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American Journal of Modern Physics

Analytical Study of the Behavioral Trend of Klein-Gordon Equation in Different Potentials

Emmanuel Ifeanyi Ugwu, Hyacinth Kevin Idu

Published Feb 28, 2024• 0 citations
Publisher

Abstract

In this work, we present the analysis of behavioral trend of Klein-Gordon Equation involving potential as regards when it comes to the study of particle, it has been observed that in every case of handling of KGE with potential of any type, it is made clear here that the equation has to first off all be transformed into a particular standard differential equation with a well-known solution which appears in form of implicitly defined transcendental equation. The equation on the other hand is to be solved analytically since the exact solution is not easily attainable without the use of mathematical tool especially when it comes to the consideration of the energy eigenvalue and the corresponding wave function because the solution is also always accompanied with a normalization constant often coupled with a condition that requires an arbitrarily chosen quantum number that come up when (l=0) and so on. In general, the analysis reveals the fact that the of trend of KGE involving potential gives a good understanding in the study of inter-molecular structure, diatomic crystals, and such case that involves inter-atomic interaction which is gives very nice idea in the study of bound state in atom.

DOI
10.11648/j.ajmp.20241301.12

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In this work, we present the analysis of behavioral trend of Klein-Gordon Equation involving potential as regards when it comes to the study of particle, it has been observed that in every case of handling of KGE with potential of any type, it is made clear here that the equation has to first off all be transformed into a particular standard differential equation with a well-known solution which appears in form of implicitly defined transcendental equation. The equation on the other hand is to be solved analytically since the exact solution is not easily attainable without the use of mathematical tool especially when it comes to the consideration of the energy eigenvalue and the corresponding wave function because the solution is also always accompanied with a normalization constant often coupled with a condition that requires an arbitrarily chosen quantum number that come up when (l=0) and so on. In general, the analysis reveals the fact that the of trend of KGE involving potential gives a good understanding in the study of inter-molecular structure, diatomic crystals, and such case that involves inter-atomic interaction which is gives very nice idea in the study of bound state in atom.
The analysis of the behavioral trend of particle using Klein-Gordon and Dirac equation that was minimally coupled to electromagnetic wave four-vector potential has been carried out. In the analysis, it was clearly observed that each of them has non-relativistic limit at one stage or the other and based on this limitation, there is a great challenge posed on the idea of single particle interpretation since in each case there is a particle and an antiparticle. It therefore reveals the fact that there is no conceptually real existence of single particle in isolation when it comes to relativistic quantum mechanics for any of the equations being used to study the interaction of particle with electromagnetic field.
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.
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.
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.