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European Journal of Physics

A trigonometric aid for the Dirac equation in hydrogen-like atoms

W Gough

Published Jan 1, 2000• 0 citations
Publisher

Abstract

A simple diagram is presented for use with the analysis of the Dirac equation for an atom with a single electron. The complicated expression for the energy of the electron takes on a very simple form. The usual theory is presented with an angle from the diagram as the energy parameter, and the radial parts of the energy eigenfunctions for n = 1 and 2 are calculated.

DOI
10.1088/0143-0807/21/1/306

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will be added. One can explain the nature of the term by examining the relativistic case. Below the formulas for the radial and angular parts of the wave function were evaluated. The term for the energetic levels of the hydrogen atom for specific quantum numbers was given. We calculated the values of several specific radial functions for different quantum numbers, as well as the values of the radial function at the origin of coordinates and wide apart. A number of mean values for different powers of r was calculated. For the continuous spectrum the radial eigenfunctions were calculated.

•Apr 1, 2013•0 citations
A simple diagram is presented for use with the analysis of the Dirac equation for an atom with a single electron. The complicated expression for the energy of the electron takes on a very simple form. The usual theory is presented with an angle from the diagram as the energy parameter, and the radial parts of the energy eigenfunctions for n = 1 and 2 are calculated.
The fine structure of hydrogen energy was calculated by using the usual momentum-wavefunction relation directly, rather than establishing the well-known Dirac wave equation. As the results, the energy levels are completely the same as that of the Dirac wave equation, while the wavefunction is single component that is quite different from Dirac's four component wavefunction, the most important thing is that the present calculation brings out electronic spin in a new way which has never been reported and indicates that the electronic spin is a kind of orbital motion.
The system of electrons bound by a charged atom chain is studied within the Dirac theory. The general analytical solution of the Dirac equation is obtained. Analytical expression for electron energy is derived from which it follows that the principal quantum number can be introduced for electron states in such a system, similar to the case of a hydrogen atom. We prove that electrons bound by the atomic chain, are fully collectivized and the energy can be calculated as a function of the occupation number. The spectral band with the principal quantum number nnn is shown to be split into nnn subbands forming the fine structure. The scale of the fine structure splitting is calculated.
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 $m=\ifmmode\pm\else\textpm\fi{}\frac{1}{2},\ifmmode\pm\else\textpm\fi{}\frac{3}{2},\ensuremath{\cdots}\ifmmode\pm\else\textpm\fi{}j$
will be added. One can explain the nature of the term by examining the relativistic case. Below the formulas for the radial and angular parts of the wave function were evaluated. The term for the energetic levels of the hydrogen atom for specific quantum numbers was given. We calculated the values of several specific radial functions for different quantum numbers, as well as the values of the radial function at the origin of coordinates and wide apart. A number of mean values for different powers of r was calculated. For the continuous spectrum the radial eigenfunctions were calculated.