Integral transform approach to solving kleingordon. Bound state solutions of the kleingordon equation for strong. In this article, we derive the scalar klein gordon equation from the. B c2 would decrease without lower bound if the charge number of nucleus zis su ciently large. Pdf the feynman problem for the kleingordon equation. These equations appear in the study of relativistic and quantum physics. Lorentz invariance of kleingordon lagrangian physics forums. Its solutions include a quantum scalar or pseudoscalar field, a field whose. Pdf numerical solution of kleingordon and sinegordon. Theoretical physics 3 phys3661 quantum mechanics epiphany 2015 alexander lenz ippp, durham university april 8, 2015 1. Mod01 lec01 introduction, the kleingordon equation youtube. Pdf in this article, we derive the scalar parametrized kleingordon equation from the formal information theory framework. Metrics are considered for an impenetrable sphere, a soft sphere of uniform density, and a soft sphere with a linear transition from constant to zero density. The latter equation represent the conservation of the number of.
Klein gordon equation article about klein gordon equation. The kleingordon equation with a scalar potential and a vector potential in ddimensions reads where represents the spatial dimensionality and represents the laplace operator in ddimensions, is the reduced planck constant, and are the speed of light and relativistic energy of the system, respectively. Kleingordondirac equation, plane harmonic waves with di. Since we want the lagrangian density to be invariant under lorentz transformations a linear wave equation, the lagrangian density can contain only the. We solve the relativistic kleingordon equation for a light particle gravitationally bound to a heavy central mass, with the gravitational interaction prescribed by the metric of a spherically symmetric spacetime. New transform iterative method for solving some kleingordon. Kleingordon equation article about kleingordon equation. The derivation of the kleingordon equation 3, 4, 5, 6, 7is essentially identical to that of the schrodinger equation namely, an action ansatz is substituted in. This equation, studied in greater detail by klein and gordon whose name it now bears. A qualitative study of the damped dung equation and. Kleingordon equation a relativistic that is satisfying the requirements of the theory of relativity quantum. Jan 30, 2008 we consider a new type of linear interaction in the klein gordon equation.
That is where cryptohermitian approach can be applied. When schrodinger developed his nonrelativistic wave equation, he also proposed a relativistic generalization. Physics, particle physics and fields, specific theories and interaction models, fundamental equations, kleingordon equation, physics publisher on behalf of the author collection flooved. Equation 12 is of course the relativistic energymomentum relation all. The f eynman problem for the klein gordon equation. The energy equation and the corresponding unnormalized wave function are obtained analytically. The klein gordon equation is a relativistic wave equation valid for spinless particles. Remark on the scattering problem for the klein gordon equation with power nonlinearity. Stationary solutions of the kleingordon equation in a potential field.
The klein gordon equation can be expressed in general possible form by. Both the kleingordon and the dirac equation are no 1particle waveequations, but relativistic. Gordon scalar field in the exterior of the domain of influence of a compact set. 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. If you like you can view the kleingordon equation as a \toy model for the maxwell equations. Solution to kleingordon equation physics stack exchange. We consider the kleingordon equation on a class of lorentzian manifolds with cauchy surface of bounded geometry, which is. Here, kleingordon equation connecting repositories. We consider the scattering problem for the nonlinear kleingordon equation. On the basis of this description, we can describe in more detail the motion of such a particle, either by the klein gordon equation, or as a component of the spin value of the particle. We solve the relativistic klein gordon equation for a light particle gravitationally bound to a heavy central mass, with the gravitational interaction prescribed by the metric of a spherically symmetric spacetime. By introducing the concept of system mass into the klein gordon equation with the scalar and vector potentials, we prove that if the.
The exact energy eigenvalues and wavefunctions are derived analytically by using the nikiforov and uvarov method. Comparison among kleingordon equation, dirac equation and relativistic stationary schr. For a free particle, the equation is p p m 0c2 0 1 where p is the fourmomentum operator. 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. For comparison, we have also discussed the dirac equation. Vortex dynamics for the nonlinear maxwellkleingordon equation. The kleingordon equation is named after oskar klein and walter gordon. We follow and modify the feshbachvillars formalism by separating the kleingordon equation into two coupled timedependent schrodinger equations for particle and. Introduction to dispersive waves, dispersion relations, uniform kleingordon equation, linear superposition and the fourier transform, the stationaryphase method for linear dispersive waves 20 extra lecture pdf. Weisskopf discovered a suitable interpretation for the equation within the scope of quantum field theory. Separation of variables and exact solution of the klein.
The goursat problem at the horizons for the kleingordon. Numerical analysis for the kleingordon equation with mass. 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. Jun 19, 2014 mod01 lec01 introduction, the kleingordon equation nptelhrd.
It is a quantized version of the relativistic energymomentum relation. The relativisticallyinvariant quantum equation describing spinless scalar or pseudoscalar particles, for example, and mesons. Comparison among kleingordon equation, dirac equation. Kleingordon equation the behavior of the kleingordon equation. For that reason it is also called the kleingordon field. The eigenfunctions of the system are given and the degeneracy of the energy eigenvalues is discussed. Such solutions if realvalued correspond to certain constant mean curvature surfaces in euclidean 3space. We follow and modify the feshbachvillars formalism by separating the kleingordon equation into two coupled timedependent. We explore kleingordon equation in the new framework of quasihermitian quantum mechanics. Historically, the kleingordon equation was the first relativistic equation in quantum mechanics for the wave function.
Comparison among kleingordon equation, dirac equation and. If you like you can view the klein gordon equation as a \toy model for the maxwell equations. Chapter propagators of the real klein gordon field. Dirac equation gordon equation dependent mass pseudospin symmetry spinless particle. Delort universite paris, institut galilee cnrs, umr 7539, laboratoire analyse geom etrie et applications 99, avenue j. Consider a meson bound in a point nucleus with in nite mass mn. Mod01 lec01 introduction, the kleingordon equation. So, none of the objections that led to discarding the klein equation held up with time. Pdf kleingordon equation in hydrodynamical form researchgate.
The general solutions found, could be used for future explorations on the study for other speci. Pdf notes on the kleingordon equation researchgate. Consider a nonlinear kleingordon equation on the unit circle, with smooth data of size. The plane waves 10 are basic solutions and the field 9 is constructed by. We consider the klein gordon equation on a class of lorentzian manifolds with cauchy surface of bounded geometry, which is. The kleingordon equation kleinfockgordon equation or sometimes kleingordonfock equation is a relativistic wave equation, related to the schrodinger equation. Oskar kleins tomb at judiska norra begravningsplatsen in solna grey stone to the right. We solve the goursat problem for fixed angular momentum n of the field with the restriction that n is non zero in the case of a massless field. Metrics are considered for an impenetrable sphere, a soft sphere of uniform density, and a soft sphere with. However, the situation becomes quite di erent in the theory of special relativity. Logical inference approach to relativistic quantum mechanics. Bound state solutions of the kleingordon equation for strong potentials.
Sep 14, 2015 solving the klein gordon equation is quite similar to solving schrodingers equation. New transform iterative method for solving some klein. This book develops a spectral theory for the integrable system of 2dimensional, simply periodic, complexvalued solutions u of the sinhgordon equation. Bound state solutions for the swave are found by separating the kleingordon equation in two parts, unlike the similar works in the literature. Exact solutions of the kleingordon equation with position. Kleingordon equation for quark pairs in color superconductor by. Greens functions in physics version 1 uw faculty web. Solutions to common problems with probability interpretation and inde. The supersymmetric approach is employed to calculate any. Kleingordon eld is characterized by \particles with rest mass mand no other structure e. Fok as a wave equation under the conditions of cyclicity in the fifth coordinate and was shortly afterwards deduced by several authors for example, w.
Fourier transforming the klein gordon equation physics. For the kleingordon equation, however, there is an exact solution. Pdf a nonlinear kleingordon equation on star graphs. We consider a modified kleingordon equation that arises at ultra high energies. Aug 23, 2019 we derive the asymptotic properties of the mmkg system maxwell coupled with a massive klein. The kleingordon equation can be expressed in general possible form by. Physics stack exchange is a question and answer site for active researchers, academics and students of physics.
Well, the problem with this idea in the case of klein gordan equation is that you end up with something that is not a continuity equation, so, if you try to use it as a probability density, you end up with something that have nonconstant normalization, and, thus, cant be used to define a probability distribution. Dec 24, 2004 quadratic nonlinear kleingordon equation in 2d, cauchy problem hayashi, nakao and naumkin, pavel i. Physics letters a vol 384, issue 12, 30 april 2020. A nekhoroshev type theorem for the nonlinear kleingordon. Relativistic quantum mechanics kleingordon equation dirac. Solving the fractional nonlinear kleingordon equation by. Parallel spectral numerical methodsthe kleingordon. Cryptohermitian approach to the kleingordon equation. Therefore, the kleingordon equation was at first rejected. The dirac equation can be thought of in terms of a square root of the kleingordon equation. A nonlinear kleingordon equation on star graphs nataliia goloshchapov a department of mathematics, imeusp, r ua do mat. Introduction the problem has originally arisen from the paper 1 and the more later one 2, where the kleingordondirac equation kgd, i. Kleingordon equation the motion of a stretched vortex filament combines both selfinduction and elasticity.
As we can see now, the apparent problems were manifestations of the new physics. For an isolated filament the solution must have the form of a helix 7, or 3, although a correction to the frequency must be made. 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. Later, the equation was revived and is now known as kleingordon equation. Solitary waves for nonlinear kleingordonmaxwell and. Mechanical interpretation of the kleingordon equation. So the klein gordon eld is physically and mathematically, too the simplest of the relativistic elds that one can study.
Numerical solution of kleingordon and sine gordon equations using the numerical method of gridless lines. Dispersive wave solutions of the kleingordon equation in. An overview over the kg propagators on minkowski spacetime is given in mikica kocic, invariant commutation and propagation functions invariant commutation and propagation functions, 2016 pdf. Klein gordon equation f or a free relativistic particle the total energy e is no longer giv en b y the equation w e used to deriv e the sc hro edinger equation. Solutions of the klein gordon equation with generalized. At the present time, there have been no numerical studies of blow up of solutions to this equation without the assumption of radial symmetry. Notice that the lefthand part of the dynamic equation 4 has the. The nonlinear term of the equation behaves like a power term. We seek to introduce a mathematical method to derive the kleingordon equation and a set of relevant laws strictly. View kleingordon equation research papers on academia. Soliton solutions of the generalized kleingordon equation.
So the kleingordon eld is physically and mathematically, too the simplest of the relativistic elds that one can study. Bound state solutions of kleingordon equation with the kratzer. Please join the simons foundation and our generous member organizations in supporting arxiv during our giving campaign september 2327. Therefore, as long as energy and momentum follows the einsteins relation eq. It is secondorder in space and time and manifestly lorentzcovariant. If the inline pdf is not rendering correctly, you can download 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.
Numerical solution of kleingordon and sinegordon equations using the numerical method of gridless lines. Gordon equation of the symmetric generalized woods. The klein gordon equation 21 is completely separable in 3, therefore we look for its solution in the form. A spectral theory for simply periodic solutions of the. The nonrelativistic equation for the energy of a free particle is. Analytical solutions of the kleingordon equation with a. Under minimal substitution the wave equation becomes quadratic in both the momentum and the coordinates,viz. Klein gordon eld is characterized by \particles with rest mass mand no other structure e. Numerical computation of kleingordon equations arising in. Stationary solutions of the kleingordon equation in a. May 14, 2011 we derive the vortex dynamics for the nonlinear maxwellkleingordon equation with the ginzburglandau type potential.
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