But its one of the most important basic questions, in my view. In chapter 2, we show that the dirac equation is the same as the klein gordon equation. Aalborg universitet generalization of the diracs equation. Its not easy to show how maxwell equation leads to the dirac equation, otherwise it would be in textbooks already. For a wave function with a given energy we have, e.
We establish local and global existence results for dirackleingordon equations in one space dimension, employing a null form estimate and a. This equation was actually discovered by erwin schrodinger before he discovered the schrodinger equation, that bears his name. Comparison among klein gordon equation, dirac equation and relativistic stationary schr. But we still have problems with the probability current. 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. The elementary solutions have a position dependence of the form eik r with the component of k in the field direction varying linearly with time. Relativistic quantum mechanics kleingordon equation dirac.
Is the dirac equation equivalent to the kleingordon. Proca equations the dirac equation predicts that the electron magnetic moment and its spin are related as 2 bs, while for normal. 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. Electromagnetic kleingordon and dirac equations in scale relativity. It is secondorder in space and time and manifestly lorentzcovariant. What is the relationship between the dirac equation and. The klein gordon equation is a relativistic version of schrodinger equation 1 that was presented as follows 2.
Mapping of the relativistic kinetic balance equations onto the kleingordon and secondorder dirac equations. Following 8,14,15, but by considering the chiral re presentation, the dirac equation in its covariant form 3 0 1 c,0 tkck k imc q t x c. Klein gordon equation dirac equation properties of. Pdf electromagnetic kleingordon and dirac equations in. The timedependent kleingordon and dirac equations are solved for the motion of a charged particle in a classical uniform electrostatic field of infinite extent. Solutions to common problems with probability interpretation and inde. It is now accepted that the klein gordon equation describes spin0 particles, whereas dirac s applies to spin12 particles. Comparison among kleingordon equation, dirac equation. Free particle in motion 295 normalization 297 example 5.
Oneparticle states and their lorentz transforma tions follow, leading to the klein gordon and the dirac equations for probability amplitudes. Since the dirac equation remains extremely important today, the derivationofthis equationisanimportantpoint. Electromagnetic kleingordon and dirac equations in scale relativity article pdf available in international journal of modern physics a 2522 september 2010. Schrodinger, kleingordon and dirac equations with different potential models 5 10. Pdf stationary solutions of the maxwelldirac and the.
Its solutions include a quantum scalar or pseudoscalar field, a field whose. 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. It is a quantized version of the relativistic energymomentum relation. Klein gordon equation with anticommutators pingback. Each of these components then satisfies a first order differential equation of time. The dirac equation can be thought of in terms of a square root of the klein gordon equation. Kleingordon and dirac equations with thermodynamic. We will come back to the question what it actually describes later on. Accordingly, the kleingordon kg equation and dirac equation are restudied and compared with the relativistic stationary schrodinger equation rsse.
The klein gordon and dirac equations in a semiinfinite lab x0, in the background metric ds2u2x. 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. Dirac equation in 1928 dirac tried to understand negative energy solutions by taking the squareroot of the klein gordon equation. Mapping of the relativistic kinetic balance equations onto. Solutions of klein gordon and dirac equations on quantum minkowski spaces. Mod01 lec01 introduction, the kleingordon equation. Aalborg universitet generalization of the diracs equation and sea. Dirac fields are in contrast to klein gordon fields for spin0 particles, such as the higgs field. The klein gordon equation is a secondorder differential equation in t and thus the initial values of.
If you have a disability and are having trouble accessing information on this website or need materials in an alternate format, contact web. The obvious way of creating a manifestly invariant wave equation is to start from the well known relativistic energy equation and. How to use the einstein energymomentum relation to derive the klein gordon wave equation. Now we want to study covariant equations of motion for spinor elds. As an application, by using the imethod of colliander, keel, staffilani, takaoka, and tao, we extend the work of tesfahun 23 on global existence below the charge class for the dirac klein gordon equation. He then tried to find solutions, but was dismayed to find that problems like nonpositivedefinite probability curre. 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. In the present work, we like to study the cauchy problem for the dirac klein gordon equations. For every solution to the dirac equation, its components will be a solution of the klein gordon equation, but the converse isnt true. Chiral dirac equation derived from quaternionic maxwell s. Relativistic quantum mechanics download ebook pdf, epub. Recall klein gordon 285 dirac equation 286 manipulation into a standard form 287 5.
Dirac and klein gordon equations with equal scalar and vector potentials a. String theory edit this section in rns string theory, the polyakov lagrangian density is added to the dirac lagrangian density, obtaining the rns, or ramondneveuschwarz lagrangian density. Iii we separate variables and solve the klein gordon equation. Let ax be the fourvector potential that accounts for electric and magnetic. Klein gordon equation is neither better nor worse than the dirac equation 1. The kleingordon equation is a relativistic version of schrodinger. Deriving the klein gordon equation relativistic quantum. The elementary solutions have a position dependence of the form eik. Comparison among kleingordon equation, dirac equation and.
In the nonrelativistic limit the generalized dirac equation gives the general. We study the thermodynamic quantities such as the helmholtz free energy, the mean energy and the specific heat for both the klein gordon, and dirac equations. The resulting equations are studied for the special. Pdf bound states of the kleingordon and dirac equations. It then introduces the dirac equation by linearizing the second order relativistic equation which reveals the spin, spin magnetic moment and the spinorbit. Separation of variables and exact solution of the klein. However, we need a form which is rst order in time. The equation describes all spinless particles with positive, negative, and zero charge. Any solution of the free dirac equation is, componentwise, a solution of the. The maxwell dirac system describes the interaction of an electron with its own electromagnetic field. Mod01 lec01 introduction, the kleingordon equation nptelhrd.
A general solution of kgd equation as a superposition of two dirac plane harmonic waves with different masses has been obtained. 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. Relativistic quantum mechanics begins with the klein gordon equation describing its features and motivating the need for a correct relativistic equation for the electron. Kleingordon equation mathematics and physics wiki fandom. The dirac equation and graphene tuesday, november 21, 2017. Klein gordon equation the klein gordon equation is the first important step from nonrelativistic quantum mechanics towards relativistic quantum mechanics. Thus it is not possible to have probability interpretation for klein gordon equation. Low regularity global solutions of the dirac klein gordon equations in one space dimension bournaveas, nikolaos and gibbeson, dominic, differential and integral equations, 2006 two energy conserving numerical schemes for the klein gordon zakharov equations chen, juan and zhang, luming, journal of applied mathematics, 20. Pdf dirac and kleingordon equations in curved space. We explore klein gordon equation in the new framework of quasihermitian quantum mechanics. Dirac and kleingordon equations with equal scalar and. For classical relativistic point particles, an interaction with these external. Kleingordon equation particles in exponentialtype molecule. The dirac equation is more restrictive than the klein gordon equation.
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