Relativistic Quantum Mechanical Equation Research Articles

On the same origin of quantum physics and general relativity from Riemannian geometry and Planck scale formalism

It has been a long time to reconcile quantum physics and general relativity. To date, no globally accepted theory has been proposed to explain all physical observations. In this work, we reformulated the Riemannian geometry in terms of curvature and energy tensors using the Planck scale formalism. The proposed equation can be transformed into Dirac equations in electrodynamic and chromodynamic fields with a reduction in the background curvature. We redefined the mass and charge of leptons in terms of the interactions between the energy of the field and the curvature of the spacetime. The obtained equation is covariant in space–time and invariant with respect to any Planck scale. Therefore, the constants of the universe can be reduced to only two quantities: Planck length and Planck time. We proved that the Einstein field equation from general relativity is actually a relativistic quantum mechanical equation. We further modeled the universe using the equation with Einstein's lambda formalism and found that the universe dynamics could be considered as harmonic oscillators entangled with lambda curvature. This equation can be used to describe the energy transfer between two entangled spacetimes between the same universe and between any two universes (ER=EPR). The singularity of black holes can be avoided at the Planck scale, because space and time are no longer entangled. This equation predicts that information of light from the entangled universe can be transferred to our universe. The gravitational wave background was predicted, and its spectrum was close to that of the observation.

Simplifying the equations of relativistic quantum mechanics

Simplifying the equations of relativistic quantum mechanics

Dunkl–Klein–Gordon Equation in Three-Dimensions: The Klein–Gordon Oscillator and Coulomb Potential

Recent studies show that deformations in quantum mechanics are inevitable. In this contribution, we consider a relativistic quantum mechanical differential equation in the presence of Dunkl operator-based deformation and we investigate solutions for two important problems in three-dimensional spatial space. To this end, after introducing the Dunkl quantum mechanics, we examine the Dunkl-Klein-Gordon oscillator solutions with the Cartesian and spherical coordinates. In both coordinate systems, we find that the differential equations are separable and their eigenfunctions can be given in terms of the associate Laguerre and Jacobi polynomials. We observe how the Dunkl formalism is affecting the eigenvalues as well as the eigenfunctions. As a second problem, we examine the Dunkl-Klein-Gordon equation with the Coulomb potential. We obtain the eigenvalue, their corresponding eigenfunctions, and the Dunkl-fine structure terms.

The Cosmic Black Hole as a Solution of the Relativistic Quantum Mechanical DIRAC Equation

For many physicists, Albert Einstein’s Theory of General Relativity is the top of Physics. A whole new concept, based on a flexible Space-Time Continuum. A wonderful New and Original insight in the Origins of Space and Time. But nowadays physics requires more than a fundamental theory about Space and Time.
 The Mathematical foundation for a “Quantum Mechanical Model of the Black Hole” is based on a 10-dimensional Space-Time Continuum. This article has been written in projections of a 10-Dimensional Space-Time Continuum within an easier to understand 4-Dimensional Space-Time Continuum. For that reason, this theory will not start with “Einstein’s famous Field Equations”, but the start will be at a very fundamental concept in Physics. Isaac Newton’s 3rd law as a fundament in Classical- and Quantum Mechanics.
 To make the theory of the “Quantum Mechanical Model of the Black Hole” as much understandable as possible, this article starts with a short comprehension of the theory.

A New Form of Quantum Mechanical Wave Equation Unifying Quantum Mechanics and Electro-Magnetics

The Schr?dinger differential equation is what we usually solve for the microscopic particles in non-relativistic quantum mechanics. Niels Bohr suggested the power two of the (usually) complex answer shows the probability of the particle’s existence at a point of space. Also, the time dependence of Schrodinger wave equation is one whereas for light in electromagnetism is two. In this paper, we show a solution for both problems. We derive a Wave Equation for the energy of every system. This electromagnetic wave equation is shown to convert to those classical (i.e. the Schrodinger) and special relativistic (i.e. Klein-Gordon) quantum mechanical equations. Also, accordingly there definitely is a physical meaning to answer to this wave equation. And therefore, switching the probabilistic interpretation of quantum mechanics to a deterministic one as (Albert) Einstein demanded.

The 4-Dimensional Dirac Equation in Relativistic Field Theory

An important milestone in quantum physics has been reachedby the publication of the Relativistic Quantum MechanicalDirac Equation in 1928. However, the Dirac equationrepresents a 1-Dimensional quantum mechanical equationwhich is unable to describe the 4-Dimensional PhysicalReality.In this article the 4-Dimensional Relativistic QuantumMechanical Dirac Equation expressed in the vectorprobability functions  and the complex conjugated vectorprobability function  *will be published. To realize this, theclassical boundaries of physics has to be changed. It isnecessary to go back in time 300 years ago. More than 200years ago before the Dirac Equation had been published.A Return to the Inception of Physics. The time of IsaacNewton who published in 1687 in the “PhilosophiaeNaturalis Principia Mathematica” a Universal FundamentalPrinciple in Physics which was in Harmony with Science andReligion. The Universal Path, the Leitmotiv, the UniversalConcept in Physics. Newton found the concept of “UniversalEquilibrium” which he formulated in his famous thirdequation Action = - Reaction. This article presents a New Kind of Physics based on this Universal Fundamental Concept in Physics which results in a New Approach in Quantum Physics and General Relativity.The physical concept of quantum mechanical probability waves has been created during the famous 1927 5th Solvay Conference. During that period there were several circumstances which came together and made it possible to create an unique idea of material waves being complex (partly real and partly imaginary) and describing the probability of the appearance of a physical object (elementary particle). The idea of complex probability waves was new in the beginning of the 20th century. Since then the New Concept has been protected carefully within the Copenhagen Interpretation. When Schrödinger published his famous material wave equation in 1926, he found spherical and elliptical solutions for the presence of the electron within the atom. The first idea of the material waves in Schrödinger’s wave equation was the concept of confined Electromagnetic Waves. But according to Maxwell this was impossible. According to Maxwell’s equations Electromagnetic Waves can only propagate along straight lines and it is impossible that Light (Electromagnetic Waves) could confine with the surface of a sphere or an ellipse. For that reason these material waves in Schrödinger’s wave equation could only be of a different origin than Electromagnetic Waves. Niels Bohr introduced the concept of “Probability Waves” as the origin of the material waves in Schrödinger’s wave equation. And defined the New Concept that the electron was still a particle but the physical presence of the electron in the Atom was equally divided by a spherical probability function.In the New Theory it will be demonstrated that because of a mistake in the Maxwell Equations, in 1927 Confined Electromagnetic waves could not be considered to be the material waves expressed in Schrödinger's wave equation. The New Theory presents a new equation describing electromagnetic field configurations which are also solutions of the Schrodinger's wave equation and the relativistic quantum mechanical Dirac Equation and carry mass, electric charge and magnetic spin at discrete values.

A Return to the Inception of Physics

A Return to the Beginning (Inception|) of Physics. The time of Isaac Newton who published in 1687 in the “Philosophiae Naturalis Principia Mathematica” a Universal Fundamental Principle in Physics which was in Harmony with Science and Religion. The Universal Path, the Leitmotiv, the Universal Concept in Physics. Newton found the concept of “Universal Equilibrium” which he formulated in his famous third equation Action = - Reaction. This article presents a New Kind of Physics based on this Universal Fundamental Concept in Physics which results in a New Approach in Quantum Physics and General Relativity. The physical concept of quantum mechanical probability waves has been created during the famous 1927 5th Solvay Conference. During that period there were several circumstances which came together and made it possible to create an unique idea of material waves being complex (partly real and partly imaginary) and describing the probability of the appearance of a physical object (elementary particle). The idea of complex probability waves was new in the beginning of the 20th century. Since then the New Concept has been protected carefully within the Copenhagen Interpretation. When Schrödinger published his famous material wave equation in 1926, he found spherical and elliptical solutions for the presence of the electron within the atom. The first idea of the material waves in Schrödinger’s wave equation was the concept of confined Electromagnetic Waves. But according to Maxwell this was impossible. According to Maxwell’s equations Electromagnetic Waves can only propagate along straight lines and it is impossible that Light (Electromagnetic Waves) could confine with the surface of a sphere or an ellipse. For that reason these material waves in Schrödinger’s wave equation could only be of a different origin than Electromagnetic Waves. Niels Bohr introduced the concept of “Probability Waves” as the origin of the material waves in Schrödinger’s wave equation. And defined the New Concept that the electron was still a particle but the physical presence of the electron in the Atom was equally divided by a spherical probability function. In the New Theory it will be demonstrated that because of a mistake in the Maxwell Equations, in 1927 Confined Electromagnetic waves could not be considered to be the material waves expressed in Schrödinger’s wave equation. The New Theory presents a new equation describing electromagnetic field configurations which are also solutions of the Schrodinger’s wave equation and the relativistic quantum mechanical Dirac Equation and carry mass, electric charge and magnetic spin at discrete values

4-Dimensional Relativistic Quantum Mechanical Equilibrium in Gravitational-Electromagnetic Confinements

An important milestone in quantum physics was reached by the publication of the Relativistic Quantum Mechanical Dirac Equation in 1928. However, the Dirac equation represents a 1-Dimensional quantum mechanical equation which is unable to describe the 4-Dimensional Physical Reality. In this article, the 4-Dimensional Relativistic Quantum Mechanical Dirac Equation expressed in the vector probability functions and the complex conjugated vector probability function will be discussed. To realize this, the classical boundaries of physics has to be changed. It is necessary to go back in time more than 200 years ago before the Dirac Equation had been published. Isaac Newton who published in 1687 in the “Philosophiae Naturalis Principia Mathematica” a Universal Fundamental Principle in Physics was in Harmony with Science and Religion. Newton found the concept of “Universal Equilibrium” which he mentioned in his famous third equation, Action = Reaction. This article presents a New Kind of Physics based on this Universal Fundamental Concept in Physics which results in a New Approach in Quantum Physics and General Relativity. The physical concept of quantum mechanical probability waves has been originated during the famous 5th Solvay Conference in 1927. During that period there were several circumstances that came together and made it possible to create a unique idea of material waves being complex (partly real and partly imaginary) and describing the probability of the appearance of a physical object (elementary particle). The idea of complex probability waves was new in the beginning of the 20th century. Since then the New Concept has been protected carefully within the Copenhagen Interpretation. When Schrödinger published his well-known material wave equation in 1926, he found spherical and elliptical solutions for the presence of the electron within the atom. The first idea of the material waves in Schrödinger's wave equation was the concept of confined Electromagnetic Waves. But according to Maxwell, this was impossible. According to Maxwell's equations, Electromagnetic Waves can only propagate along straight lines and it is impossible that Light (Electromagnetic Waves) could confine with the surface of a sphere or an ellipse. For that reason, these material waves in Schrödinger's wave equation could only be of a different origin than Electromagnetic Waves. Niels Bohr introduced the concept of “Probability Waves” as the origin of the material waves in Schrödinger’s wave equation. And defined the New Concept that the electron was still a particle but the physical presence of the electron in the Atom was equally divided by a spherical probability function. In the New Theory, it will be demonstrated that because of a mistake in the Maxwell Equations, in 1927 Confined Electromagnetic waves could not be considered to be the material waves expressed in Schrödinger's wave equation. The New Theory presents a new equation describing electromagnetic field configurations which are also solutions of the Schrodinger's wave equation and the relativistic quantum mechanical Dirac Equation and carry mass, electric charge, and magnetic spin at discrete values.

The Illusion of Quantum Mechanical Probability Waves

An important milestone in quantum physics has been reached by the publication of the Relativistic Quantum Mechanical Dirac Equation in 1928. However, the Dirac equation represents a 1-Dimensional quantum mechanical equation which is unable to describe the 4-Dimensional Physical Reality.
 In this article the 4-Dimensional Relativistic Quantum Mechanical Dirac Equation expressed in the vector probability functions and the complex conjugated vector probability function will be published. To realize this, the classical boundaries of physics has to be changed. It is necessary to go back in time 300 years ago. More than 200 years ago before the Dirac Equation had been published.
 A Return to the Inception of Physics. The time of Isaac Newton who published in 1687 in the “Philosophiae Naturalis Principia Mathematica” a Universal Fundamental Principle in Physics which was in Harmony with Science and Religion. The Universal Path, the Leitmotiv, the Universal Concept in Physics. Newton found the concept of “Universal Equilibrium” which he formulated in his famous third equation Action = - Reaction. This article presents a New Kind of Physics based on this Universal Fundamental Concept in Physics which results in a New Approach in Quantum Physics and General Relativity.
 The physical concept of quantum mechanical probability waves has been created during the famous 1927 5th Solvay Conference. During that period there were several circumstances which came together and made it possible to create an unique idea of material waves being complex (partly real and partly imaginary) and describing the probability of the appearance of a physical object (elementary particle). The idea of complex probability waves was new in the beginning of the 20th century. Since then the New Concept has been protected carefully within the Copenhagen Interpretation. When Schrödinger published his famous material wave equation in 1926, he found spherical and elliptical solutions for the presence of the electron within the atom. The first idea of the material waves in Schrödinger’s wave equation was the concept of confined Electromagnetic Waves. But according to Maxwell this was impossible. According to Maxwell’s equations Electromagnetic Waves can only propagate along straight lines and it is impossible that Light (Electromagnetic Waves) could confine with the surface of a sphere or an ellipse. For that reason, these material waves in Schrödinger’s wave equation could only be of a different origin than Electromagnetic Waves. Niels Bohr introduced the concept of “Probability Waves” as the origin of the material waves in Schrödinger’s wave equation. And defined the New Concept that the electron was still a particle but the physical presence of the electron in the Atom was equally divided by a spherical probability function.
 In the New Theory it will be demonstrated that because of a mistake in the Maxwell Equations, in 1927 Confined Electromagnetic waves could not be considered to be the material waves expressed in Schrödinger's wave equation. The New Theory presents a new equation describing electromagnetic field configurations which are also solutions of the Schrodinger's wave equation and the relativistic quantum mechanical Dirac Equation and carry mass, electric charge and magnetic spin at discrete values.

The Illusion of Quantum Mechanical Probability Waves

An important milestone in quantum physics has been reached by the publication of the Relativistic Quantum Mechanical Dirac Equation in 1928. However, the Dirac equation represents a 1-Dimensional quantum mechanical equation which is unable to describe the 4-Dimensional Physical Reality.
 In this article the 4-Dimensional Relativistic Quantum Mechanical Dirac Equation expressed in the vector probability functions and the complex conjugated vector probability function will be published. To realize this, the classical boundaries of physics has to be changed. It is necessary to go back in time 300 years ago. More than 200 years ago before the Dirac Equation had been published.
 A Return to the Inception of Physics. The time of Isaac Newton who published in 1687 in the “Philosophiae Naturalis Principia Mathematica” a Universal Fundamental Principle in Physics which was in Harmony with Science and Religion. The Universal Path, the Leitmotiv, the Universal Concept in Physics. Newton found the concept of “Universal Equilibrium” which he formulated in his famous third equation Action = - Reaction. This article presents a New Kind of Physics based on this Universal Fundamental Concept in Physics which results in a New Approach in Quantum Physics and General Relativity.
 The physical concept of quantum mechanical probability waves has been created during the famous 1927 5th Solvay Conference. During that period there were several circumstances which came together and made it possible to create an unique idea of material waves being complex (partly real and partly imaginary) and describing the probability of the appearance of a physical object (elementary particle). The idea of complex probability waves was new in the beginning of the 20th century. Since then the New Concept has been protected carefully within the Copenhagen Interpretation. When Schrödinger published his famous material wave equation in 1926, he found spherical and elliptical solutions for the presence of the electron within the atom. The first idea of the material waves in Schrödinger’s wave equation was the concept of confined Electromagnetic Waves. But according to Maxwell this was impossible. According to Maxwell’s equations Electromagnetic Waves can only propagate along straight lines and it is impossible that Light (Electromagnetic Waves) could confine with the surface of a sphere or an ellipse. For that reason, these material waves in Schrödinger’s wave equation could only be of a different origin than Electromagnetic Waves. Niels Bohr introduced the concept of “Probability Waves” as the origin of the material waves in Schrödinger’s wave equation. And defined the New Concept that the electron was still a particle but the physical presence of the electron in the Atom was equally divided by a spherical probability function.
 In the New Theory it will be demonstrated that because of a mistake in the Maxwell Equations, in 1927 Confined Electromagnetic waves could not be considered to be the material waves expressed in Schrödinger's wave equation. The New Theory presents a new equation describing electromagnetic field configurations which are also solutions of the Schrodinger's wave equation and the relativistic quantum mechanical Dirac Equation and carry mass, electric charge and magnetic spin at discrete values.

Discreteness of space from GUP in strong gravitational fields

A large class of quantum theories of gravity show that the Heisenberg's uncertainty principle is modified to the “Generalised Uncertainty Principle” (GUP) near the Planckian scale. It has also been shown that the GUP induces perturbative corrections to all quantum mechanical Hamiltonians, even at low energies, and thereby introduces Planck scale corrections to the Schrödinger equation and to the relativistic quantum mechanical equations. Some of these corrections give rise to potentially measurable effects in the low-energy laboratory. Another prediction of these corrections is that a measured length must be quantized, as seen by studying the solutions of the GUP modified Schrödinger, Klein-Gordon, and Dirac equations in a one, two, and three dimensional box. This result was subsequently extended to spacetimes with weak gravitational fields. In this work, we further extend this length quantization to spacetimes with strong gravitational fields and show that this result continues to hold, thereby showing that it is robust.

Action-reaction in mass-charge quaternions

Dirac’s relativistic quantum mechanical equation has been rewritten in such a way that both the Dirac equation and the structure of the fermion can be derived from a novel mathematical arrangement of the four fundamental parameters of physics, space, time, mass and charge, using quaternions. Mass and charge are conserved quantities and their products are found in the inverse-square laws of Newton, F = Gm1m2 / r2 and Coulomb, kQ1Q2/r2. Peter Rowlands gives this product the name ‘interaction’ and interprets the product, a squaring, to mean that it is universal and nonlocal. This paper will investigate the relationship between zero-point energy, quaternion operations and mass-charge interactions in relation to Newton’s third law of motion.

A new matrix formulation of the Maxwell and Dirac equations

Presented in this paper is a new matrix formulation of both the classical electromagnetic Maxwell equations and the relativistic quantum mechanical Dirac equation. These new matrix representations will be referred to as the Maxwell spacetime matrix equation and the Dirac spacetime matrix equation. Both are Lorentz invariant. Key to these new matrix formulations is an 8-by-8 matrix operator referred to here as the spacetime matrix operator. As it turns out, the Dirac spacetime matrix equation is equivalent to four new vector equations, which are similar in form to the four Maxwell vector equations. These new equations will be referred to as the Dirac spacetime vector equations. This allows these new vector equations to be as readily solved as solving a set of Maxwell vector equations. Based on these two new matrix approaches, two computer programs, encoded using Matlab software, have been developed and tested for determining the reflection and transmission characteristics of multilayer optical thin-film structures and multilayer quantum well-and-barrier structures. A listing of these software programs may be found in supplemental material associated with this article. Numerical results obtained based on the use of these computer programs are presented in the results section of this article.

Relativistic quantum mechanical spin-1 wave equation in 2 + 1 dimensional spacetime

In this study, we introduce a relativistic quantum mechanical wave equation of the spin-1 particle as an excited state of the zitterbewegung and show that it is consistent with the 2 + 1 dimensional Proca theory. At the same time, we see that in the rest frame, this equation has two eigenstates; particle and antiparticle states or negative and positive energy eigenstates, respectively, and satisfy SO(2,1) spin algebra. As practical applications, we derive the exact solutions of the equation in the presence of a constant magnetic field and a curved spacetime background. From these solutions, we find Noether charge by integrating the zeroth component of the spin-1 particle current, called charge density, on hyper surface and discuss pair production from this charge. And, we see that the discussion on the Noether charge is a useful tool for understanding the pair production phenomenon because it is derived from a probabilistic particle current.

Matrix Continued Fraction Solution to the Relativistic Spin-0 Feshbach–Villars Equations

The Feshbach–Villars equations, like the Klein–Gordon equation, are relativistic quantum mechanical equations for spin-0 particles.We write the Feshbach–Villars equations into an integral equation form and solve them by applying the Coulomb–Sturmian potential separable expansion method. We consider boundstate problems in a Coulomb plus short range potential. The corresponding Feshbach–Villars CoulombGreen’s operator is represented by a matrix continued fraction.

The Light-Front Schrödinger Equation and the Determination of the Perturbative QCD Scale from Color Confinement: A First Approximation to QCD

The valence Fock-state wavefunctions of the light-front QCD Hamiltonian satisfy a relativistic equation of motion with an effective confining potential $U$ which systematically incorporates the effects of higher quark and gluon Fock states. If one requires that the effective action which underlies the QCD Lagrangian remains conformally invariant and extends the formalism of de Alfaro, Fubini and Furlan to light front Hamiltonian theory, the potential $U$ has a unique form of a harmonic oscillator potential, and a mass gap arises. The result is a nonperturbative relativistic light-front quantum mechanical wave equation which incorporates color confinement and other essential spectroscopic and dynamical features of hadron physics, including a massless pion for zero quark mass and linear Regge trajectories with the same slope in the radial quantum number $n$ and orbital angular momentum $L$. Only one mass parameter $\kappa$ appears. Light-front holography thus provides a precise relation between the bound-state amplitudes in the fifth dimension of AdS space and the boost-invariant light-front wavefunctions describing the internal structure of hadrons in physical space-time. We also show how the mass scale $\kappa$ underlying confinement and hadron masses determines the scale $\Lambda_{\overline{MS}} $ controlling the evolution of the perturbative QCD coupling. The relation between scales is obtained by matching the nonperturbative dynamics, as described by an effective conformal theory mapped to the light-front and its embedding in AdS space, to the perturbative QCD regime computed to four-loop order. The result is an effective coupling defined at all momenta. The predicted value $\Lambda_{\overline{MS}} = 0.328 \pm 0.034$ GeV is in agreement with the world average $0.339 \pm 0.010$ GeV. The analysis applies to any renormalization scheme.

Light-front holography and superconformal quantum mechanics: A new approach to hadron structure and color confinement

A primary question in hadron physics is how the mass scale for hadrons consisting of light quarks, such as the proton, emerges from the QCD Lagrangian even in the limit of zero quark mass. If one requires the effective action which underlies the QCD Lagrangian to remain conformally invariant and extends the formalism of de Alfaro, Fubini and Furlan to light-front Hamiltonian theory, then a unique, color-confining potential with a mass parameter [Formula: see text] emerges. The actual value of the parameter [Formula: see text] is not set by the model – only ratios of hadron masses and other hadronic mass scales are predicted. The result is a nonperturbative, relativistic light-front quantum mechanical wave equation, the Light-Front Schrödinger Equation which incorporates color confinement and other essential spectroscopic and dynamical features of hadron physics, including a massless pion for zero quark mass and linear Regge trajectories with the identical slope in the radial quantum number [Formula: see text] and orbital angular momentum [Formula: see text]. The same light-front equations for mesons with spin [Formula: see text] also can be derived from the holographic mapping to QCD (3+1) at fixed light-front time from the soft-wall model modification of AdS5 space with a specific dilaton profile. Light-front holography thus provides a precise relation between the bound-state amplitudes in the fifth dimension of AdS space and the boost-invariant light-front wavefunctions describing the internal structure of hadrons in physical space-time. One can also extend the analysis to baryons using superconformal algebra – [Formula: see text] supersymmetric representations of the conformal group. The resulting fermionic LF bound-state equations predict striking similarities between the meson and baryon spectra. In fact, the holographic QCD light-front Hamiltonians for the states on the meson and baryon trajectories are identical if one shifts the internal angular momenta of the meson ([Formula: see text]) and baryon ([Formula: see text]) by one unit: [Formula: see text]. We also show how the mass scale [Formula: see text] underlying confinement and the masses of light-quark hadrons determines the scale [Formula: see text] controlling the evolution of the perturbative QCD coupling. The relation between scales is obtained by matching the nonperturbative dynamics, as described by an effective conformal theory mapped to the light-front and its embedding in AdS space, to the perturbative QCD regime. The data for the effective coupling defined from the Bjorken sum rule [Formula: see text] are remarkably consistent with the Gaussian form predicted by LF holographic QCD. The result is an effective coupling defined at all momenta. The predicted value [Formula: see text] is in agreement with the world average [Formula: see text]. We thus can connect [Formula: see text] to hadron masses. The analysis applies to any renormalization scheme.

Sedeonic Equations of Massive Fields

Prior work on space-time sedeon algebra models relativistic quantum mechanical equation of motion with corresponding field equations, mediated by massive or massless spin-1 or spin-1/2 particles. In the massless spin-1 case, such exchange particles mediate fields in analogy to Maxwell’s equations in Lorentz gauge. This paper demonstrates fundamental aspects of massive field’s theory, such as gauge invariance, charge conservation, Poynting’s theorem, potential of a stationary scalar point source, plane wave solution, and interaction between point sources. We briefly discuss some aspects of sedeonic algebra and their potential physical applications.

κ state solutions for the fermionic massive spin-½ particles interacting with double ring-shaped Kratzer and oscillator potentials

In recent years, an extensive survey on various wave equations of relativistic quantum mechanics with different types of potential interactions has been a line of great interest. In this regime, special attention has been given to the Dirac equation because the spin-½ fermions represent the most frequent building blocks of the molecules and atoms. Motivated by the considerable interest in this equation and its relativistic symmetries (spin and pseudospin), in the presence of solvable potential model, we attempt to obtain the relativistic bound states solution of the Dirac equation with double ring-shaped Kratzer and oscillator potentials under the condition of spin and pseudospin symmetries. The solutions are reported for arbitrary quantum number in a compact form. The analytic bound state energy eigenvalues and the associated upper- and lower-spinor components of two Dirac particles have been found. Several typical numerical results of the relativistic eigenenergies have also been presented. We found that the existence or absence of the ring shaped potential has strong effects on the eigenstates of the Kratzer and oscillator particles, with a wide band spectrum except for the pseudospin-oscillator particles, where there exist a narrow band gap.

Quantum-mechanical description of Lense-Thirring effect for relativistic scalar particles

Exact expression for the Foldy-Wouthuysen Hamiltonian of scalar particles is used for a quantum-mechanical description of the relativistic Lense-Thirring effect. The exact evolution of the angular momentum operator in the Kerr field approximated by a spatially isotropic metric is found. The quantum-mechanical description of the full Lense-Thirring effect based on the Laplace-Runge-Lenz vector is given in the nonrelativistic and weak-field approximation. Relativistic quantum-mechanical equations for the velocity and acceleration operators are obtained. The equation for the acceleration defines the Coriolis-like and centrifugal-like accelerations and presents the quantum-mechanical description of the frame-dragging effect.