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