Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Scientific Reports

Exact quasi-relativistic wavefunctions of Hydrogen-like atoms

Luis Grave de Peralta

Published Sep 10, 2020• 18 citations
Publisher

Abstract

Exact solutions of a novel quasi-relativistic quantum mechanical wave equation are found for Hydrogen-like atoms. This includes both, an exact analytical expression for the energies of the bound states, and exact analytical expressions for the wavefunctions, which successfully describe quantum particles with mass and spin-0 up to energies comparable to the energy associated to the mass of the particle. These quasi-relativistic atomic orbitals may be used for improving ab-initio software packages dedicated to numerical simulations in physical-chemistry and atomic and solid-state physics.

DOI
10.1038/s41598-020-71505-w

Related papers

Quasi-Relativistic Description of Hydrogen-Like Atoms

Luis Grave de Peralta

Using a novel wave equation, which is Galileo invariant but can give precise results up to energies as high as mc2, exact quasi-relativistic quantum mechanical solutions are found for the Hydrogen atom. It is shown that the exact solutions of the Grave de Peralta equation include the relativistic correction to the non-relativistic kinetic energies calculated using the Schrödinger equation.

Journal of Modern Physics•Jan 1, 2020•13 citations

Did Schrödinger have other options?

Luis Grave de Peralta

Abstract Wave mechanics triumphed when Schrödinger published his now famous equation and showed how to describe hydrogen-like atoms using it. However, while looking for the right equation, Schrödinger first explored, but did not publish, the equation that we today call the Klein–Gordon equation. An alternative possible choice is explored in this work. It is shown a quasi-relativistic wave equation which solutions match the Schrödinger’s results at electron energies much smaller than the energy associated to the electron’s mass, but include, at higher energies, the relativity energy’s correction calculated in traditional first order perturbation theory. A discussion is presented about several consequences that would follow from using this quasi-relativistic wave equation as a quantum mechanics foundational equation. It is also suggested the academic use of this equation for introducing the students to the implications of the special theory of relativity in introductory quantum mechanics courses.

European Journal of Physics•Jul 21, 2020•18 citations

Analytical and Numerical Solutions of the Relativistic Schrödinger Differential Wave Equation

James McNeill, James E. Bidlack

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

Proceedings of the Oklahoma Academy of Science•Mar 6, 2026•0 citations

A Notable Quasi-Relativistic Wave Equation and Its Relation to the Schrödinger, Klein-Gordon, and Dirac Equations

Luis Grave de Peralta, Hira Farooq

An intriguing quasi-relativistic wave equation, which is useful between the range of applications of the Schrödinger and the Klein-Gordon equations, is discussed. This equation allows for a quantum description of a constant number of spin-0 particles moving at quasi-relativistic energies. It is shown how to obtain a Pauli-like version of this equation from the Dirac equation. This Pauli-like quasi-relativistic wave equation allows for a quantum description of a constant number of spin-1/2 particles moving at quasi-relativistic energies and interacting with an external electromagnetic field. In addition, it was found an excellent agreement between the energies of the electron in heavy Hydrogen-like atoms obtained using the Dirac equation, and the energies calculated using a perturbation approach based on the quasi-relativistic wave equation. Finally, it is argued that the notable quasi-relativistic wave equation discussed in this work provides interesting pedagogical opportunities for a fresh approach to the introduction to relativistic effects in introductory quantum mechanics courses.

Journal of Modern Physics•Jan 1, 2021•7 citations

Quasi-relativistic description of a quantum particle moving through one-dimensional piecewise constant potentials

Luis Grave de Peralta

Using a novel quasi-relativistic wave equation, which can give precise results up to energies ~mc2, exact quantum mechanical solutions are found which corresponds to a particle with mass moving through one-dimensional piecewise constant potentials. As expected, at low particle’s speeds, the found solutions coincide with the solutions of the same problems calculated using the Schrödinger equation; however, as it should be, both solutions have a significative difference at quasi-relativistic speeds. Then, it is argued that the quasi-relativistic wave equation provides a simpler description than a fully relativistic theory or the perturbation approach for a quantum particle moving at quasi-relativistic energies through piecewise constant potentials.

Results in Physics•Aug 18, 2020•13 citations

Discrete wave mechanics: The hydrogen atom

Frederick T. Wall

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.

Proceedings of the National Academy of Sciences•Aug 1, 1986•9 citations
Exact solutions of a novel quasi-relativistic quantum mechanical wave equation are found for Hydrogen-like atoms. This includes both, an exact analytical expression for the energies of the bound states, and exact analytical expressions for the wavefunctions, which successfully describe quantum particles with mass and spin-0 up to energies comparable to the energy associated to the mass of the particle. These quasi-relativistic atomic orbitals may be used for improving ab-initio software packages dedicated to numerical simulations in physical-chemistry and atomic and solid-state physics.
Abstract Wave mechanics triumphed when Schrödinger published his now famous equation and showed how to describe hydrogen-like atoms using it. However, while looking for the right equation, Schrödinger first explored, but did not publish, the equation that we today call the Klein–Gordon equation. An alternative possible choice is explored in this work. It is shown a quasi-relativistic wave equation which solutions match the Schrödinger’s results at electron energies much smaller than the energy associated to the electron’s mass, but include, at higher energies, the relativity energy’s correction calculated in traditional first order perturbation theory. A discussion is presented about several consequences that would follow from using this quasi-relativistic wave equation as a quantum mechanics foundational equation. It is also suggested the academic use of this equation for introducing the students to the implications of the special theory of relativity in introductory quantum mechanics courses.
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
An intriguing quasi-relativistic wave equation, which is useful between the range of applications of the Schrödinger and the Klein-Gordon equations, is discussed. This equation allows for a quantum description of a constant number of spin-0 particles moving at quasi-relativistic energies. It is shown how to obtain a Pauli-like version of this equation from the Dirac equation. This Pauli-like quasi-relativistic wave equation allows for a quantum description of a constant number of spin-1/2 particles moving at quasi-relativistic energies and interacting with an external electromagnetic field. In addition, it was found an excellent agreement between the energies of the electron in heavy Hydrogen-like atoms obtained using the Dirac equation, and the energies calculated using a perturbation approach based on the quasi-relativistic wave equation. Finally, it is argued that the notable quasi-relativistic wave equation discussed in this work provides interesting pedagogical opportunities for a fresh approach to the introduction to relativistic effects in introductory quantum mechanics courses.
Using a novel quasi-relativistic wave equation, which can give precise results up to energies ~mc2, exact quantum mechanical solutions are found which corresponds to a particle with mass moving through one-dimensional piecewise constant potentials. As expected, at low particle’s speeds, the found solutions coincide with the solutions of the same problems calculated using the Schrödinger equation; however, as it should be, both solutions have a significative difference at quasi-relativistic speeds. Then, it is argued that the quasi-relativistic wave equation provides a simpler description than a fully relativistic theory or the perturbation approach for a quantum particle moving at quasi-relativistic energies through piecewise constant potentials.
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.