Deriving the klein gordon equation relativistic quantum. Mod01 lec01 introduction, the kleingordon equation. The klein gordon equation is a secondorder differential equation in t and thus the initial values of. It is secondorder in space and time and manifestly lorentzcovariant. For classical relativistic point particles, an interaction with these external.
However, we need a form which is rst order in time. The dirac equation can be thought of in terms of a square root of the klein gordon equation. Accordingly, the kleingordon kg equation and dirac equation are restudied and compared with the relativistic stationary schrodinger equation rsse. Pdf stationary solutions of the maxwelldirac and the. Large and small spinor components 298 pauli limit 300 higher order terms 301. Comparison among kleingordon equation, dirac equation. Relativistic quantum mechanics begins with the klein gordon equation describing its features and motivating the need for a correct relativistic equation for the electron. Its not easy to show how maxwell equation leads to the dirac equation, otherwise it would be in textbooks already. Klein gordon equation dirac equation properties of. Solutions of klein gordon and dirac equations on quantum minkowski spaces. Bound states of the kleingordon and dirac equations for potential v r ar2br1. Iii we separate variables and solve the klein gordon equation.
The dirac equation is more restrictive than the 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. Mapping of the relativistic kinetic balance equations onto the kleingordon and secondorder dirac equations. Klein gordon equation is neither better nor worse than the dirac equation 1. I go over fourvector notation, fourgradient, and writing the equation. Any solution of the free dirac equation is, componentwise, a solution of the.
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. He then tried to find solutions, but was dismayed to find that problems like nonpositivedefinite probability curre. The klein gordon and dirac equations in a semiinfinite lab x0, in the background metric ds2u2x. The kleingordon equation is a relativistic version of schrodinger. Aalborg universitet generalization of the diracs equation. Comparison among kleingordon equation, dirac equation and. Because of this reason, the klein gordon equation was abandoned for a while. 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. How to use the einstein energymomentum relation to derive the klein gordon wave equation. But we still have problems with the probability current. Relativistic quantum mechanics download ebook pdf, epub. The kleingordon equation is a relativistic generalisation of the schrordinger equation to relativistic quantum mechanics. Dirac and kleingordon equations with equal scalar and. We explore klein gordon equation in the new framework of quasihermitian quantum mechanics.
Thus it is not possible to have probability interpretation for klein gordon equation. Solutions to common problems with probability interpretation and inde. 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. The dirac equation and graphene tuesday, november 21, 2017. In the present work, we like to study the cauchy problem for the dirac klein gordon equations. Oneparticle states and their lorentz transforma tions follow, leading to the klein gordon and the dirac equations for probability amplitudes. The elementary solutions have a position dependence of the form eik. Klein gordon equation the klein gordon equation is the first important step from nonrelativistic quantum mechanics towards relativistic quantum mechanics. This equation was actually discovered by erwin schrodinger before he discovered the schrodinger equation, that bears his name. Electromagnetic kleingordon and dirac equations in scale relativity article pdf available in international journal of modern physics a 2522 september 2010. Relativistic quantum mechanics kleingordon equation dirac.
Aalborg universitet generalization of the diracs equation and sea. What is the relationship between the dirac equation and. 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. It is a quantized version of the relativistic energymomentum relation. Pdf dirac and kleingordon equations in curved space.
We establish local and global existence results for dirackleingordon equations in one space dimension, employing a null form estimate and a. Each of these components then satisfies a first order differential equation of time. In the nonrelativistic limit the generalized dirac equation gives the general. The resulting equations are studied for the special. The maxwell dirac system describes the interaction of an electron with its own electromagnetic field. 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.
Kleingordon and dirac equations with thermodynamic. 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. 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. In chapter 2, we show that the dirac equation is the same as the klein gordon equation. Chiral dirac equation derived from quaternionic maxwell s. Since the dirac equation remains extremely important today, the derivationofthis equationisanimportantpoint. For every solution to the dirac equation, its components will be a solution of the klein gordon equation, but the converse isnt true. 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.
It is now accepted that the klein gordon equation describes spin0 particles, whereas dirac s applies to spin12 particles. The timedependent kleingordon and dirac equations are solved for the motion of a charged particle in a classical uniform electrostatic field of infinite extent. Dirac fine structure increased 3p32, 3d32 2sl2 12 lamb shift increased hyperfine structure increased bohr levels. For a wave function with a given energy we have, e. Its solutions include a quantum scalar or pseudoscalar field, a field whose. The obvious way of creating a manifestly invariant wave equation is to start from the well known relativistic energy equation and.
But its one of the most important basic questions, in my view. It then introduces the dirac equation by linearizing the second order relativistic equation which reveals the spin, spin magnetic moment and the spinorbit. Also, the recent study of the relativistic wave equation in the recent years. 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. Free particle in motion 295 normalization 297 example 5. Dirac and klein gordon equations with equal scalar and vector potentials a. If you have a disability and are having trouble accessing information on this website or need materials in an alternate format, contact web.
The klein gordon equation is a relativistic version of schrodinger equation 1 that was presented as follows 2. Recall klein gordon 285 dirac equation 286 manipulation into a standard form 287 5. Pdf electromagnetic kleingordon and dirac equations in. Comparison among klein gordon equation, dirac equation and relativistic stationary schr. Is the dirac equation equivalent to the kleingordon. 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. In quantum field theory, it is a relativistic field equation for a spin0 field klein gordon field. The equation describes all spinless particles with positive, negative, and zero charge. We will come back to the question what it actually describes later on. Dirac equation in 1928 dirac tried to understand negative energy solutions by taking the squareroot of the klein gordon equation. 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. Separation of variables and exact solution of the klein. Schrodinger, kleingordon and dirac equations with different potential models 5 10.
Proca equations the dirac equation predicts that the electron magnetic moment and its spin are related as 2 bs, while for normal. Let ax be the fourvector potential that accounts for electric and magnetic. 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. Kleingordon equation particles in exponentialtype molecule. Now we want to study covariant equations of motion for spinor elds.
Dirac fields are in contrast to klein gordon fields for spin0 particles, such as the higgs field. Generalized dirac and kleingordon equations for spinor. Mapping of the relativistic kinetic balance equations onto. Kleingordon equation mathematics and physics wiki fandom.