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Scientific Bulletin

ANALYTICAL SOLUTION OF THE KLEIN-FOCK-GORDON EQUATION FOR A FIELD WITH HÜLTEN POTENTIAL IN ORDINARY RELATIVISTIC QUANTUM MECHANICS

Aghaali Ali AHMADOV

Published Apr 1, 2026• 0 citations
Publisher

Abstract

In this article, the analytical solution of the Klein-Fock-Gordon equation in the Hülten potential domain is investigated within the framework of ordinary quantum mechanics. The Klein-FockGordon equation, which describes the dynamics of relativistic scalar particles, is of particular interest in the case of interactions with short-range potentials of the central type. The Hülten potential, on the other hand, has an exponentially decreasing form and is widely used in modeling atomic, molecular, and nuclear systems due to its approximation to the short-range Coulomb potential and its rapid decay at large distances. In this work, the radial Klein-Fock-Gordon equation is separated in spherical coordinates and the differential equation for the Hülten potential is reduced to a hypergeometric type equation. By applying the appropriate boundary conditions, the quantization condition for the energy spectrum is obtained and the analytical expression of the energy levels for the bound states is derived. The normalized form of the wave functions is expressed by means of special functions. Analysis of the obtained results shows that the analytical solution of the Klein-Fock-Gordon equation in the Hülten potential leads to a clear manifestation of relativistic effects in the energy spectrum of the system and the structure of the wave functions. The energy spectrum depends not only on the parameters and quantum numbers of the potential, but also on relativistic factors introduced through the speed of light c. A comparison of the wave functions shows that the radial solutions obtained by the Klein-Fock-Gordon equation differ somewhat from those obtained by the Schrödinger equation. The shape and maximum of the radial wave function, which characterizes the probability of a particle being at a certain distance from the center, may change slightly compared to the non-relativistic case. These changes are especially noticeable in regions where the potential is deep and the interaction is stronger. Thus, the analytical solution of the Klein-Fock-Gordon equation in the Hülten potential field both clarifies the nature of relativistic effects and demonstrates agreement with the results of nonrelativistic quantum mechanics in terms of theoretical consistency and accuracy of the limit transition. This indicates that the model is a suitable theoretical framework for a more fundamental and accurate description of short-range interactions in molecular systems. Keywords: Klein-Fock-Gordon equation, Hülten potential, analytical solutions, energy spectrum, relativistic quantum mechanics.

DOI
10.54414/jliz6803

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In this article, the analytical solution of the Klein-Fock-Gordon equation in the Hülten potential domain is investigated within the framework of ordinary quantum mechanics. The Klein-FockGordon equation, which describes the dynamics of relativistic scalar particles, is of particular interest in the case of interactions with short-range potentials of the central type. The Hülten potential, on the other hand, has an exponentially decreasing form and is widely used in modeling atomic, molecular, and nuclear systems due to its approximation to the short-range Coulomb potential and its rapid decay at large distances. In this work, the radial Klein-Fock-Gordon equation is separated in spherical coordinates and the differential equation for the Hülten potential is reduced to a hypergeometric type equation. By applying the appropriate boundary conditions, the quantization condition for the energy spectrum is obtained and the analytical expression of the energy levels for the bound states is derived. The normalized form of the wave functions is expressed by means of special functions. Analysis of the obtained results shows that the analytical solution of the Klein-Fock-Gordon equation in the Hülten potential leads to a clear manifestation of relativistic effects in the energy spectrum of the system and the structure of the wave functions. The energy spectrum depends not only on the parameters and quantum numbers of the potential, but also on relativistic factors introduced through the speed of light c. A comparison of the wave functions shows that the radial solutions obtained by the Klein-Fock-Gordon equation differ somewhat from those obtained by the Schrödinger equation. The shape and maximum of the radial wave function, which characterizes the probability of a particle being at a certain distance from the center, may change slightly compared to the non-relativistic case. These changes are especially noticeable in regions where the potential is deep and the interaction is stronger. Thus, the analytical solution of the Klein-Fock-Gordon equation in the Hülten potential field both clarifies the nature of relativistic effects and demonstrates agreement with the results of nonrelativistic quantum mechanics in terms of theoretical consistency and accuracy of the limit transition. This indicates that the model is a suitable theoretical framework for a more fundamental and accurate description of short-range interactions in molecular systems. Keywords: Klein-Fock-Gordon equation, Hülten potential, analytical solutions, energy spectrum, relativistic 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.
In this paper, the bound state solution of the modified Klein–Fock–Gordon equation is obtained for the Hulthén plus ring-shaped-like potential by using the developed scheme to overcome the centrifugal part. The energy eigenvalues and corresponding radial and azimuthal wave functions are defined for any [Formula: see text] angular momentum case on the conditions that scalar potential is whether equal and nonequal to vector potential, the bound state solutions of the Klein–Fock–Gordon equation of the Hulthén plus ring-shaped-like potential are obtained by Nikiforov–Uvarov (NU) and supersymmetric quantum mechanics (SUSY QM) methods. The equivalent expressions are obtained for the energy eigenvalues, and the expression of radial wave functions transformations to each other is revealed owing to both methods. The energy levels and the corresponding normalized eigenfunctions are represented in terms of the Jacobi polynomials for arbitrary [Formula: see text] states. A closed form of the normalization constant of the wave functions is also found. It is shown that the energy eigenvalues and eigenfunctions are sensitive to [Formula: see text] radial and [Formula: see text] orbital quantum numbers.
Quantum relativistic solutions for a particle in a one-dimensional Morse potential are presented using methods previously proposed and applied to a particle in a harmonic oscillator. The methods lead to both numerical and analytical solutions, with the latter allowing smooth variation of the system parameters from non-relativistic to ultrarelativistic limits. Analytical expressions for the energy levels and wavefunctions are obtained, as solutions to a Schrödinger-type equation, including relativistic effects through a state dependent rescaled mass. The eigenstates of the Morse potential exhibit suitable and smooth behavior, and approach the corresponding harmonic oscillator solutions as the depth of the Morse potential well increases, as expected. A comparison is also presented between the relativistic harmonic oscillator obtained with this method, and the so-called Klein-Gordon oscillator.
We seek to introduce a mathematical method to derive the Klein-Gordon equation and a set of relevant laws strictly, which combines the relativistic wave functions in two inertial frames of reference. If we define the stationary state wave functions as special solutions like Ψ(r,t)=ψ(r)e−iEt/ℏΨ(\mathbf{r},t)=ψ(\mathbf{r})e^{-iEt/\hbar}Ψ(r,t)=ψ(r)e−iEt/ℏ, and define m=E/c2m=E/c^2m=E/c2, which is called the mass of the system, then the Klein-Gordon equation can clearly be expressed in a better form when compared with the non-relativistic limit, which not only allows us to transplant the solving approach of the Schrödinger equation into the relativistic wave equations, but also proves that the stationary solutions of the Klein-Gordon equation in a potential field have the probability significance. For comparison, we have also discussed the Dirac equation. By introducing the concept of system mass into the Klein-Gordon equation with the scalar and vector potentials, we prove that if the Schr\"{o dinger equation in a certain potential field can be solved exactly, then under the condition that the scalar and vector potentials are equal, the Klein-Gordon equation in the same potential field can also be solved exactly by using the same method.
In this study, we replace the standard partial derivatives in the Klein–Gordon equation with Dunkl derivatives and obtain exact analytical solutions for the eigenvalues and eigenfunctions of the Dunkl–Klein–Gordon equation in higher dimensions. We apply this formalism to two key quantum mechanical systems: the d-dimensional harmonic oscillator and the Coulomb potential. First, we introduce Dunkl quantum mechanics in d-dimensional hyperspherical coordinates, followed by an analysis of the d-dimensional Dunkl–Klein–Gordon oscillator. Subsequently, we derive the energy spectrum and eigenfunctions, which are expressed using confluent hypergeometric functions. Furthermore, we examine the impact of the Dunkl formalism on both the eigenvalues and eigenfunctions. In the second case, we explore both the bound-state solutions and scattering scenarios of the Dunkl–Klein–Gordon equation with the Coulomb potential. The bound-state solutions are represented in terms of confluent hypergeometric functions, while the scattering states enable us to compute the particle creation density and probability using the Bogoliubov transformation method.
This study examined the relativistic Klein–Gordon equation and highlighted the characteristics of positive and negative energies. Positive energies correspond to real particles, as they exhibit stable motion forward in time, while negative energies correspond to non-real systems due to antiparticles, highlighting the understanding of particles in relativistic quantum theory. The system’s stability is more pronounced at small screening parameters as the creation of particles (indicated by negative energies) also enlarged. The study further explores the contributions of component potentials in an interacting potential, with a focus on the Pöschl–Teller and symmetric trigonometric Rosen–Morse potentials. The duality of positive and negative energy solutions exemplifies the symmetry between matter and antimatter. The solution of the Schrödinger equation with the same potential is also studied and applied to diatomic models, specifically Cl2, Br2, I2, K2, N2 and F2, comparing analytic results with tested data for various quantum states. The analytic results demonstrate excellent agreement with experimental results, with deviations attributed to experimental methods or theoretical model limitations. The study underscores the reliability of the proposed analytic methods for predicting molecular properties.