In particle physics, the dirac equation is a relativistic wave equation derived by british physicist paul dirac in 1928. Both the klein gordon and the dirac equation are no 1particle wave equations, but relativistic. Now one of the most important models of physics is a lorentzinvariant model that has solutions in the form of plane waves and is a. Formally this corresponds the upper solution corresponds to a particle with momentum p and energy e while the lower solution has. Solutions of klein gordon equation free particle solutions for u 0. More new exact travelling wave solutions are obtained for the quadratic nonlinear kleingordon equation, the combined kdv and mkdv equation, the sinegordon equation and the whithambroer. The variational equations governing linear stability are derived and solved. We look for plane wave solutions with both positive and negative energy. We interpret this as an equation of continuity for probability with j. A bidirectional traveling plane wave representation of. We focus our attention, once again, on the klein gordon and dirac equations with a plane wave field.
If you substitute the free particle solution into the klein gordon equation you get, unsurpisingly, the relation e 22p m. We will come back to the question what it actually describes later on. Secondorder hyperbolic partial differential equations wave equation linear wave equation 2. The nonlocality of the wave equation does not disturb the light cone structure, and the relativity principle of special. Notice that for photons the kleingordon equation becomes the wave equation.
Multiply the nonconjugated dirac equation by the conjugated wave function from the left and multiply the conjugated equation by the wave function from right and subtract the equations. Analytical study on the balancing principle for the nonlinear klein gordon equation with a fractional power potential. How to derive general solution to the klein gordon equation a. The analysis is applied to the special cases of x and sine gordon wave equations. Kleingordon equation, phase plane trajectories, hamiltonian value, traveling wave solutions some. Lets turn this around and ask, given the kleingordon equation, what is the most general plane wave solution. Kleingordon equation and wave velocity stack exchange. Now the solution obeys well the free field condition. Exact planewave solutions of the coupled maxwellklein. 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. The interaction of pseudo scalar particles with the electromagnetic field is described by the minimal substitution. Therefore is a vector in the particles direction with magnitude equal to the flux.
The klein gordon equation for a free particle has a simple planewave solution. For this case the right hand sides of the wave equations are zero. For particles per unit volume moving at velocity, have passing through a unit area per unit time particle flux. Dirac plane wave solution we now have simple solutions for spin up and spin down for both positive energy and. The kleingordon equation for a free particle has a simple planewave solution.
This is a plane wave multiplied by a four component spinor up note that the spinor depends on four momentum p. Substitution of this solution into the equation for the probability density yields. The theory has a classical solution which exhibits black hole formation by a soliton. A wave described by such a solution in one spatial dimension is sometimes called a \rightmover. Moreover, the conservation of the energy non relativistic e p2. Kg2 using kg can be expressed compactly as kg3 for plane wave solutions, the kg equation gives not surprisingly, the kg equation has negative energy solutions this is just what we started with in eq. Nonlinear wave equations with explicit periodic plane wave solutions are considered. Analytical study on the balancing principle for the. Solutions to common problems with probability interpretation and inde. Later, the equation was revived and is now known as klein gordon equation. Underlying physics the same for klein gordon equation, but paradox not so visible. This is recognized as the plane wave solution of the klein gordon equation in the. This is a lorentzinvariant property because we are dealing. Gordon and dirac equations are obtained for a particle moving in an arbitrary plane electromagnetic wave in vacuo plus a uniform static magnetic field parallel to the direction of propagation of the electromagnetic wave.
Klein gordon equation wave packet construction physics 452 justin peatross 32. As we can see now, the apparent problems were manifestations of the new physics. Fock also determined the gauge theory for the wave equation. The nonrelativistic equation for the energy of a free particle is. Stability, nonlinear waves, sine gordon equation, a wave equation. Black holes in two dimensional dilaton gravity and. The klein gordon equation applying to the relativistic equation for energy. In this paper, inspired by the observation of the structure of the solutions in a plane wave field, we develop a new method and obtain the analytical solution for the klein gordon equation and equivalently the action function of the solution for the dirac equation in this field, under a largest dynamical parameter condition that there exists an.
We will see below that the general solution of the equation u. It is acquired constraint conditions for the existence of solitons. Therefore, as long as energy and momentum follows the einsteins relation eq. Request pdf travelling wave solution for nonlinear kleingordon equation in this work, we construct explicit exact solutions for the nonlinear kleingordon equation by using a g g. Apr 06, 2020 the result we have here is the electromagnetic wave equation in 3dimensions. To leave a comment or report an error, please use the auxiliary blog. Topological and nontopological soliton solutions are obtained as well as jacobi elliptic function solutions. Solutions of the classical maxwell klein gordon equations are investigated for which the klein gordon field is assumed to be. How to derive the schrodinger equation plane wave solutions to the wave equation. As a result, the klein gordon equation predicts that a given plane wave solution can have either a positive or negative energy. In its free form, or including electromagnetic interactions, it describes all spin1 2 massive particles such as electrons and quarks for which parity is a symmetry. Relativistic quantum mechanics klein gordon equation youtube. Each component satisfies the wave klein gordon equation and a solution can be written as a constant spinor times the usual exponential representing a wave. If you substitute the free particle solution into the kleingordon equation you get, unsurpisingly, the relation e 22p m.
In section 2, during the course of the derivation of inces equation from the kleingordon equation. Exact solutions are presented of the kleingordon equation of a charged particle. 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. Analytical solution for kleingordon equation and action. Relativistic quantum mechanics kleingordon equation dirac. Stability of periodic plane wave solutions of wave equations. Stationary solutions of the kleingordon equation in a. Substituting this solution back into 6 leads back to the energy equation 1, which gives e2 and not just eitself.
The solutions for nonzero momentum are not as simple. The result we have here is the electromagnetic wave equation in 3dimensions. Klauber, student friendly quantum field theory, sandtrove press, 20. Is the kleingordon equation the same in all reference frames. Wangperiodic wave solutions to coupled kdv equations with. Pdf a new class of exact solutions of the kleingordon. How to derive general solution to the kleingordon equation. Later, the equation was revived and is now known as kleingordon equation. The kleingordon equation was one of the first attempts at producing a relativistic quantum theory.
The pseudodifferential operator square root of the klein gordon equation. A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation. Plane wave solutions of the free dirac equation assume solutions of the form. 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 section 2, during the course of the derivation of inces equation from the klein gordon equation. In other words, the klein gordon equation, if useful at all, does not describe the probability wave, which the schrodinger equation does, but describes something else. Besieris the bradley department of electrical engineering, virginia polytechnic institute and state university, blacksburg, virginia 24061 richard w. A novel approach to the synthesis of nondispersive wave packet solutions to the klein gordon and dirac equations amr m. Exact travelling wave solutions of the coupled kleingordon. Exact solutions are presented of the kleingordon equation of a charged particle moving in a transverse monochromatic plasmon wave of arbitrary high amplitude, which propagates in an underdense plasma. This equation is manifested not only in an electromagnetic wave but has also shown in up acoustics, seismic waves, sound waves, water waves, and fluid dynamics. Introduction the nonlinear kleingordon equation is one of the most important equations that arises in the study of theoretical physics 1. The mps pulse, for example, can be optimized so that it. However, pauli and weisskopf suggested 6 years after the development of diracs equation to interpret the 4current as a 4current charge.
Note that the probability of the system being in the. The wave equation is often encountered in elasticity, aerodynamics, acoustics, and electrodynamics. Jan 30, 2017 this paper obtains exact travelling wave solutions of five various forms of the generalized nonlinear kleingordon equations using jacobi elliptic functions. Thus, in general, we will have to resort to an approximate procedure. So, none of the objections that led to discarding the klein equation held up with time. In 1934 pauli and weisskopf 46 solved the prob lem by reinterpreting kleingordon equation in the context of quantum. New exact travelling wave solutions of nonlinear kleingordon. Because we have eliminated the matrices from the equation, this is an equation for each component of the dirac spinor. A novel approach to the synthesis of nondispersive wave. Travelling wave solution for nonlinear kleingordon equation. Aug 22, 2016 strictly speaking, the klein gordon equation should really be considered as the equation of motion for a classical or quantum field, not a wave equation, which is why doing all of this seems handwavy if we interpret it as the latter. Relativistic quantum mechanics klein gordon equation plane wave solution and equation of continuity m. 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. In what follows, we outline the derivation of kleingordon equation as a.
The dirac equation can be thought of in terms of a square root of the klein gordon equation. The geometry of black hole induced by a soliton is investigated. We recall that for the first time a set of solutions of these equations was found by volkov. Shortly after it was published in t he fall of 1925 pieter debye, a. Since we want the lagrangian density to be invariant under lorentz transformations a linear wave equation, the lagrangian density can contain only the. The pseudodifferential operator square root of the klein. Planewave solutions are readily found by inspection. Varro s, a new class of exact solutions of the kleingordon equation of a charged particle interacting with an electromagnetic plane wave in a medium. Apr 12, 2016 how to derive general solution to the klein gordon equation.
Travelling wave solutions of generalized kleingordon. A new class of exact solutions of the kleingordon equation. Qualitative analysis and exact traveling wave solutions for. Negative energy solutions given the plane wave 3, we found the kleingordon equation as the equation that it satis. Twodimensional dilaton gravity coupled to a klein gordon matter field with a quartic interaction term is considered. Because of this reason, the klein gordon equation was abandoned for a while. We explore kleingordon equation in the new framework of quasihermitian quantum mechanics. The general solution to the klein gordon equation would then be given by. Conclusion the infinite series method has been successfully applied here for solving the coupled klein gordon equation. The klein gordon equation can be generalized to describe a field in some potential v. A new class of exact solutions of the klein gordon equation of a charged particle interacting with an electromagnetic plane wave in a medium article pdf available in laser physics letters 111.
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