All books are in clear copy here, and all files are secure so dont worry about it. Kleingordon and squareroot operator equations for two. Numerical solution of klein gordon and sine gordon equations using the numerical method of gridless lines. Hydrodynamic limits of the nonlinear klein gordon equation chikun lin1 and kungchien wu2 1department of applied mathematics and center of mathematical modeling and scienti c computing national chiao tung university hsinchu 30010, taiwan 2department of pure mathematics and mathematical statistics university of cambridge, wilberforce road. Relativisticquantum mechanics 1 university of oxford. This kleingordon kg equation can of course be used for dynamic system analysis. Covariant notation for the klein gordon 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.
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. In particular, the klein gordon field has to be a scalar field, by definition. This site is like a library, you could find million book here by using search box in the header. How to derive general solution to the kleingordon equation. However, it has some problems and due to these problems, there is dirac equation that handles these problems. It is a quantized version of the relativistic energymomentum relation. It is secondorder in space and time and manifestly lorentzcovariant. Separation of variables and exact solution of the klein. So, none of the objections that led to discarding the klein equation held up with time. This paper presents the derivation of the schrodinger, klein gordon and dirac equations of particle physics, for free particles, using classical methods. It is described in nonrelativistic quantum mechanics by the schr. 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.
Kleingordon eld is characterized by \particles with rest mass mand no other structure e. Kleingordon equation relativistic quantum mechanics youtube. The non relativistic limit of the real kleingordon equation. In order to solve the dirac equation we apply the algebraic method of separationofvariables2024. Scattering solutions of the klein gordon equation for a step potential with hyperbolic tangent potential article pdf available in modern physics letters a 2928. Numerical solution of nonlinear kleingordon equation. Read online exact solutions of the klein gordon equation with. Stationary solutions of the kleingordon equation in a. Exact solutions nonlinear partial differential equations secondorder hyperbolic partial differential equations nonlinear klein gordon equation 7. Separation of variables pde on kleingordon equation. Several anomalies are highlighted and resolutions proposed. The kleingordon equation kleinfockgordon equation or sometimes kleingordonfock equation is a relativistic wave equation, related to the schrodinger equation. Hydrodynamic limits of the nonlinear kleingordon equation chikun lin1 and kungchien wu2 1department of applied mathematics and center of mathematical modeling and scienti c computing national chiao tung university. Accordingly, the kleingordon kg equation and dirac equation are restudied and compared with the relativistic stationary schr.
Schroeder, addisonwesley 3quantum field theory in a nutshell, a. For classical relativistic point particles, an interaction with these external. Comparison among kleingordon equation, dirac equation and. Pdf dirac and kleingordon equations in curved space. Convergence of the kleingordon equation to the wave map equation with magnetic field kungchien wu1 department of applied mathematics center of mathematical modeling and scienti c computing national chiao tung university hsinchu 30010, taiwan abstract this paper is devoted to the proof of the convergence from the modulated. The aim of this chapter is to introduce a relativistic formalism which can. Sep 14, 2015 solving the klein gordon equation is quite similar to solving schrodingers equation.
Solitary waves for nonlinear kleingordonmaxwell and. So the klein gordon eld is physically and mathematically, too the simplest of the relativistic elds that one can study. If you like you can view the klein gordon equation as a \toy model for the maxwell equations. We explore klein gordon equation in the new framework of quasihermitian quantum mechanics. Pdf a search on the kleingordon equation researchgate. Hylomorphic solitons in the nonlinear kleingordon equation j. 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. An approach tothekleingordon equation for a dynamic study.
Separation of variables and exact solution of the kleingordon and. Quantum field theory i babis anastasiou institute for theoretical physics, eth zurich, 8093 zurich, switzerland. Hydrodynamic limits of the nonlinear kleingordon equation. At the very least it should be consistent with the theory of spe cial relativity that had revolutionized classical physics. Now we want to study covariant equations of motion for spinor elds. Kleingordon equation for simplicity, we begin from a particle with mass m 0 but without spin. The general solution to the klein gordon equation would then be given by. Comparison among kleingordon equation, dirac equation. If you like you can view the kleingordon equation as a \toy model for the maxwell equations. Apr 12, 2016 how to derive general solution to the klein gordon equation. Then its kinetic energy reads 1 2 m 0v 2 with velocity vbeing unlimited and m 0 unchanged. We use the differential transform method for the solution of the initial value problem. Modern service and trade college, yunnan university of finance and economics kunming china. The kleingordon equation is applied in the description of mesons and corresponding fields.
We know that there is a relativistic version of schrodinger equation called klein gordon equation. How to derive general solution to the klein gordon equation a. If the equation contains a field with a nonzero spin, then it is not the klein gordon equation. 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 klein gordon equation is proposed.
Both the klein gordon and the dirac equation are no 1particle wave equations, but relativistic. In this paper, we practiced relatively new, analytical method known as the variational homotopy perturbation method for solving klein gordon and sine gordon equations. We will come back to the question what it actually describes later on. School of mathematical science and computing technology, central south university changsha china. A fourier spectral method for the nonlinear coupled space. Derivation of solutions of the kleingordon equation from. 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. Convergence of the kleingordon equation to the wave map. The paper also shows that in classical limit the hydrodynamic kleingordon equations lead to the madelung pseudopotential 1 as well as to the quantum. Later, the equation was revived and is now known as kleingordon equation. Pdf numerical solution of kleingordon and sinegordon. The kleingordon equation is a linear homogeneous secondorder partial differential equation with constant coefficients.
Jun 19, 2014 mod01 lec01 introduction, the kleingordon equation nptelhrd. Approximate solutions to the nonlinear kleingordon. On the spectral stability of periodic waves of the klein gordon equation, joint with a. Roughly speaking a solitary wave is a solution of a. So, the question is, if there is a solution that is allowed by klein gordon, but not by dirac, can this solution be considered valid. Exact solutions of the kleingordon equation with hylleraas potential akpan n. Sharp decay estimates for the klein gordon equation on kerrads jacques smulevici universit e parissud orsay joint work with gustav holzegel imperial college. In these equations we have omitted the hat on top of the.
The kleingordon equation is a relativistic wave equation, related to the schrodinger equation. Fourier transforming the klein gordon equation physics. Klein gordon equation the quest for a mathematical theory of quantum mechanics began with great am bition. Since we want the lagrangian density to be invariant under lorentz transformations a linear wave equation, the lagrangian density can contain only the. Sharp decay estimates for the klein gordon equation on. Sinibaldi communicated by giovanni gallavotti, received april 17, 2009. A complex field remains a complex field as expected for nonrelativistic problems where we are all familiar with wave functions and their interpretation. To view pdf files on an android tablet, i use the free foxit pdf app, which.
Klein gordon and squareroot operator equations for twospinors and scalars. An approach tothekleingordon equation for a dynamic study in. Wecomparethesolutionswiththoseofobtainedaftersolving. Klein gordon equation f eynman stuec k elburg in terpretation dirac equation an tiparticles f ermion spin co v arian t notation massless fermions learning outcomes be able to deriv e k g equation and explain ph ysical meaning of v e e solutions. Kapitza spring 2018 second quantization of the klein. As we can see now, the apparent problems were manifestations of the new physics. Let ax be the fourvector potential that accounts for electric and magnetic. Can the kleingordon equation represent particles with non. The late sidney coleman taught the quantum field theory course at harvard for many years, influencing a generation of physicists in the way they view and teach qft. The fourier spectral method is used to discretize the ncsfkgs equations. Relativistic quantum mechanics kleingordon equation dirac. It is the purpose of the present article to solve the kleingordon and dirac equations in the friedman universe associated with the metric 3 in the presence of a time dependent magnetic eld. To present the present methods effectiveness many examples are given.
Klein gordon eld is characterized by \particles with rest mass mand no other structure e. The plane waves 10 are basic solutions and the field 9 is constructed by. View kleingordon equation research papers on academia. Introduction to relativistic quantum mechanics 249 1. As the field equations are developed, it is found that invariance of the scalar of curvature under the appropriate gauge and coordinate transformations implies the.
Numerical solution of nonlinear kleingordon equation using lattice boltzmann method qiaojie li1, zong ji2, zhoushun zheng1, hongjuan liu1. In order to show the accuracy of results for the solutions, we use the variational iteration method with adomians polynomials for the nonlinearity. However, the klein gordon equation is, by definition, the wave equation for a spinless, scalar field. Iii we separate variables and solve the klein gordon equation. Geometrical derivation of the kleingordon equation springerlink. Iterating the equation and using the algebra leads to the kleingordon equation in curved space in its canonical from. With the taylor and chapmanenskog expansion, the nonlinear klein gordon equation is recovered correctly from the lattice boltzmann equation. Pdf in this article, we derive the scalar parametrized kleingordon equation from the formal information theory framework. Kleingordon equation article about kleingordon equation. The aim of this chapter is to introduce a relativistic formalism which can be used to describe particles and their interactions. Pdf approximate solutions for solving the kleingordon. The relativistic quantum hydrodynamic representation of klein. Accordingly, the kleingordon kg equation and dirac equation are restudied and compared with the relativistic stationary schrodinger equation rsse.
Pdf notes on the kleingordon equation researchgate. When schrodinger developed his nonrelativistic wave equation, he also proposed a relativistic generalization. The non relativistic limit of the klein gordon equation kg is the schrodinger equation s. Because of this reason, the kleingordon equation was abandoned for a while. Sec iv, using the algebraic method of separation of variables, we reduce the dirac equation to a system of first order coupled differential equations that we solve. Kleingordon equation for quark pairs in color superconductor.
Both the kleingordon and the dirac equation are no 1particle waveequations, but relativistic. 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. Convergence of the klein gordon equation to the wave map equation with magnetic field kungchien wu1 department of applied mathematics center of mathematical modeling and scienti c computing national chiao tung university hsinchu 30010, taiwan abstract this paper is devoted to the proof of the convergence from the modulated. Historically, the kleingordon equation was the first relativistic equation in quantum mechanics for the wave function. Pdf the swave kleingordon equation for the bound states is separated in two parts to see clearly the relativistic contributions to the solution in. So the kleingordon eld is physically and mathematically, too the simplest of the relativistic elds that one can study. Suppose w wx,t is a solution of the nonlinear klein gordon equation. 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. Later, the equation was revived and is now known as klein gordon equation. The wave equation is derived for quark pairs in color superconductor in the regime of low density strong coupling.
Solutions to common problems with probability interpretation and inde. The method relies on a variational approach and the solutions are obtained as mountainpass critical points for the associated energy functional. 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. Pdf scattering solutions of the klein gordon equation. Below you can find the pdf files of handwritten lecture notes for colemans course transcribed by brian hill. What links here related changes upload file special pages permanent link page. Hylomorphic solitons in the nonlinear kleingordon equation. Media in category solutions of pde the following 92 files are in this category, out of 92 total. The derivations are based on the assumption that these wave equations are homogeneous and soluble via separation of variables. In sec iv, using the algebraic method of separation of variables, we reduce the dirac equation. Kleingordon equation with coulomb potential 3 from this point onwards, the solution is quite similar to that for the hydrogen atom and you can see greiners exercise 1.
1058 636 8 860 156 1525 635 417 856 744 467 1130 542 660 625 1229 1458 816 1541 1015 725 414 1282 432 641 284 1232 564 1330 212 338 1184 1003 1419 186 1044 1341