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Physical Review

Stereophotographic Models of Electron Motion in Stark Effect

Robert Williams Wood

Published Jul 15, 1931• 5 citations
Publisher

Abstract

By means of multiple exposures with a stereoscopic camera of a rotating wire frame representing the curved quadrangle formed by the intersection of the parabolic coordinates of the Epstein-Schwarzchild treatment of the Stark effect, within which quadrangle the electron is progressively advanced along a properly curved wire, the orbital motion within the anchor-ring is very clearly brought out. The pictures when viewed with a stereoscope show the anchor-ring as a wire cage within which the precessing orbit appears as a white dotted line. The change in orientation and eccentricity of the orbit, with the constancy of the distance of the electric center from the $\mathrm{YZ}$ plane (field parallel to $X$) is easily seen. Photographs are shown illustrating various assignments of the quantum numbers ${n}_{1}$ and ${n}_{2}$.

DOI
10.1103/physrev.38.346

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By means of multiple exposures with a stereoscopic camera of a rotating wire frame representing the curved quadrangle formed by the intersection of the parabolic coordinates of the Epstein-Schwarzchild treatment of the Stark effect, within which quadrangle the electron is progressively advanced along a properly curved wire, the orbital motion within the anchor-ring is very clearly brought out. The pictures when viewed with a stereoscope show the anchor-ring as a wire cage within which the precessing orbit appears as a white dotted line. The change in orientation and eccentricity of the orbit, with the constancy of the distance of the electric center from the YZ\mathrm{YZ}YZ plane (field parallel to XXX) is easily seen. Photographs are shown illustrating various assignments of the quantum numbers n1{n}_{1}n1​ and n2{n}_{2}n2​.
A model which I wish to show you was constructed to illustrate the motion of the electrons around the nucleus of the Rutherford atom, and the changing dimensions and varying plane of their orbit. More violent disturbance produces a series of secondary waves in each orbit, which divide the orbit in the manner suggested by the wave and quantum theories. The model is a very simple one, and my only excuse for bringing it to your notice is the pleasure which it has given to myself, and the fact that it has enabled me to realise as I could not previously do, Rutherford's theory of atomic structure. Rutherford's atom, which at present holds the field, consists essentially of a nucleus around which rotate a number of electrons. This number is the atomic number of the atom, so that with hydrogen there is one circulating electron, with helium there are two, and the number of electrons steadily rises with the atomic number until we reach uranium, which has ninety-two circulating electrons. Each of these electrons rotates independently, and its orbit is subject to very great variation. Indeed the motion of the electron is so complex that it is exceedingly difficult to form any concrete idea of what a variable orbit really means, or what perturbations in the orbit signify. It is with the idea of assisting the imagination that I have constructed this model, consisting of two prisms so placed that a horizontal beam of light is reflected downwards by the first prism on to the second, by which it is again reflected horizontally.
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 $m=\ifmmode\pm\else\textpm\fi{}\frac{1}{2},\ifmmode\pm\else\textpm\fi{}\frac{3}{2},\ensuremath{\cdots}\ifmmode\pm\else\textpm\fi{}j$
The linear Stark effect is investigated, using the old quantum theory and the equations of motion of the electron in orbital variables. In contrast to the well-known solution in terms of parabolic coordinates, the new form has a simple geometric interpretation, namely, perturbation by the uniform electric field leads in the first approximation to a slow regular precession of the electron orbit around the direction of the field (subject to a single additional condition). This geometric picture outwardly resembles electron orbits precessing around a uniform magnetic field in the normal Zeeman effect.
L'analyse des spectres de photoionisation d'atomes à un électron optique en présence d'un champ électrique peut être faite en partant du modèle hydrogénoïde. Le problème de l'effet Stark de l'hydrogène peut être résolu de façon exacte, l'hamiltonien correspondant étant séparable en coordonnées paraboliques. On définit des densités partielles d'états et on montre qu'au-dessus de la limite classique d'ionisation par champ le spectre de l'hydrogène comporte des états quasi-discrets superposés à des états continus, les différents états n'étant pas couplés. Deux expériences concernant les spectres de photoionisation Stark du rubidium montrent l'importance des perturbations liées à l'interaction spin-orbite, comme par exemple la stabilisation d'un état Stark par le champ électrique.
The path integral evaluation of the hydrogen atom by Duru and Kleinert is studied by recognizing it as a special case of the general treatment of the separable Hamiltonian of Liouville-type. The basic dynamical principle involved is identified as the Jacobi’s principle of least action for given energy which is reparametrization invariant, and thus the appearance of a gauge freedom is naturally understood. Unlike the conventional Feynman path integral,which deals with a space-time picture of particle motion, the path integral on the basis of the Jacobi’s principle sums over orbits in space. We illustrate these properties by evaluating an exact path integral of the hydrogen atom in parabolic coordinates. In 3-dimensions the parabolic coordinates do not quite lead to a Liouville-type Hamiltonian, but we can devise asimple trick which converts it to a Liouville-type Hamiltonian and thus avoiding the use of the Kustaanheimo-Stiefel transformation. In the present formulation, the Hamiltonian for Stark effect is converted to the one for anharmonic oscillators with an unstable quartic coupling. We also study the hydrogen atom path integral from a view point of one-dimensional quantum gravity coupled to matter fields representing the electron coordinates. A simple BRST analysis of the problem is presented. 1 1
This manuscript explores the fundamental mechanisms of electron positioning through an integrated framework of atomic structure, wave mechanics, and optical physics. Rejecting rigid classical planetary trajectories, the work models electron orbitals as three-dimensional standing waveforms regulated by constructive interference and quantized nodes. It details how discrete changes in electron positions dictate photon absorption, emission, and macro-scale chemical bonding. Serving as a conceptual baseline, this text establishes the foundational structural pillars for an upcoming, broader unified theory of material and optical behaviors.