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arXiv:hep-th/0605245 · International Journal of Theoretical Physics

Relativistic Wave Equations and Hydrogenic Atoms

Brian Albert Robson, Sylvia Hastuti Sutanto

Published Aug 1, 2001• 7 citations
arXivHTMLPDF
DOI
10.1023/a:1017557528265

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This paper discusses the relativistic Schrödinger differential wave equation for the hydrogen atom and hydrogen-like cations, which have measured ionization energies from hydrogen to the Cu 28+ hydrogen-like cation.The analytical solution to the relativistic Schrödinger differential wave equation correlates with experimental data if the angular momentum quantum number l >0.For quantum levels with l = 0 , the analytical solutions predict ionization energy values larger than that measured for the hydrogen atom and hydrogen-like cations.Also, beyond the element erbium, Z > 68 for l=0 quantum states, the analytical solution yields energy values that are complex numbers.This is similar for the Dirac relativistic differential wave equation which fails for nuclei that have an atomic number larger than 137.Using measured 1s ionization energy values of hydrogen-like cations, a numerical solution to the relativistic Schrödinger differential wave equation is possible since an analytical one is not when the measured ionization energy is different from the analytical result.In addition, since the analytical solutions to the radial part of the relativistic Schrödinger differential wave equation are complicated, normalized numerical solutions are less difficult to obtain using Microsoft Excel.Most differential equations do not have analytical solutions, and, therefore, only numerical solutions are possible utilizing boundary or initial conditions. Classical Schrödinger Differential Wave Equation and Wave-Particle Duality in Quantum Mechanics
In this paper, we look further into one of the relativistic wave equations we have introduced recently. Our relativistic wave equation is a PDE that is rooted in the relativistic energy Compton momentum relation, rather than the standard energy momentum relation. They are two sides of the same coin, but the standard momentum is just a derivative of the Compton momentum, so this simplifies things considerably. Here the main focus is to rewrite our relativistic PDE wave equation in spherical polar coordinates, then by some separation of variables we end up with three ordinary differential equations (ODEs) for which we present possible solutions, and we also provide a table where we compare our ODEs with the ODEs one gets from the Schro dinger equation [1]. This approach is used to describe hydrogen-like atoms. We encourage other researchers to check our calculations and the predictions from our solutions and see how they fit compared to observations from hydrogen-like atoms.
The author has previously derived an energy-momentum relationship applicable in a hydrogen atom. Since this relationship is taken as a departure point, there is a similarity with the Dirac’s relativistic wave equation, but an equation more profound than the Dirac equation is derived. When determining the coefficients and β of the Dirac equation, Dirac assumed that the equation satisfies the Klein-Gordon equation. The Klein-Gordon equation is an equation which quantizes Einstein's energy-momentum relationship. This paper derives an equation similar to the Klein-Gordon equation by quantizing the relationship between energy and momentum of the electron in a hydrogen atom. By looking to the Dirac equation, it is predicted that there is a relativistic wave equation which satisfies that equation, and its coefficients are determined. With the Dirac equation it is necessary to insert a term for potential energy into the equation when describing the state of the electron in a hydrogen atom. However, in this paper, a potential energy term is not introduced into the relativistic wave equation. Instead, potential energy is incorporated into the equation by changing the coefficient of the Dirac equation.