}$ $\left\{\frac{\ensuremath{\pi}{2}^{2l+4}l!}{{(\ensuremath{\gamma}h)}^{\frac{3}{2}}}{\left(\frac{n(n\ensuremath{-}l\ensuremath{-}1)!}{(n+l)!}\right)}^{\frac{1}{2}}\frac{{\ensuremath{\zeta}}^{l}}{{({\ensuremath{\zeta}}^{2}+1)}^{l+2}}{C}_{n\ensuremath{-}l\ensuremath{-}1}^{l+1}\left(\frac{{\ensuremath{\zeta}}^{2}\ensuremath{-}1}{{\ensuremath{\zeta}}^{2}+1}\right)\right\}$ in which $\ensuremath{\zeta}=(\frac{2\ensuremath{\pi}}{\ensuremath{\gamma}h})P$, with $\ensuremath{\gamma}=(\frac{4{\ensuremath{\pi}}^{2}\ensuremath{\mu}{e}^{2}Z}{n{h}^{2}})=(\frac{Z}{n{a}_{0}})$. The probability ${\ensuremath{\Xi}}_{\mathrm{nl}}(P)\mathrm{dP}$ that the electron have a total momentum lying within the limits $P$ and $P+dP$ is also evaluated, and it is shown that the root mean square of the total momentum is equal to the momentum of the electron in a circular Bohr orbit with the same total quantum number.

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

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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 $m=\ifmmode\pm\else\textpm\fi{}\frac{1}{2},\ifmmode\pm\else\textpm\fi{}\frac{3}{2},\ensuremath{\cdots}\ifmmode\pm\else\textpm\fi{}j$

Physical Review•Aug 1, 1931•34 citations
It is well known that the solutions of the wave-equation for hydrogen-like atoms may be represented graphically by interpreting \ensuremathΨ\ensuremathΨ∗\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}\ensuremathΨ\ensuremathΨ∗ as a probability density. The probability density factors \ensuremathΦm\ensuremathΦm∗\ifmmode⋅\else\textperiodcentered\fi[\ensuremathΘm,l]2\ifmmode⋅\else\textperiodcentered\fi[Rn,l]2=\ensuremathΨ\ensuremathΨ∗{\ensuremath{\Phi}}_{m}{{\ensuremath{\Phi}}_{m}}^{*}\ifmmode\cdot\else\textperiodcentered\fi{}{[{\ensuremath{\Theta}}_{m,l}]}^{2}\ifmmode\cdot\else\textperiodcentered\fi{}{[{R}_{n,l}]}^{2}=\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}\ensuremathΦm​\ensuremathΦm​∗\ifmmode⋅\else\textperiodcentered\fi[\ensuremathΘm,l​]2\ifmmode⋅\else\textperiodcentered\fi[Rn,l​]2=\ensuremathΨ\ensuremathΨ∗ are represented graphically and briefly discussed and compared with the electron orbits of four classical models. Graphs for sss, ppp, ddd, fff, ggg, and hhh electrons are given. An attempt to combine the probability density factors and form some graphical representation of \ensuremathΨ\ensuremathΨ∗\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}\ensuremathΨ\ensuremathΨ∗ 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,\ifmmode±\else\textpm\fi1,\ifmmode±\else\textpm\fi2,\ifmmode±\else\textpm\fi3,\ensuremath⋯m=0,\ifmmode\pm\else\textpm\fi{}1,\ifmmode\pm\else\textpm\fi{}2,\ifmmode\pm\else\textpm\fi{}3,\ensuremath{\cdots}m=0,\ifmmode±\else\textpm\fi1,\ifmmode±\else\textpm\fi2,\ifmmode±\else\textpm\fi3,\ensuremath⋯ are given for 1s1s1s, 2p2p2p, 3d3d3d, 4f4f4f, 2s2s2s, 3p3p3p, 4d4d4d, 5f5f5f, 3s3s3s, 4p4p4p, and 5d5d5d electrons, see Fig. 6.
The mere existence of stable atoms, and their arrangement into many-atom nanoscale structures, can only be explained by the laws of quantum mechanics. Indeed, the accumulated evidence regarding the internal structure of atoms motivated the formulation of quantum mechanics at the early twentieth century. Here we focus on single-electron (hydrogen-like) atoms. The dominance of the Coulomb central potential between the electron and nucleus, translates into a simple Schrödinger equation for their relative position distribution. Solving rigorously this equation reveals quantized energy levels, which explains the emission spectra of hydrogen-like atoms (Rydberg formula). The corresponding stationary probability density distributions for the electron around the nucleus are reminiscent of the Bohr model, where classical trajectories (orbits) are replaced by single-electron wave functions (orbitals). The degeneracies of different orbitals, associated with three quantum numbers, will later turn out to be critical for understanding many-electron atoms and the periodic table of the elements.
The electronic transition energies from the electron clouds are the same as predicted by the Bohr model. The hydrogen atom, most probable radius of the first quantum mechanical electron cloud is identical to the radius of the first Bohr orbit. The quantum mechanical description goes much further, however. Not only does it predict a series of increasingly larger spherical electron clouds, but other electron cloud series as well. Because of their three dimensional shapes, these electron clouds sometimes are said to describe space states. The analogy to planetary orbits ends when we grasp the fact that quantum mechanical electron clouds are three-dimensional, where a Bohr orbit would be a circle lying entirely in a plane. Quantum mechanics also predicts the ionization energy of the hydrogen atom.
We introduce an atomic model that avoids some of the worst dissonances. According to this model, the electron is not point-like but extended. The square of the wave function is interpreted as the density of a fluid, the electronium. On the basis of the electronium model suggestive pictures and animations of the atom can be generated. We shall see and discuss pictures of the various states of a hydrogen atom as well as animations of transitions from one stationary state to another. We also shall represent the internal movement of the electronium, that is responsible of the angular momentum and the magnetic moment of the atom. From the pictures we shall directly read properties of the various states and the various transitions by only using arguments of classical physics.
A point-like picture of the Schrödinger solution for hydrogen atom is worked to emphasize that "point-like particles" may describe as "probability wave function". In each case, the three-dimensional shape of the |Ψnlm(rn, cos θ)|2 is plotted and the paths of the point-like electron (it is better to say reduced mass of the pair particles) are described in each closed shell. Finally, the orbital shape of the molecules are given according to the present simple model. In our opinion, "interpretations of the Correspondence Principle", which is a basic principle in all elementary quantum text, seems to be reviewed again!
The probability density that an electron have certain momenta is given by the square of the absolute magnitude of a momentum eigenfunction ${\ensuremath{\Upsilon}}_{\mathrm{nlm}}(P, \ensuremath{\Theta}, \ensuremath{\Phi})$, in which PPP, $\ensuremath{\Theta}$, and $\ensuremath{\Phi}$ are spatial polar coordinates of the total momentum vector referred to the same axes as the coordinates rrr, $\ensuremath{\theta}$, and $\ensuremath{\varphi}$ of the electron. The following general expression for these functions for a hydrogen-like atom is obtained: ${\ensuremath{\Upsilon}}_{\mathrm{nlm}}(P, \ensuremath{\Theta}, \ensuremath{\Phi})=\left\{\frac{1}{{(2\ensuremath{\pi})}^{\frac{1}{2}}}{e}^{\ifmmode\pm\else\textpm\fi{}\mathrm{im}\ensuremath{\Phi}}\right\} \left\{{\left(\frac{(2l+1)(l\ensuremath{-}m)!}{2(l+m)!}\right)}^{\frac{1}{2}}{{P}_{l}}^{m}(cos\ensuremath{\Theta})\right\}$ $\left\{\frac{\ensuremath{\pi}{2}^{2l+4}l!}{{(\ensuremath{\gamma}h)}^{\frac{3}{2}}}{\left(\frac{n(n\ensuremath{-}l\ensuremath{-}1)!}{(n+l)!}\right)}^{\frac{1}{2}}\frac{{\ensuremath{\zeta}}^{l}}{{({\ensuremath{\zeta}}^{2}+1)}^{l+2}}{C}_{n\ensuremath{-}l\ensuremath{-}1}^{l+1}\left(\frac{{\ensuremath{\zeta}}^{2}\ensuremath{-}1}{{\ensuremath{\zeta}}^{2}+1}\right)\right\}$ in which $\ensuremath{\zeta}=(\frac{2\ensuremath{\pi}}{\ensuremath{\gamma}h})P$, with $\ensuremath{\gamma}=(\frac{4{\ensuremath{\pi}}^{2}\ensuremath{\mu}{e}^{2}Z}{n{h}^{2}})=(\frac{Z}{n{a}_{0}})$. The probability ${\ensuremath{\Xi}}_{\mathrm{nl}}(P)\mathrm{dP}$ that the electron have a total momentum lying within the limits PPP and P+dPP+dPP+dP is also evaluated, and it is shown that the root mean square of the total momentum is equal to the momentum of the electron in a circular Bohr orbit with the same total quantum number.
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 rrr 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 12S121^{2}S_{\frac{1}{2}}12S21​​, 22P12,322^{2}P_{\frac{1}{2}, \frac{3}{2}}22P21​,23​​, 32D32,523^{2}D_{\frac{3}{2}, \frac{5}{2}}32D23​,25​​ 42F52,724^{2}F_{\frac{5}{2}, \frac{7}{2}}42F25​,27​​, and 52G725^{2}G_{\frac{7}{2}}52G27​​, with $m=\ifmmode\pm\else\textpm\fi{}\frac{1}{2},\ifmmode\pm\else\textpm\fi{}\frac{3}{2},\ensuremath{\cdots}\ifmmode\pm\else\textpm\fi{}j$