Kleingordon equation with a powerlaw nonlinearity 1, kleingordon equation with a powerlaw nonlinearity 2, modi. So the klein gordon eld is physically and mathematically, too the simplest of the relativistic elds that one can study. Comparison among kleingordon equation, dirac equation. The general solutions found, could be used for future explorations on the study for other speci. It is a quantized version of the relativistic energymomentum relation. Combining this equation and 57 or equivalently 60 shows that the position wave. 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. Maxwells equations and here we see the kaluza miracle at work.
The klein gordon equation is a relativistic wave equation valid for spinless particles. At the very least it should be consistent with the theory of spe cial relativity that had revolutionized classical physics. The kleingordon equation dates to the legendary early period of. Equivalently, is a distributional solution to klein gordon if it annihilates any schwartz function in the image of the klein gordon. Kleingordon equation article about kleingordon equation. Stationary solutions of the kleingordon equation in a. The kleingordon equation is a linear homogeneous secondorder partial differential equation with constant coefficients. Pdf dirac and kleingordon equations in curved space. Small data blowup for semilinear kleingordon equations. 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. Since we want the lagrangian density to be invariant under lorentz transformations a linear wave equation, the lagrangian density can contain only the. As the derivation of 1 as a limiting case of 3 requires.
Thus maxwells theory of electromagnetism is an inevitable consequence of einsteins general theory of relativity, given that one is willing to buy the idea of a. Later, the equation was revived and is now known as kleingordon equation. This chapter deals with the theory and applications of nonlinear kleingordon and sinegordon equations. A qualitative study of the damped dung equation and applications. Innovative merge of cryptohermitian quantum theory and perturbation the. The equation originated with attempts to combine relativity theory and quantum mechanics. The klein gordon equation klein fock gordon equation or sometimes klein gordon fock 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. The kleingordon equation is applied in the description of mesons and corresponding fields. Pdf we solve the d dimensional kleingordon equation with a newly proposed generalized hyperbolic potential model, under the condition. Gaussian and airy wave packets of massive particles with orbital.
So in his derivation he didnt have to make a choice between four vector model or six. Pdf notes on the kleingordon equation researchgate. Exact solutions nonlinear partial differential equations secondorder hyperbolic partial differential equations nonlinear klein gordon equation 7. 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. Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. Pdf the kleingordon equation is considered for the kratzer. To present the present methods effectiveness many examples are given. As we can see now, the apparent problems were manifestations of the new physics. How to derive general solution to the kleingordon equation. The fourier spectral method is used to discretize the ncsfkgs equations. Proca equations the dirac equation predicts that the electron magnetic moment and its spin are related as 2 bs, while for normal.
In the dirac case, the discrete solution converges to the continuum propagator and it can also be rewritten in the form. Lecture notes massachusetts institute of technology. Accordingly, the kleingordon kg equation and dirac equation are restudied and compared with the relativistic stationary schr. Jun 19, 2014 mod01 lec01 introduction, the kleingordon equation nptelhrd. For discrete 1d, 2d and 3d schrodinger and kleingordon equations the.
In this note i will show how the quantum version of the kleingordon equation. In this section we combine the spectral properties of the perturbed resolvent and. 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. 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. The klein gordon field gives a spin 0 representation, while the dirac equation gives two spin 12 representations which merge to a single representation if one also accounts for discrete symmetries.
Analytical solutions of the kleingordon equation with a. Iii we separate variables and solve the klein gordon equation. That is where cryptohermitian approach can be applied. Suppose w wx,t is a solution of the nonlinear klein gordon equation. Deriving the klein gordon equation relativistic quantum. Relativistic wave equation for spino particles the kleingordon. Both the klein gordon and the dirac equation are no 1particle wave equations, but relativistic. 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. Firstly, let us illustrate how the kleingordon equation could be reduced to the 2d. Relativistic quantum mechanics kleingordon equation dirac. Kleingordon and squareroot operator equations for two.
Klein gordon equation the quest for a mathematical theory of quantum mechanics began with great am bition. Boundary value problems for the kleingordon equation. 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. Em lorentz force derived from klein gordons equation 11.
Sep 14, 2015 solving the klein gordon equation is quite similar to solving schrodingers equation. Historically, the kleingordon equation was the first relativistic equation in quantum mechanics for the wave function. Discretizing the maxwellkleingordon equation by the. I go over fourvector notation, fourgradient, and writing the equation.
The kleingordon equation is a relativistic wave equation valid for spinless particles. Special emphasis is given to various methods of solutions of these equations. For a free particle, the equation is p p m 0c2 0 1 where p is the fourmomentum operator. Pdf approximate solutions for solving the kleingordon. The klein gordon equation is the first attempt to covariantize quantum mechanics but, as we have just shown, fails due to some problems that arise when we try to link special relativity concepts with quantum mechanics ones.
Solving the klein gordon equation with a fourier transform. The supersymmetric approach is employed to calculate any. Combining equations18 and 20 to include both the even and odd states, we obtain. Klein gordon eld is characterized by \particles with rest mass mand no other structure e. Relativistic supersymmetric quantum mechanics based on klein. Introduction to dispersive waves, dispersion relations, uniform klein gordon equation, linear superposition and the fourier transform, the stationaryphase method for linear dispersive waves 20 extra lecture pdf. The kleingordon 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. The plane waves 10 are basic solutions and the field 9 is constructed by. The elementary solutions have a position dependence of the form eik.
So, none of the objections that led to discarding the klein equation held up with time. The energy equation and the corresponding unnormalized wave function are obtained analytically. Iterating the equation and using the algebra leads to the kleingordon equation in curved space in its canonical from. The interest of the einsteinscalar eld system is that it is one of the simplest model that incorporates dynamical \matter and gravitational energy in the sense that some components. The timedependent klein gordon and dirac equations are solved for the motion of a charged particle in a classical uniform electrostatic field of infinite extent. Apr 12, 2016 how to derive general solution to the klein gordon equation. Note that in the case of the wave equation, the fundamental solution is positive in dimensions n 2,3, a fact central to the arguments of 12, 16 and indirectly 32. Iii we separate variables and solve the kleingordon equation. The deeper reason for this is that the kleingordon equation is of second order in time, so that.
We will come back to the question what it actually describes later on. Sauga it is shown that, in case of special classes of static metrics, the fundamental solution greens function for the klein gordon equation can be derived by means of the massless fundamental solution. A discrete scheme for the dirac and kleingordon equations. Moreover, the probabilities to be in any state and the energies of those states. A couple things you might note about the klein gordon equation right o the bat. The greens function method combined with integral transforms is employed to solve the linear kleingordon equation. Quantum field theory i babis anastasiou institute for theoretical physics, eth zurich, 8093 zurich, switzerland. 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. Comparison among kleingordon equation, dirac equation and. A fourier spectral method for the nonlinear coupled space. Boundary value problems for the klein gordon equation. 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.
We represent the scalar feynman propagator with the symbol d f x y. The kleingordon equation kleinfockgordon equation or sometimes kleingordonfock equation is a relativistic wave equation, related to the schrodinger equation. In this paper, we practiced relatively new, analytical method known as the variational homotopy perturbation method for solving klein gordon and sine gordon equations. How to use the einstein energymomentum relation to derive the klein gordon wave equation. Introduction to relativistic quantum mechanics 249 1. Contradictory solutions to the klein gordon equation as. It is secondorder in space and time and manifestly lorentzcovariant. Klein gordon equation that includes this dirac current probability.
Schroeder, addisonwesley 3quantum field theory in a nutshell, a. Klein gordon equation dirac equation maxwell equation equation for massive vector particles shrodinger equation 2. Klein gordon and squareroot operator equations for twospinors and scalars. Chapter 11 em lorentz force derived from klein gordons. In applying the obtained result to the weak gravitational. Long time decay for 2d kleingordon equation fakultat fur. The math in this article should be merged with klein gordon equation and schroedinger equation, and then the notion of a free particle should be discussed in a less equation intense fashion i think thats right. Kapitza spring 2018 second quantization of the klein. Laues tensordynamics merged the energy and momentum with the. Numerical solution of nonlinear kleingordon equation. The lagrangian in the standard model gives rise to the evolution equations of the gauge fields, called the yang mills equations actually the name yangmills is most often used in the context of. Fourier transforming the klein gordon equation physics. Mod01 lec01 introduction, the kleingordon equation youtube. The signs in the formula for w1are taken arbitrarily.
For the klein gordon equation, we do not have such positivity. From this global structure, some qualitative behaviors of exact solutions of equation 1. If you like you can view the klein gordon equation as a \toy model for the maxwell equations. The decay in the lp norms for wave and kleingordon equations was obtained in. How to derive general solution to the klein gordon equation a.
Separation of variables and exact solution of the klein. These equations appear in the study of relativistic and quantum physics. See also special cases of the nonlinear kleingordon equation. Hence, the probability current can take on both positive and negative values. In what follows, we outline the derivation of kleingordon equation as a. Pdf solutions of the klein gordon equation with generalized. Traveling wave solutions of the kleingordon equation 1. Gordon 2 without this requirement on the fifth coordinate. A generalized klein gordon equation with a closed system. 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. Kleingordon equation 2 may be rewritten in schrodingerlike form i. By squaring his equation, dirac obtained a differential equation that included the.
Because of this reason, the kleingordon equation was abandoned for a while. Both the kleingordon and the dirac equation are no 1particle waveequations, but relativistic. Klein gordon equation has attracted much attention in studying solutions and condensed mater physics, in in vestigating the interaction of solitons in a collisionless plasma, the recurrence of initial states, and in examining the nonlinear wave equations 1,2. Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless. This equation is known as the klein gordon equation and was one of the rst attempts to merge special relativity with the schr odinger equation.
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