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Journal of Condensed Matter Nuclear Science

Highly Relativistic Deep Electrons and the Dirac Equation

J.-L. Paillet, A. Meulenberg

Published Aug 1, 2020• 1 citation
Publisher

Abstract

After analyzing, in the literature, deep orbit results of relativistic quantum equations, we studied them in a semi-classical way, by looking for a local minimum of total energy of an electron near the nucleus, while respecting the Heisenberg Uncertainty Relation (HUR). Now, while using new information thanks to semi-classical computations, we come back to deep electrons as solutions of the Dirac equation, to solve several important and subtle outstanding issues, such as the continuity of derivatives of wave functions, a spectral problem about the energy levels associated with the wave functions to compute, as well as essential relativistic and energy parameters of the solutions. We thus obtain a better completeness of the solutions. Finally, we give some approaches on the probability of the presence of Electron Deep Orbit (EDO) states in H atom.

DOI
10.70923/001c.72561

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After analyzing, in the literature, deep orbit results of relativistic quantum equations, we studied them in a semi-classical way, by looking for a local minimum of total energy of an electron near the nucleus, while respecting the Heisenberg Uncertainty Relation (HUR). Now, while using new information thanks to semi-classical computations, we come back to deep electrons as solutions of the Dirac equation, to solve several important and subtle outstanding issues, such as the continuity of derivatives of wave functions, a spectral problem about the energy levels associated with the wave functions to compute, as well as essential relativistic and energy parameters of the solutions. We thus obtain a better completeness of the solutions. Finally, we give some approaches on the probability of the presence of Electron Deep Orbit (EDO) states in H atom.
The chapter covers some of the essential aspects of relativistic quantum mechanics. The Klein-Gordon equation and its solution in coulombic fields, how to interpret probability density and probability current derived from the Klein-Gordon equation, derivation of the Dirac equation and its covariant form, probability density and probability current for Dirac equation, positive and negative energy eigenvalues in the case of a free particle, constants of motion for relativistic treatment, presence of spin magnetic moment and spin-orbit interaction energy in the Dirac equation, and the solution of the Dirac equation for central potentials and for the hydrogen atom are presented with elegant mathematical treatment and physical interpretations. A modern and reader friendly approach is adopted for derivation of the Dirac equation, constants of motion, and for the solution of the Dirac equation for central potentials. Solved examples and unsolved exercises relating to fundamental concepts of relativistic quantum mechanics are added towards the end of chapter.
In this paper, it is presented a historical account of the formulation of the quantum relativistic wave equation of an electron – the Dirac equation, issues regarding its interpretation that arose from the very beginning, and the later formulation of this equation in relation to a quantized electron-positron field, which implies a new interpretation. The way in which solutions obtained under each interpretation of the equation relate to one another is also considered for the simple case of hydrogen-like problems.
The events observable at the atomic scale cannot be adequately explained by classical physics, hence a new conceptual framework for physics must be developed. "Quantum mechanics" is the name given to this new theory of the physical cosmos. The basic formula of relativistic quantum mechanics is the problem of Klein-Gordon. In relativistic quantum physics, the wave function is crucial since it contains all the data describing a quantum system, in relativity quantum physics, they are crucial. How to solve Klein Gordon's equation has been the topic of intense debate in recent years. To determine the system's spectrum using numerous potentials and methodologies. To obtain the Klein-Gordon issues solution Through determining the harmonic and perturbed Coulomb potentials' energies, The recent study employed an entirely distinct methodology. using the Semi-Inverse Variational Technique to aid in the process, we were able to identify the eigen energies for Harmonic- and Perturbed Coulomb Potentials for a variety of quantum numbers. A further benefit of the technique was that it enabled us to obtain the wave eigenfunctions for various potentials, demonstrating the viability of this approach. We are excited to deal with additional challenges in our upcoming works, the first being the use of nonlinear potentials to solve the Dirac equation due to the effectiveness of this approach.
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.