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Proceedings of the Oklahoma Academy of Science

Analytical and Numerical Solutions of the Relativistic Schrödinger Differential Wave Equation

James McNeill, James E. Bidlack

Published Mar 6, 2026• 0 citations

Abstract

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

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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 investigation a simple method developed by introducing spin to Schrodinger equation to study the relativistic hydrogen atom. By separating Schrodinger equation to radial and angular parts, we modify these parts to the associated Laguerre and Jacobi differential equations, respectively. Bound state Energy levels and wave functions of relativistic Schrodinger equation for Hydrogen atom have been obtained. Calculated results well matched to the results of Dirac’s relativistic theory. Finally the factorization method and supersymmetry approaches in quantum mechanics, give us some first order raising and lowering operators, which help us to obtain all quantum states and energy levels for different values of the quantum numbers n and m.

In this work the Schrödinger equation of the hydrogen-like atom is analytically solved. Three sets of analytical solution are obtained if the factor r−l is not neglected. The first solution is the same as the traditional radial wave function; another one diverges; the last one is far different from the traditional solution. On the consideration of the finite size of the nucleus, the third wave function does not diverge while r approaches to zero. Its radial wave function has below characteristics: (1) the angular-momentum quantum number l must be greater than the principal quantum number n; (2) l must not be 0 or 1; (3) the electron-cloud distribution differs from the traditional one; (4) the electron is closer to the nucleus by comparison with that in traditional results. On the other hand, the validity of solutions needs to be verified experimentally. 1.

The quantum mechanical problem of the hydrogen atom is treated by use of a finite difference equation in place of Schrödinger's differential equation. The exact solution leads to a wave vector energy expression that is readily converted to the Bohr-Rydberg formula. (The calculations here reported are limited to spherically symmetric states.) The wave vectors reduce to the familiar solutions of Schrödinger's equation as c --> infinity. The internal consistency and limiting behavior provide support for the view that the equations employed could well constitute an approach to a relativistic formulation of wave mechanics.

In 1928, the brilliance of Paul Dirac granted him the insight to develop a particle equation that was fully consistent with both the principles of quantum mechanics and the theory of special relativity. In this bachelor thesis we have studied the fundamentals of relativistic quantum mechanics including the derivation of the Dirac equation in a similar fashion to the way originally done by Paul Dirac. Additionally we have studied the analytical solution for hydrogen-like atoms. Furthermore we have numerically analysed the energy spectrum, radial characteristics and correctional terms using programs written in Python and MATLAB. Finally we have investigated the implications and applications ofthe Dirac Hamiltonian and its use to effectively describe the attributes of gold atoms.We find that relativistic contribution scales approximately by Z2 for large Z. In addition, we conclude that the Dirac Hamiltonian is of vital importance for an accurate description of chemical and physical properties of large atoms. Furthermore, we have completed calculations of the radial solution for the Coulomb potential and also for a free particle.

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 lightest and simplest arrangement of protons and electrons is a hydrogen atom. One of the wave characteristics of an atom can be known using the Schrodinger equation. Schrodinger's equations of spherical coordinates consist of radial equations and angular equations. Quantum numbers 4≤ n ≤ 5from the Schrodinger equation sphere coordinates produce different angular wave functions. To find out the angular wave equation using normalization conditions . Based on the normalization conditions for n = 4 a wave function is produced , while for n = 5 a wave function is .