The kleingordon equation is a relativistic wave equation, related to the schrodinger equation. Klein gordon equation article about klein gordon equation. If you like you can view the kleingordon equation as a \toy model for the maxwell equations. Introduction in the model that describes the interaction of scalar nucleons interacting with neutral scalar mesons, the dynamics of these fields through yukawa coupling are given by the following klein gordon schro.
Accordingly, the kleingordon kg equation and dirac equation are restudied and compared with the relativistic stationary schr. These equations, together with the existence and stability of their soliton solutions1. The general solutions found, could be used for future explorations on the study for other speci. This method, which clearly depends on the choice of the initial values, is based on the new iteration method nim and the elzaki transformation. The klein gordon equation kg is nowadays regarded as the relativistic form of the. Then he considered the special case of a point source. Ita 2 1 theoretical physics group, department of physics, university of uyonigeria. Numerical solution of klein gordon and sine gordon equations using the numerical method of gridless lines. New transform iterative method for solving some kleingordon. In this paper, in order to extend the lattice boltzmann method to deal with more nonlinear equations, a onedimensional 1d lattice boltzmann scheme with an amending function for the nonlinear kleingordon equation is proposed. A sharp bilinear estimate for the klein gordon equation in arbitrary spacetime dimensions jeavons, chris, differential and integral equations, 2014.
A nekhoroshev type theorem for the nonlinear kleingordon. Therefore, as long as energy and momentum follows the einsteins relation eq. On the basis of this description, we can describe in more detail the motion of such a particle, either by the kleingordon equation, or as a component of the spin value of the particle. Kleingordon eld is characterized by \particles with rest mass mand no other structure e. The non relativistic limit of the kleingordon equation kg is the schrodinger equation s. This paper is concerned with strong blowup instability definition 1. The inverse of the wave operator linearizes the nonlinear equation. Solving the fractional nonlinear kleingordon equation by. We consider a modified kleingordon equation that arises at ultra high energies. New transform iterative method for solving some klein. Pdf bound states of kleingordon equation and dirac equation. This interpretation of the relativistic elliptical wave equation and klein. Sep 14, 2015 solving the klein gordon equation is quite similar to solving schrodingers equation. Its solutions include a quantum scalar or pseudoscalar field, a field whose.
Covariant notation for the kleingordon equation if an equation is consistent with the principles of special relativity, it should be possible to write it in covariant form, that is, in a form that is the same in all lorentz frames. Relativistic qm the klein gordon equation 1926 1 relativistic qm the klein gordon equation 1926 scalar particle field j0 energy eigenvalues 1927 dirac tried to eliminate negative solutions by writing a relativistic equation linear in e a theory of fermions 1934 pauli and weisskopf revived kg equation with. Stationary solutions of the kleingordon equation in a. Because you must solve the general poisson equation. In this paper, the homotopy analysis method is applied to obtain the solution of nonlinear fractional partial differential equations. The wave equation is derived for quark pairs in color superconductor in the regime of low density strong coupling. Solving the fractional nonlinear kleingordon equation by means of. Kleingordon equation relativistic quantum mechanics. Solving the kleingordon equation with a fourier transform 1 contradictory solutions to the kleingordon equation as an initial value problem canonical formulation.
The klein gordon equation in curved spacetime from within the proper time formalism of schwinger and dewitt has been treated, at least from the gaussian approximation, by beken stein 2, 3. In other words, the kleingordon equation, if useful at all, does not describe the probability wave, which the schrodinger equation does, but describes something else. As we can see now, the apparent problems were manifestations of the new physics. The general solution to the kleingordon equation would then be given by. Especially, almost no one has studied the asymptotic theory of the initial value problem for the nonlinear perturbed kleingordon equations in the classical sense of c 2, and a lot of problems need to be solved 21. All structured data from the file and property namespaces is available under the creative commons cc0 license. It is secondorder in space and time and manifestly lorentzcovariant. The kleingordon equation can be expressed in general possible form by. How to derive general solution to the kleingordon equation. In this study, we treat some klein gordon equations kges. Parallel spectral numerical methodsthe kleingordon equation.
Since we want the lagrangian density to be invariant under lorentz transformations a linear wave equation, the lagrangian density can contain only the. Therefore, a system that can be described by a complex solution to the klein gordon equation also be described by a system of two independent particles with equal mass that have real solutions to the klein klein gordon equation. Kleingordon equation the quest for a mathematical theory of quantum mechanics began with great am bition. If you like you can view the klein gordon equation as a \toy model for the maxwell equations. The homotopy perturbation method hpm has been successively applied for finding approximate analytical solutions of the fractional nonlinear kleingordon equation which can be used as a numerical algorithm. In a suitable approximation it is shown that for the linear potential which is of interest in quark interactions, their confinement for example,we get solutions that mimic the harmonic oscillator energy levels, surprisingly. Klein gordon equation the quest for a mathematical theory of quantum mechanics began with great am bition. Exact solutions of the klein gordon equation with hylleraas potential akpan n. Ppt relativistic qm the klein gordon equation 1926.
We will come back to the question what it actually describes later on. Employing a transformation to hyperbolic space, we derive in a simple way exact solutions for the kleingordon equation in an infinite squarewell potential with one boundary moving at constant velocity, for the massless as well as for the massive case. The klein gordon equation plays an important role in mathematical physics. The solution just like in the case with the wave function in the schrodinger equation just comes out to be some scalar function of x, y, z, and t. The klein gordon equation klein fock gordon equation or sometimes klein gordon fock equation is a relativistic wave equation, related to the schrodinger equation. Dec 24, 2004 the nonlinear klein gordon equation with mass decreasing to zero catalano, fabio, advances in differential equations, 2002. Em lorentz force derived from klein gordons equation 11.
All structured data from the file and property namespaces is available under the. Pdf in this article, we derive the scalar parametrized kleingordon equation from the formal information theory framework. The relativistic quantum hydrodynamic representation of klein. Large time behavior of solutions to the kleingordon equation with nonlinear dissipative terms sunagawa, hideaki, journal of. See also special cases of the nonlinear kleingordon equation. The plane waves 10 are basic solutions and the field 9 is constructed by. The kleingordon equation kleinfockgordon equation or sometimes kleingordonfock equation is a relativistic wave equation, related to the schrodinger equation. In sec iv, using the algebraic method of separation of variables, we reduce the dirac equation to a system of rst order coupled di erential equations that we solve in terms of special functions.
The covariant kleingordon equation requires twice the boundary conditions of the schrodinger equation and does not have an accepted. Introduction to relativistic quantum mechanics 249 1. The linear kleingordon equation occurs as a modification of the linear schrodinger equation that is consistent with special relativity, see for example landau or grenier. The form of the kleingordon equation is derived by requiring that planewave solutions. It is a quantized version of the relativistic energymomentum relation. Numerical solution of nonlinear kleingordon equation. We present an approximate formula of the fractional derivative.
Media in category solutions of pde the following 92 files are in this category, out of 92 total. The homotopy perturbation method hpm has been successively applied for finding approximate analytical solutions of the fractional nonlinear klein gordon equation which can be used as a numerical algorithm. Remark on the scattering problem for the kleingordon equation with power nonlinearity. Exact solutions of the kleingordon equation with hylleraas potential akpan n. When an electron is bound in a hydrogenlikeatom with pointlikenucleus havingcharge number z, thecritical value of z, zc, equals to 7 in dirac equation whereas. At the very least it should be consistent with the theory of spe cial relativity that had revolutionized classical physics. The non relativistic limit of the real kleingordon equation.
So the klein gordon eld is physically and mathematically, too the simplest of the relativistic elds that one can study. How to derive general solution to the kleingordon equation a. A sharp bilinear estimate for the kleingordon equation in arbitrary spacetime dimensions jeavons, chris, differential and integral equations, 2014. Large time behavior of solutions to the klein gordon equation with nonlinear dissipative terms sunagawa, hideaki, journal of the mathematical society of japan, 2006. The kleingordon equation in the spacetime representation 9 the quantity q2 is represented by the differential operator dalembert operator from the mass shell condition 6 this results in the kleingordon equation as the basic field equation of the scalar field. View kleingordon equation research papers on academia. So the kleingordon eld is physically and mathematically, too the simplest of the relativistic elds that one can study. Kleingordon equation article about kleingordon equation. Employing a transformation to hyperbolic space, we derive in a simple way exact solutions for the klein gordon equation in an infinite squarewell potential with one boundary moving at constant velocity, for the massless as well as for the massive case. On the cauchy problem for the nonlinear kleingordon equation with a cubic convolution motai, takahiro, tsukuba journal of mathematics, 1988.
Historically, the kleingordon equation was the first relativistic equation in quantum mechanics for the wave function. On the cauchy problem for the nonlinear klein gordon equation with a cubic convolution motai, takahiro, tsukuba journal of mathematics, 1988. The kleingordon equation plays an important role in mathematical physics. Solution of the klein gordon equation the covariant generalization of the klein gordon equation in curved spacetime has the form 12 g r. Parallel spectral numerical methodsthe kleingordon. Relativistic quantum mechanics kleingordon equation dirac. Kleingordon equation a relativistic that is satisfying the requirements of the theory of relativity quantum. The kleingordon equation is the linear partial differential equation which is the equation of motion of a free scalar field of possibly nonvanishing mass m m on some possibly curved spacetime lorentzian manifold. Solving the klein gordon equation is quite similar to solving schrodingers equation.
The interaction of pseudo scalar particles with the electromagnetic field is described by the minimal substitution. The nonlinear kleingordon equation with mass decreasing to zero catalano, fabio, advances in differential equations, 2002. These equations appear in the study of relativistic and quantum physics. This paper presents the formulation of timefractional kleingordon equation using the eulerlagrange. The legendre collocation method is used to reduce fkge to the solution of system of odes which is solved by using finite difference method. Geometrical derivation of the kleingordon equation springerlink. China weimin zhouy and jun yanz department of physics, new york university, 4 washington place, new york, ny, 3 a particle is always not pure. The supersymmetric approach is employed to calculate any. Comparison among kleingordon equation, dirac equation. Pdf solving the kleingordon equation and dirac equation with ringshaped nonspherical oscillator.
Later, the equation was revived and is now known as kleingordon equation. Klein gordon eld is characterized by \particles with rest mass mand no other structure e. Each component of the wave function of particles of any spin also satisfies the kleingordon equation, but only for the case where the spin is 0 is the function invariant with respect to the lorentzpoincare group. In the late 1890s arnold sommerfeld12 18681951 developed a tech. Kleingordon equation with a powerlaw nonlinearity 1, kleingordon equation with a powerlaw nonlinearity 2, modi. Apr 12, 2016 how to derive general solution to the klein gordon equation. Approximate solution of nonlinear fractional kleingordon.
For instance, quantum features of molecular interactions which are subjected to ringshaped. Pdf numerical solution of kleingordon and sinegordon. So, none of the objections that led to discarding the klein equation held up with time. The klein gordon equation is the linear partial differential equation which is the equation of motion of a free scalar field of possibly nonvanishing mass m m on some possibly curved spacetime lorentzian manifold. In the single case, namely the nonlinear klein gordon equation with power type nonlinearity, stability and instability for standing wave solutions have been extensively studied. The klein gordon equation in the spacetime representation 9 the quantity q2 is represented by the differential operator dalembert operator from the mass shell condition 6 this results in the klein gordon equation as the basic field equation of the scalar field. Pdf notes on the kleingordon equation researchgate. Chapter 11 em lorentz force derived from klein gordons.
Kleingordon equation relativistic quantum mechanics youtube. Asymptotic theory of initial value problems for nonlinear. By means of this method, we are able to solve the klein gordon equation just like solving the schr. We propose a novel iterative approach called the elzaki iterative method eim. Both the kleingordon and the dirac equation are no 1particle waveequations, but relativistic. We prove for such an equation that there is a neighbourhood of zero in a hilbert space of initial conditions for which the cauchy problem has global solutions and on which there is asymptotic completeness. A complex field remains a complex field as expected for nonrelativistic problems where we are all familiar with wave functions and their interpretation. Negative energy is one of those concepts which if can be detected physically then the most. Iii we separate variables and solve the kleingordon equation. At the present time, there have been no numerical studies of blow up of solutions to this equation without the assumption of radial symmetry.
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