The klein gordon equation is then derived by a stochastic variational principle with the classical einstein action. We introduce some changes of variables, we show that a onedimensional linear potential for the relativistic system in a space deformed can be equivalent to the. There seems to be a formal analogy between our problem and a certain. Is the dirac equation equivalent to the kleingordon equation. As in the comments, fulfilment of the klein gordon equation is only a necessary condition for a field and it is fulfilled by all fields. The plane waves 10 are basic solutions and the field 9 is constructed by. Low regularity global solutions of the dirac klein gordon equations in one space dimension bournaveas, nikolaos and gibbeson, dominic, differential and integral equations, 2006. In 1934 pauli and weisskopf 46 solved the problem by reinterpreting kleingordon equation in the context of quantum. Let us now discuss the construction of the ground state and of the excitation spectrum. Dirac tried to get round this by finding a field equation which was linear in the operators.
Convergence of scattering operators for the klein gordon equation with a nonlocal nonlinearity sasaki, hironobu, differential and integral equations, 2006. Any solution of the free dirac equation is, componentwise, a solution of the free klein gordon equation. The spinor of mass minimally coupled to gravity is described by the dirac equation with. 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. Any solution of the free dirac equation is, componentwise, a solution of the free. Indeed, the klein gordon equation only expresses the mass shell constraint and nothing else. The discovery of dirac equation and its impact on presentday physics reproduced with permission from dirac cmd feynman. Fourmomentum a free particle in the nonrelativistic limit obeys. Comparison among klein gordon equation, dirac equation and relativistic stationary schr. Kleingordon and dirac equations with thermodynamic. Iii we separate variables and solve the klein gordon equation. Solutions of klein gordon equation free particle solutions for u 0. Klein gordon equation and particles with spin physics forums. Derivation of klein gordon equation or relativistic schrodinger equation, klein gordan equation is relativistic wave equation.
Klein gordon equations in one space dimension, employing a null form estimate and a. However, the klein gordon equation has issues when interpreted as a standard wave equation. Comparison among kleingordon equation, dirac equation and. The green function for klein gordon dirac equation is obtained. Pdf dirac and kleingordon equations in curved space. In contrast to the klein gordon case, the situation is more complicated in the case of the dirac equation and this complication is basically due to the square root, which appears in the expressions of and. Lorentzinvariant differential equations which take the place of the schrodinger equation of a par. 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. For a free particle, the kleingordon equation is written. Dirac equation and quantum relativistic effects in a single. We compare the solutions and the energy spectra obtained with the corresponding ones from the dirac equation for a spin onehalf relativistic particle.
We study the thermodynamic quantities such as the helmholtz free energy, the mean energy and the specific heat for both the. Derivation of dirac, kleingordon, schrodinger, diffusion. We will come back to the question what it actually describes later on. Pdf kleingordon type equations with a singular time. For example, the dirac equation for an electron implies the klein gordon equation, but not conversely. The components of every free field satistfy the klein gordon equation, irrespective of their spin. The background to the story of the dirac equation is the story of. Generalized klein gordon and dirac equations from nonlocal kinetic approach. Separation of variables and exact solution of the klein. The kleingordon equation is a relativistic wave equation, related to the schrodinger equation. He then tried to find solutions, but was dismayed to find that problems like nonpositivedefinite probability curre. This generalizes to particles of any spin due extension to the bargmannwigner equations. Dirac equation and quantum relativistic effects in a single trapped ion l.
Inequivalence of kleingordon and dirac equation hans detlef hutten. A universal quantum wave equation that yields dirac, klein gordon, schrodinger and quantum heat equations is derived. Quantization of the free dirac field university of illinois. As seen in the dispersion relation for graphene, for low energies near the dirac point, electrons obey a dirac equation with m 0 and c v f, the fermi velocity. Kleingordon equation article about kleingordon equation. We explore klein gordon equation in the new framework of quasihermitian quantum mechanics. The problems with the klein gordon equation led dirac to search for an alternative relativistic wave equation in 1928, in which the time and space derivatives are. Dirac fine structure increased 3p32, 3d32 2sl2 12 lamb shift increased hyperfine structure increased bohr levels. Kleingordon and dirac equations with thermodynamic quantities. The discovery of dirac equation and its impact on present. 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. Is the dirac equation equivalent to the kleingordon. Relativistic qm the klein gordon equation 1926 1 relativistic qm the klein gordon equation 1926 scalar particle field j0 energy eigenvalues 1927 dirac tried to eliminate negative solutions by writing a relativistic equation linear in e a theory of fermions 1934 pauli and weisskopf revived kg equation with.
This equation is associated with the relativistic relationship between the energy. Electromagnetic kleingordon and dirac equations in scale. For a free fermion the wavefunction is the product of a plane wave and a. Relativistic quantum mechanics download ebook pdf, epub. But we have four dirac operators and only three pauli operators. In the present work, we like to study the cauchy problem for the dirac klein gordon equations. Previously, quantum motion equations schrodinger, klein gordon, dirac, pauli have been derived as geodesic equations written with a quantumcovariant derivative operator. The equation does not form the basis of a consistent quantum relativistic oneparticle theory. The dirac equation the hydrogen atom why do we need the dirac equation. The development of quantum mechanics is presented from a historical perspective. Solutions to common problems with probability interpretation and inde. The case with the dominating klein gordon term is considered. Solving the klein gordon equation is quite similar to solving schrodingers equation.
We show here that if one tries to derive the electromagnetic dirac equation from the kleingordon one as for the free particle motion, i. Ppt relativistic qm the klein gordon equation 1926. The overall fourmomentum transfer is still q, but it has to be divided between the two. 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. The interacting klein gordon equation in the rst section is in fact diracs starting point in his 1928 paper where he introduces his famous dirac equation.
Then, the nature of gauge transformations, of gauge fields and of conserved charges have been given a geometric meaning in terms of a scalecovariant derivative tool. Since we want the lagrangian density to be invariant under lorentz transformations a linear wave equation, the lagrangian density can contain only the. In this study, we investigate the relativistic klein gordon equation analytically for the dengfan potential and hulthen plus eckart potential under the equal vector and scalar potential conditions. Separation of variables for the kleingordon and dirac. Chapter 11 em lorentz force derived from klein gordons. Dirac and kleingordon equations with equal scalar and vector potentials a. In its free form, or including electromagnetic interactions, it describes all. These equations are related by complex transformation of space, time and mass. For the love of physics walter lewin may 16, 2011 duration.
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. We establish local and global existence results for dirackleingordon equations in one space dimension, employing a null form estimate and a. Plane wave solutions of the free dirac equation assume solutions of the form. The discovery of dirac equation and its impact on presentday physics. The elementary solutions have a position dependence of the form eik. We show here that if one tries to derive the electromagnetic dirac equation from the klein gordon one as for the free particle motion, i.
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. Since it is secondorder in time like the classical wave equation from classical lorentzinvariant em, probability density is not explicitly always conserved. The problems with the kleingordon equation led dirac to search for an. This is a case where an incautious assumption at the outset led to misinterpretation. The dirac equation was an attempt to solve this issue.
In particle physics, the dirac equation is a relativistic wave equation derived by british physicist paul dirac in 1928. Furthermore, any component of any solution to the free dirac equation for a spin12 particle is automatically a solution to the free kleingordon equation. Klein gordon for spin1 particle photon stack exchange. 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. Bound state solutions of threedimensional kleingordon. A direct proof of global existence for the dirac klein gordon equations in one space dimension fang, yungfu, taiwanese journal of mathematics, 2004. Exact solution of kleingordon and dirac equations with. Klein gordon equation derivation and continuity equations 3 energies, were taken to be major problems with the klein gordon equation which led to it being disregarded initially as a valid relativistic 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. Stochastic derivation of the kleingordon equation from. Now, the fourcurrent density fj g3 0 associated with the dirac equation for a.
We show that these coordinates can be associated with a system with constant acceleration. The dirac equation is one of the two factors, and is conventionally taken to be p m 0 31 making the standard substitution, p. Relativistic quantum mechanics kleingordon equation dirac. There is no problem at least at the level of the free field equation. It should be added, however, that it was dirac who found most of the additional insights. Dec 31, 2011 it may be worth mentioning that just as the 1component non relativistic schrodinger equation can be tweaked to accommodate spin by adding on a itex \sigma. If youve never seen this, try working out the following. Mod01 lec01 introduction, the kleingordon equation. A complex field remains a complex field as expected for nonrelativistic problems where we are all familiar with wave functions and their interpretation.
Klein gordon equation dirac equation the hydrogen atom. Click download or read online button to get relativistic quantum mechanics book now. 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. 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. 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. The kleingordon equation is a linear homogeneous secondorder partial differential equation with constant coefficients. Download fulltext pdf stationary solutions of the maxwell dirac and the kleingordondirac equations article pdf available in calculus of variations 43.
The non relativistic limit of the klein gordon equation kg is the schrodinger equation s. Boththe klein gordon and dirac wave equations arediscussed in terms of the spherical harmonics with spinweight. 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. Download fulltext pdf klein gordon type equations with a singular timedependent potential article pdf available in rendiconti dellistituto di matematica delluniversita di trieste xxxix. Comparison among kleingordon equation, dirac equation. Dirac himself remarked in one of his talks that his equation was more intelligent than its author. Relativistic quantum mechanics is developed, including the klein gordon equation and up to the dirac equation. The relativistic wave equation of the electron ranks amo. Introduction to relativistic quantum mechanics 250 an equation in relativistic quantum field theory.
Regularity results for the dirackleingordon equations. There is no known such theory for particles of any spin. The fact that these two equations, for free particles, are constructed using two objects. A direct proof of global existence for the dirackleingordon equations in one space dimension yungfu fang1 abstract. Now, suppose a solution to the klein gordon equation is a free particle with energy eand momentum p ip ne x 11 1. 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. For every solution to the dirac equation, its components will be a solution of the klein gordon equation, but the converse isnt true. The dirac equation has several signi cant consequences, for instance, the existence of antiparticles and spin. The dirac and kleingordon equations with equal scalar and. The separation of variables for the klein gordon and dirac equations, in the presence of electromagnetic fields, for a class of curvilinear coordinate systems with a null coordinate is presented. What is the relationship between the dirac equation and. The dirac equation the problems with the klein gordon equation all came about because of the square root required to get the energy. Gordon equation in terms of differential operators. Kleingordon equation mathematics and physics wiki fandom.
Pdf stationary solutions of the maxwelldirac and the. To handle this problem, dirac appealed to the fact that electrons are fermions and no two electrons could take the same quantum state the pauli exclusion principle. 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. Spin comes in when one looks at what happens to the components. The non relativistic limit of the real kleingordon equation. Introduction to relativistic quantum mechanics smokey. The kleingordon equation is applied in the description of mesons and corresponding fields. It is the purpose of the present article to solve the klein gordon and dirac equations in. What is the relationship between the dirac equation and the. Physics 221b spring 2020 notes 45 introduction to the dirac. This site is like a library, use search box in the widget to get ebook that you want. The new symmetry exhibited by these equations is investigated.
Lorentz transformations the free particle kleingordon equation 10. Accordingly, we obtain the energy eigenvalues of the molecular systems in different states as well as the normalized wave function in terms of the generalized laguerre polynomials function through. The dirac equation provides a description of elementary spinhalf particles, such as electrons. A direct proof of global existence for the dirackleingordon. Inequivalence of klein gordon and dirac equation hans detlef hutten bach abstract. We say the charge carriers in this case are \emergent dirac fermions. A critical case on the dirac klein gordon equations in one space dimension fang, yungfu and huang, hsiuchuan, taiwanese journal of mathematics, 2008. 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. Thus we study a system where we have two independent spins, one with the spin. A new proof of global existence for the dirac kleingordon. The dirac equation a great deal more was hidden in the dirac equation than the author had expected when he wrote it down in 1928.
We obtain exact solutions of the dirac and kleingordon equations for a static diagonal metric. Nevertheless, somehow the physical importance of the potentials, rather then only the e and b elds, wasnt wider spread to the broader community. Electromagnetic kleingordon and dirac equations in scale relativity. Toobtain the dirac wave equation we use the factorizationof the secondorder invariant casimir operatorassociated to the fantappiede sitter group. Pio neers in quantum mechanics, edited by ranabir dull and asim k ray, wiley eastern limited, 1993. In particular, every component of the dirac equations solves the klein gordon equation. The interaction of pseudo scalar particles with the electromagnetic field is described by the minimal substitution. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was.