The convergence of solutions of the dirac and kleingordon equations to solutions of the pauli and schrodinger equations in the nonrelativistic limit is discussed. Generalized dirac and kleingordon equations for spinor. We show here that if one tries to derive the electromagnetic dirac equation from the kleingordon one as for the free particle motion, i. Mod01 lec01 introduction, the kleingordon equation. I was wondering if there is any way to interpret the fact that the klein gordon equation is a 2nd order pde in time. There is no known such theory for particles of any spin. Dirac s equation would then need to begin in the form. He then tried to find solutions, but was dismayed to find that problems like nonpositivedefinite probability curre. The 4vector coulomb, the scalar linear, and the equally mixed scalar and 4vector powerlaw potentials are used in kg and dirac equations. Overall, the kleingordon equation appears to be a good relativistic replacement for the. Jun 21, 2016 the solutions to the dirac equation are also solutions of the klein gordon equation, which is the equation of motion for the real scalar field.
I go over fourvector notation, fourgradient, and writing the equation in terms of the dalambertian. However, we need a form which is rst order in time. In particle physics, the dirac equation is a relativistic wave equation derived by british physicist paul dirac in 1928. Chapter 11 em lorentz force derived from klein gordons. What is the relationship between the dirac equation and. We compare the solutions and the energy spectra obtained with the corresponding ones from the dirac equation for a spin onehalf relativistic particle. For a free particle, each component of the dirac spinor satisfies the kleingordon equation. 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. 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. Stationary solutions of the kleingordon equation in a. Start from the dirac equation and attempt to develop an equation to show that each component has the free particle exponential. Now, suppose a solution to the kleingordon equation is a free particle. Klein gordon and dirac equations with thermodynamic quantities article pdf available in fewbody systems 572 october 2015 with 256 reads how we measure reads. It is immediate that if the mass m vanishes, as in the case of a photon, kge reduces to the standard electromagnetic wave equation 15 1 c2.
Notes 49 solutionsof the dirac equation and their properties 1. For a free particle, the kleingordon equation is written this equation is associated with the relativistic relationship between the energy. How to use the einstein energymomentum relation to derive the klein gordon wave equation. A novel approach to the synthesis of nondispersive wave. But the topic was left with lots of room left for deeper exploration. A parameter \lambda0,1 is introduced in such a way that one can obtain the klein gordon kg bound states from dirac bound states. The dirac equation provides a description of elementary spinhalf particles, such as electrons.
The klein gordon and dirac equations in a semiinfinite lab x0, in the background metric ds2u2x. Physics 221b spring 2020 notes 49 solutionsof the dirac. This paper presents the derivation of the schrodinger, kleingordon and dirac equations of particle physics, for free particles, using classical methods. It is secondorder in space and time and manifestly lorentzcovariant. We will do this by making a second order differential equation, which turns out to be the kleingordon equation.
The elementary solutions have a position dependence of the form eik. The new generalized dirac and klein gorodon equations are obtained by using nonzero phase factors, and these equations reduce to the standard dirac and klein gordon equations once the phase function is set to. Free particle in motion 295 normalization 297 example 5. Quantum mechanics of kleingordon equation katedra fyziky fjfi. Large and small spinor components 298 pauli limit 300. Initially, it was assumed that the 4dimensional space introduced by dirac. The dirac equation and the prediction of antimatter. Dirac pictures, the time evolution of the field operators is given by heisenberg equations of. Why do dirac spinors obey the kleingordon equation. The klein gordon equation is a secondorder differential equation in t and thus the initial values of. The dirac equation we will try to find a relativistic quantum mechanical description of the electron. The dirac equation can be thought of in terms of a square root of the kleingordon equation. We will come back to the question what it actually describes later on.
The derivations are based on the assumption that these wave equations are homogeneous and soluble via separation of variables. I can see that the converse is not true, but why do spinors follow the equation for realfield particles. The dirac and kleingordon equations with equal scalar and. Its solutions include a quantum scalar or pseudoscalar field, a field whose. Regularity results for the dirackleingordon equations. Klein gordon equation dirac equation properties of. Is the dirac equation equivalent to the kleingordon equation. The kleingordon and dirac equations in a semiinfinite lab x0, in the background metric ds2u2x. 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.
Dirac fine structure increased 3p32, 3d32 2sl2 12 lamb shift increased hyperfine structure increased bohr levels. Oct 21, 2015 kleingordon and dirac equations with thermodynamic quantities article pdf available in fewbody systems 572 october 2015 with 256 reads how we measure reads. Introduction in notes 45 we introduced the dirac equation in much the same manner as dirac himself did, with the motivation of curing the problems of the klein gordon equation. The resulting equations are studied for the special. Historically, dirac 1928 was looking for a covariant wave equation that was. Note that any solution to the free dirac equation is automatically a solution to the free klein gordon equation but the converse is not true. The four normalized solutions for a dirac particle at rest are. Free particle at rest look for free particle solutions to the dirac equation of form. Dirac equation in 1928 dirac tried to understand negative energy solutions by taking the squareroot of the klein gordon equation.
Em lorentz force derived from klein gordon s equation 11. Nov 16, 2017 to illustrate this point, we solve the problem of a spin zero relativistic particle in a one and threedimensional box using the klein gordon equation in the feshbachvillars formalism. For this to be a reasonable formulation of relativistic qm, a free particle must also obey, i. For a free particle, the equation is p p m 0c2 0 1 where p is the fourmomentum operator. 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. Several publications concerning this subject appeared 39, 43, 44, 40 or 56, 57, 67. 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. Relativistic quantum mechanics kleingordon equation dirac. A generalized klein gordon equation with a closed system. Dirac equation and hydrogen atom physics department. A novel approach to the synthesis of nondispersive wave packet solutions to the klein gordon and dirac equations amr m.
The dirac equation and the kleingordon kg equation are the most frequently used wave equations for the description of particle dynamics in relativistic quantum mechanics. Comparison among kleingordon equation, dirac equation and relativistic stationary schr. Hence for the dirac equation to be consistent with the kg equation require. It is a quantized version of the relativistic energymomentum relation. Pdf electromagnetic kleingordon and dirac equations in scale. Physics 221b spring 2020 notes 50 holetheory and second. Dirac current, the dirac adjoint and the use of the dirac equation to prove the continuity equation for the dirac current, its generalization into a dirac probability or. Holetheory and second quantization of the dirac equation 1. The maxwell dirac system describes the interaction of an electron with its own electromagnetic field. For solutions in various potentials, see the literature. What is the relationship between the dirac equation and the. Dirac and kleingordon equations with equal scalar and.
This is the kleingordon equation which describes spin zero massive. The kleingordon equation successfully describes spin 0 particles in relativistic. So, in a typical physics curriculum, it is through such courses. An abstract theory of these equations is developed which is general enough to allow physical space to be an arbitrary complete riemannian manifold.
For every solution to the dirac equation, its components will be a solution of the klein gordon equation, but the converse isnt true. Solutions to common problems with probability interpretation and inde. Dirac and kleingordon equations with equal scalar and vector. In the dirac case, the discrete solution converges to the continuum propagator and it can also be rewritten in the form. Comparison among kleingordon equation, dirac equation. Historical background on klein gordon and dirac equations klein gordon and dirac equations are two of the most important eld equations arising in relativistic quantum mechanics. Relativistic shiftedl expansion technique for dirac and. Deriving the klein gordon equation relativistic quantum. Kleingordon equation nonrelativistic limit pingback. 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.
Comparison among klein gordon equation, dirac equation and relativistic stationary schr. Lorentz transformations the free particle kleingordon equation 10. The dirac equation can be thought of in terms of a square root of the klein gordon equation. The dirac equation and the klein gordon kg equation are the most frequently used wave equations for the description of particle dynamics in relativistic quantum mechanics. Pdf kleingordon and dirac equations with thermodynamic. The fact that these two equations, for free particles, are constructed using two objects.
By introducing the concept of system mass into the klein gordon equation with the scalar and vector potentials, we prove that if the. Introduction in these notes we provide the ultimate resolution of the di. This paper presents the derivation of the schrodinger, klein gordon and dirac equations of particle physics, for free particles, using classical methods. Proca equations the dirac equation predicts that the electron magnetic moment and its spin are related as 2 bs, while for normal. Global wellposedness and parametrices for critical. Nov 12, 2015 we study the thermodynamic quantities such as the helmholtz free energy, the mean energy and the specific heat for both the kleingordon, and dirac equations. Pdf dirac and kleingordon equations in curved space. Plane wave solutions of the free dirac equation assume solutions of the form. For mkg, this work extends to the general case m 2 0 the results of kriegersterbenz. When sr and vr are both coulombic, we obtain all the exact solutions.
We interpret this as an equation of continuity for probability with j. Klein gordon equation with scalar sr and vector potentials vr. Quantization of the free dirac field eduardo fradkin. We explore klein gordon equation in the new framework of quasihermitian quantum mechanics.
Unlike schrodinger equation, the free kleingordon equation is invariant. Jun 04, 2018 how to use the einstein energymomentum relation to derive the klein gordon wave equation. I mean, normally you would expect that as soon as you fix the initial wavefunction, then the evolution of your system is fixed for all further moments in time. Dirac in his effort to discover an equation that was free of the problems seen with the. The dirac equation is more restrictive than the klein gordon equation. Kleingordon and dirac equations with thermodynamic. In its free form, or including electromagnetic interactions, it describes all spin 1 2 massive particles such as electrons and quarks for which parity is a symmetry. Comparison among kleingordon equation, dirac equation and. Besieris the bradley department of electrical engineering, virginia polytechnic institute and state university, blacksburg, virginia 24061 richard w.
Quantum field theory i babis anastasiou institute for theoretical physics, eth zurich, 8093 zurich, switzerland email. The new generalized dirac and klein gorodon equations are obtained by using nonzero phase factors, and these equations reduce to the standard dirac and klein gordon equations once the phase function is set to zero. This is consistent with the relativistic energy relation. Apoorva d patel,department of physics,iisc bangalore.
Kleingordon equation derivation and continuity equations 3 energies, were taken to be major problems with the kleingordon equation which led to it being disregarded initially as a valid relativistic equation. Several anomalies are highlighted and resolutions proposed. 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. Pdf a discrete scheme for the dirac and kleingordon. The solutions to the dirac equation are also solutions of the klein gordon equation, which is the equation of motion for the real scalar field. I go over fourvector notation, fourgradient, and writing the equation. The equation does not form the basis of a consistent quantum relativistic oneparticle theory.