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Semi-inverse variational approach to solve the Klein-Gordon equation for harmonic- and perturbed Coulomb potentials

Khalid Reggab, Ahmed Gueddim, Abdelkrim Naas

Published Aug 30, 2023• 1 citation

Abstract

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.

DOI: 10.54021/sesv4n1-017

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Incorporating relativity into quantum mechanics presents challenges, one of which is the inconsistent possibility of multiple particles within a single-particle theory. This chapter describes the traditional, semi-analytic bound state in a 1-D box problem that requires a trial-and-error search algorithm. It examines techniques for finding the bound states for an arbitrary potential as a numerical eigenvalue problem and employs similar techniques for the relativistic hydrogen states of the Klein-Gordon equation. The chapter also employs a Monte Carlo algorithm to simulate the spontaneous decay of atoms or nuclei. It presents the numerical solution of the time-independent Schrodinger equation and also examines partial wave expansions, which leads to methods of computing the spherical harmonics and the associated Legendre functions, and ultimately the 3-D hydrogenic wave functions. The chapter deals with the Feynman path integral formulation of quantum mechanics, and its solution via the Metropolis algorithm.

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(\mathbf{r},t)=ψ(\mathbf{r})e^{-iEt/\hbar}

, and define

m=E/c^2

, 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.

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.

Chapter Preview: We begin at the beginning with Plank's black-body radiation spectrum applied to the cosmic background. We move on to bound states; first finding them semi-analytically with a trial-and-error search, and then finding them numerically as an eigenvalue problem. We include relativistic effects in hydrogen bound states by solving the Klein-Gordon equation. We then examine the numerical solution of the time- independent Schrödinger equation, and various types of bound state wave functions. We move on to wave packet solutions of the time- dependent Schrödinger equation. Then we study scattering from: a 3-D square well, the Coulomb potential, three stationary disks, and on a billiard table (see also classical billiards in § 3.8 ). We next look at quantum mechanics in momentum space: integral equations, bound states, the delta-shell potential, hyperfine structure of hydrogen, SU3 symmetry of quarks. Next to last we examine several examples of coherent states. We end the chapter with the Feynman path integral formulation of quantum mechanics, and its solution via the Metropolis algorithm. If not enough, Chapter 8 discussion of quantum entanglement fits in well here too.

This chapter introduces the concepts of the relativistic theory to basic problems of quantum physics, in particular, the extension of the Schrodinger equation to a relativistic form. Starting from the fundamental laws of relativity, the chapter evaluates the Klein–Gordon equation and investigates if the equation can be considered a generalization of the non-relativistic Schrodinger equation to the case of the relativistic energy. The chapter shows that the wave function, which is a solution to the Klein–Gordon equation, cannot be connected with the probability wave function. For this reason, the Dirac equation is used to solve the problems faced by the Klein–Gordon equation. The Dirac equation also includes the spin. The advantage of the Dirac equation over the Schrodinger and Klein–Gordon equations is that it naturally includes the spin. In the Schrodinger and Klein–Gordon equations, the spin is not present and has to be added to the wave function manually whenever the wave function determines a system with the spin.