Relativistic Wave Equations Research Articles

Diffraction in time in the relativistic domain

The diffraction of matter waves in time is investigated by using the kinetic energy based relativistic wave equation for the quantum shutter problem. The differential equation is solved with the aid of the Fourier integral transform according to the spatial coordinate. A new Green’s function is obtained and a scattering integral is composed for the wave function. The boundary condition of the quantum shutter problem is expressed in the kernel of the integral. Two field components are derived by the asymptotic evaluation of the scattering integral. The behavior of the scattered matter waves is studied in the relativistic and non-relativistic domains numerically.

The Reduced Spinless Salpeter Wave Equation for Bound State Systems

In this work, the reduced spinless Salpeter equation is transformed into a relativistic second-order Schrodinger-like wave equation. The converted equation is so simple that one can solve it by using our proposed perturbation technique within the framework of the asymptotic iteration method. As an example, the Coulomb and Oscillator potential models are tested within this method. It is found that this method yields quite simple analytical solutions that will give rapidly the bound state energies with good accuracy compared with the available theoretical predictions. Furthermore, we predicted the relativistic energy eigenvalues of muonium, as well as, for the positronium bound states arising from the one-photon exchange in QED.

Relativistic time-dependent quantum dynamics across supercritical barriers for Klein-Gordon and Dirac particles

We investigate wave-packet dynamics across supercritical barriers for the Klein-Gordon and Dirac equations. Our treatment is based on a multiple scattering expansion (MSE). For spin-0 particles, the MSE diverges, rendering invalid the use of the usual connection formulas for the scattering basis functions. In a time-dependent formulation, the divergent character of the MSE naturally accounts for charge creation at the barrier boundaries. In the Dirac case, the MSE converges and no charge is created. We show that this time-dependent charge behavior dynamics can adequately explain the Klein paradox in a first-quantized setting. We further compare our semianalytical wave-packet approach to exact finite-difference solutions of the relativistic wave equations.

New solutions of the Dirac, Maxwell, and Weyl equations from the fractional Fourier transform

New solutions of relativistic wave equations are obtained in a unified manner from generating functions of spinorial variables. The choice of generating functions as Gaussians leads to representations in the form of generalized fractional Fourier transforms. Wave functions satisfying the Dirac, Maxwell, and Weyl equations are constructed by simple differentiations with respect to spinorial arguments. In the simplest case, one obtains Maxwell and Dirac hopfion solutions.

Мeson resonances in the relativistic quark model

In this paper, the relativistic quark model is developed for the study of mesons and resonances as quasi-bound quark states. A classic analogue of the spinless Salpeter equation is analyzed. It is shown that the potential for a conservative isolated two-particle system is the Lorentz-scalar function of the distance between quarks and can be included into the particle mass, which leads to the position-dependent quark mass. The funnel-type potential is modified with taking into account the dependence of the strong coupling αS on the distance. The concept of free motion of particles in a bound state is developed. The eigenvalue problem for the bound state is defined by the relativistic quasiclassical wave equation for the scalar potential. Two exact asymptotic solutions of the equation for the Coulomb and linear parts of the potential are obtained analytically; on this basis, the complex-mass formula for mesons and resonances is written. The efficiency of the model is demonstrated by comparison of the calculation results with the data for the masses of ρ and D mesons.

Horizons and the wave function of Planckian quantum black holes

At the Planck scale the distinction between elementary particles and black holes becomes fuzzy. The very definition of a “quantum black hole” (QBH) is an open issue. Starting from the idea that, at the Planck scale, the radius of the event horizon undergoes quantum oscillations, we introduce a black hole mass-radius Generalized Uncertainty Principle (GUP) and derive a corresponding gravitational wavelength. Next we recover a GUP encoding effective geometry. This semi-classical gravitational description admits black hole configurations only for masses higher than the Planck mass. Quantum corrections lead to a vanishing Hawking temperature when the Planck mass is approached from above. Finally we replace our semi-classical model by a relativistic wave equation for the “horizon wave function”. The solution admits a discrete mass spectrum which is bounded from below by a stable ground state with energy close to the Planck mass. Interestingly higher angular momentum states fit onto Regge trajectories indicating their stringy nature.

Analytical Bound State Solutions of the Klein-Fock-Gordon Equation for the Sum of Hulthén and Yukawa Potential within SUSY Quantum Mechanics

The relativistic wave equations determine the dynamics of quantum fields in the context of quantum field theory. One of the conventional tools for dealing with the relativistic bound state problem is the Klein-Fock-Gordon equation. In this work, using a developed scheme, we present how to surmount the centrifugal part and solve the modified Klein-Fock-Gordon equation for the linear combination of Hulthén and Yukawa potentials. In particular, we show that the relativistic energy eigenvalues and corresponding radial wave functions are obtained from supersymmetric quantum mechanics by applying the shape invariance concept. Here, both scalar potential conditions, which are whether equal and nonequal to vector potential, are considered in the calculation. The energy levels and corresponding normalized eigenfunctions are represented as a recursion relation regarding the Jacobi polynomials for arbitrary l states. Beyond that, a closed form of the normalization constant of the wave functions is found. Furthermore, we state that the energy eigenvalues are quite sensitive with potential parameters for the quantum states. The nonrelativistic and relativistic results obtained within SUSY QM overlap entirely with the results obtained by ordinary quantum mechanics, and it displays that the mathematical implementation of SUSY quantum mechanics is quite perfect.

A relativistic electron interacting with electromagnetic waves

A relativistic electron, in an electromagnetic wave is studied. The relativistic energy-momentum relation is taken into consideration for the derivation of a wave equation for spin-1/2 particle. The electromagnetic wave is represented by the electric scalar and magnetic vector potentials. Thus the spin information of the electron is integrated into the relativistic differential equation. A relativistic quantum wave equation is derived by using the derivatives with respect to time and spatial coordinates with respect to the kinetic energy and momentum of the electron. It is shown that the obtained relativistic equation directly reduces to the Pauli equation in the non-relativistic limit.

Частица со спином 1/2 и двумя массовыми параметрами во внешнем кулоновском поле

A generalized relativistic wave equation for a fermion with two mass parameters is studied in the presence of an external Coulomb field. After separating the variables, the problem is reduced to a system of eight differential equations of the first order. Taking into account the diagonalization of the spatial reflection operator, we derive two independent systems of four equations, referring to states of opposite parity. At a large distance from the center, they take the form of two subsystems for two ordinary Dirac particles in external Coulomb field, with masses M1 and M2 respectively. To simplify the problem, we perform transition to the nonrelativistic description of the system. In this way, we derive two systems of linked second order equations, referring to states with different parities, from which the corresponding fourth order differential equations for separate functions are found. Solutionsof the Frobenius type are constructed, they involve power series with 10-term recurrent relations. Two of the four solutions are appropriate for describing bound states. As a quantization rule, we apply the known transcendency condition, in this way we obtained two analytical formulas for energy spectra. They are similar to nonrelativistic spectra for ordinary spin 1/2 particle, but being governed by massesM1 and M2.

Quantum cellular automata and quantum field theory in two spatial dimensions

Quantum walks on lattices can give rise to one-particle relativistic wave equations in the long-wavelength limit. In going to multiple particles, quantum cellular automata (QCA) are natural generalizations of quantum walks. In one spatial dimension, the quantum walk can be "promoted" to a QCA that, in the long-wavelength limit, gives rise to the Dirac quantum field theory (QFT) for noninteracting fermions. This QCA/QFT correspondence has both theoretical and practical applications, but there are obstacles to similar constructions in two or more spatial dimensions. Here we show that a method of construction employing distinguishable particles confined to the completely antisymmetric subspace yields a QCA in two spatial dimensions that gives rise to the 2D Dirac QFT. Generalizing to 3D will entail some additional complications, but no conceptual barriers. We examine how this construction evades the "no go" results in earlier work.

Foundations of the Quaternion Quantum Mechanics.

We show that quaternion quantum mechanics has well-founded mathematical roots and can be derived from the model of the elastic continuum by French mathematician Augustin Cauchy, i.e., it can be regarded as representing the physical reality of elastic continuum. Starting from the Cauchy theory (classical balance equations for isotropic Cauchy-elastic material) and using the Hamilton quaternion algebra, we present a rigorous derivation of the quaternion form of the non- and relativistic wave equations. The family of the wave equations and the Poisson equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic continuum. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. The problem of the Schrödinger equation, where imaginary ‘i’ should emerge, is solved. This interpretation is a serious attempt to describe the ontology of quantum mechanics, and demonstrates that, besides Bohmian mechanics, the complete ontological interpretations of quantum theory exists. The model can be generalized and falsified. To ensure this theory to be true, we specified problems, allowing exposing its falsity.

Solutions of Klein–Gordon and Dirac Equations for Non-pure Dipole Potential in 2D Systems

We study the quantum relativistic wave equations (Klein–Gordon and Dirac) for the non-pure dipole potential $$V(r)=-Ze/r+D\cos \theta /r^{2}$$ , in the case of two-dimensional systems. We consider either spin symmetry or anti-spin symmetry cases in our computations. We give the analytical expressions of the eigenfunctions, compute the exact values of the energies and study their dependence according to the dipole moment D. Our study generalizes the energies of the Kratzer potential as well as the magnetic quantum number m, which is replaced with the Mathieu characteristic values obtained during the resolution of the angular equations. For each magnetic quantum number, we demonstrate the existence of a critical value for the dipole moment, beyond which the corresponding bound state can no longer exist. We find that the critical value is null when $$m=0$$ ; this means that these s-states cannot exist for this system and this is in agreement with non-relativistic studies.

Spin 3/2 particle: Puali – Fierz theory, non-relativistic approximation

The relativistic wave equation is well-known for a spin 3/2 particle proposed by W. E. Pauli and M. E. Fierz and based on the 16-component wave function with the transformation properties of the vector-bispinor. In this paper, we investigated the nonrelativistic approximation in this theory. Starting with the first-order equation formalism and representation of Pauli – Fierz equation in the Petras basis, also applying the method of generalized Kronecker symbols and elements of the complete matrix algebras, and decomposing the wave function into large and small nonrelativistic constituents with the help of projective operators, we have derived a Pauli-like equation for the 4-component wave function describing the non-relativistic particle with a 3/2 spin.

Quantum field theory from a quantum cellular automaton in one spatial dimension and a no-go theorem in higher dimensions

It has been shown that certain quantum walks give rise to relativistic wave equations, such as the Dirac and Weyl equations, in their long-wavelength limits. This intriguing result raises the question of whether something similar can happen in the multi-particle case. We construct a one-dimensional quantum cellular automaton (QCA) model which matches the quantum walk in the single particle case, and which approaches the quantum field theory of free fermions in the long-wavelength limit. However, we show that this class of constructions does not generalize to higher spatial dimensions in any straightforward way, and that no construction with similar properties is possible in two or more spatial dimensions. This rules out the most common approaches based on QCAs. We suggest possible methods to overcome this barrier while retaining locality.

Klein-Gordon Equation and Wave Function for Free Particle in Rindler Space-Time

Klein-Gordon equation is a relativistic wave equation. It treats spinless particle. The wave function cannot use as a probability amplitude. We made Klein-Gordon equation in Rindler space-time. In this paper, we make free particle’s wave function as the solution of Klein-Gordon equation in Rindler space-time.

Some Remarks About Virtual Annihilation Interactions in Relativistic n-Body Wave Equations in QED

The existence of entities consisting of electrons and positrons was predicted in 1946 by J. A. Wheeler and he called them polyelectrons. The simplest bound-state is called positronium (Ps, e−e+). Wheeler speculated that two Ps atoms may combine to form the di-positronium molecule (Ps2, e−e+e−e+) which was finally observed in 2007. He also conjectured the existence of larger systems such as (Ps3, e−e+e−e+e−e+). In a previous work the author has formulated the relativistic wave equations in quantum electrodynamics (QED) for a system consisting of n fermions and antifermions of equal masses where n can be any natural number and the equations contain relativistic effects up to the order of O(α4), α is the coupling. The kernels of the n-body wave equations in QED include one-photon exchange and virtual annihilation interactions. The latter interactions occur among pairs of fermions and antifermions. The equations have the Schrodinger non relativistic limit. In this manuscript, some points have been provided with respect to the virtual annihilation interactions for the few-body systems in addition to the particular cases of five- (e−e+e−e+e−) and six-body (e−e+e−e+e−e+) QED systems.

Theoretical investigation of the elastic scattering of by the ions of nitrogen isonuclear series

This work reports on differential, integrated and momentum transfer cross sections for the elastic scattering of electrons and positrons off the ions of nitrogen isonuclear series (N+ - N7+) over the energy range 1–500 eV. Calculations of these observable quantities involve the solution of Dirac relativistic wave equation employing a pure Coulomb potential and a short range potential. The latter comprises static, exchange, polarization and absorption potentials. Effects of these individual potentials on angular distribution of elastically scattered electrons are investigated throughout this ionic series. Differential cross sections for electron and positron scattering are compared with each other to demonstrate the difference of their collision dynamics. A satisfactory agreement between our predictions and other available data, experimental and theoretical, is observed.

Was Sommerfeld wrong? (To the history of the appearance of spin in relativistic wave equations)

This article presents a brief history of the appearance of electron spin in relativistic wave equations. Dirac derived his wave equation in 1928 with the intention to obtain an equation for the ‘simplest’ particle with spin zero. But, as Dirac later announced at the European Conference on Particle Physics (Budapest, 4–9 July 1977), it was a big surprise for him that the equation described the states of a particle with spin 1/2.

Analytical solutions of the Proca equation for modified Manning–Rosen potential with centrifugal term and direct coupling approaches by hypergeometric method

The Proca equation is one of the fundamental relativistic wave equations of massive vector (spin 1) boson fields. Proca equation is used to explain a relatively small particle that has a speed approaching the speed of light. The effect of photon imaginary mass to the quantum system that was coupled by noncentral potential consisting of radial, polar, and azimuthal modified Manning–Rosen potentials was investigated. The vector potential was coupled to the photon relativistic energy and the quadratic scalar potential was coupled directly to the quadratic photon mass. In the case of free space and when the vector potential was equal to scalar potential, therefore, the time-independent Proca equation reduced to a three-dimensional variable separable Schrodinger-like equation. By using variable separation method, the three-dimensional Schrodinger-like equation was resolved into three one-dimensional radial, polar, and azimuthal parts of Schrodinger-like equations. Each part of one-dimensional Schrodinger-like equation was solved using the hypergeometric differential equation through variables and wave function substitutions and resulted in the energy and wave function equations of massive photons. The relativistic energy equation of massive photons was a function of imaginary mass photon, the potential parameters, and the quantum numbers in radial, polar, and azimuthal variables. By using the Maxwell field approach, we got the correction factor for the energy function of this spin-1 particle.

Noncommutative Integration of the Klein–Gordon Equation in Electromagnetic Fields Admitting Functional Arbitrariness

A generalization of the noncommutative reduction of the relativistic wave equations in external electromagnetic fields admitting functional arbitrariness is proposed. Electromagnetic fields with functional arbitrariness, admitting noncommutative integration of the relativistic Klein–Gordon wave equation with respect to symmetry algebras containing the operator P0 + P3 are constructed.