The non relativistic limit of the klein gordon equation kg is the schrodinger equation s. I just have a question regarding assessing the non relativistic limit of the klein gordon equation. Lax introduction in this paper we give a mathematical proof that the kleingordon and dirac equations of relativistic quantum mechanics have. We now want to reconcile the principles of quantum mechanics with special relativity. However, the equation does not carry a probabilistic interpretation. There is a close algebraic analogy between 1 the helmholtz equation governing scalar optics and the kleingordon equation and 2 the matrix form of maxwells equations governing. Nonrelativistic limit of the nonlinear kleingordon. On timesplitting pseudospectral discretization for nonlinear klein gordon equation in nonrelativistic limit regime volume 16 issue 2 xuanchun dong, zhiguo xu, xiaofei zhao. In this paper we give a mathematical proof that the kleingordon and dirac equations of relativistic quantum mechanics have the correct nonrelativistic limits. For a long time, it was believed that the dirac equation was. The non relativistic limit of the dirac equation one important requirement for the dirac equation is that it reproduces what we know from non relativistic quantum mechanics. Numerical methods for computing ground states and dynamics of nonlinear relativistic hartree equation for boson stars, j.
The schrodinger newton equation as nonrelativistic. Pdf singular limits of the kleingordon equation researchgate. Overall, the kleingordon equation appears to be a good relativistic replacement for the nonrelativistic schr. I just have a question regarding assessing the nonrelativistic limit of the kleingordon equation. The full consistency with quantum mechanics will eventually be recovered by. 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 conclusion that the kleingordon equation is a relativistic extension of the schrodinger. Pauli equation in the nonrelativistic limit, spin can be introduced using the pauli equation which describes the interaction of a spin 12 particle with an external electromagnetic. Pasquali 1universit a dellaquila 2universit a degli studi di milano 2016, march 22 d.
Nonexistence of a probability for the klein gordon equation. A complex field remains a complex field as expected for non relativistic problems where we are all familiar with wave functions and their interpretation. Nonrelativistic limit of klein gordon and massless limits. Note that we have derived this equation from something that did give the right answers so. The paper also shows that in classical limit the hydrodynamic kleingordon equations lead to the madelung pseudopotential 1 as well as to the quantum pseudo potential for a. The non relativistic viewpoint implicit in the form of the coupling in eq. For the complete study of the non relativistic limit of the cauchy problem for the nonlinear klein gordon equation we will refer to machiharanakanishiozawa 20 see also 24, 26 and references. Nonrelativistic limit of klein gordon equation kit. Limits used to find nonrel limit of the kleingordon equation. Nonrelativistic limit of the nonlinear kleingordon equation. We study the nonrelativistic limit of the cauchy problem for the nonlinear kleingordon equation and prove that any finite energy solution converges to the corresponding solution of the nonlinear schrodinger equation in the energy space, after the infinite oscillation in time is removed. In page 42 of david tongs lectures on quantum field theory, he says that one can also derive the schrodinger lagrangian by taking the nonrelativistic limit of the complex.
Relativistic quantum mechanics janos polonyi university of strasbourg dated. 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. Apparently, schrodinger tried out klein equation before proceeding with his nonrelativistic equations, but dropped it seeing many problems and never published it this equation was discarded in the community as faulty in addition, it did not leave any room for spi n. In the non relativistic limit, the formalism reduces naturally to that of the schr\odinger equation. Fourmomentum a free particle in the non relativistic limit obeys. On timesplitting pseudospectral discretization for nonlinear. About the nonrelativistic limit of the phase velocity of. The schrodinger equation is not relativistically invariant. Cant the negative probabilities of klein gordon equation be avoided. On the nonrelativistic limits of the kleingordon and. On the nonrelativistic limits of the kleingordon and dirac. Journal of mathematical analysis and applications 71, 3647 1979 on the nonrelativistic limits of the klein gordon and dirac equations andrew y.
Pauli and weisskopf in 1934 showed that kleingordon equation describes a spin0 scalar eld. Nonrelativistic limit in the energy space for nonlinear. Note that we have derived this equation from something that did give the right answers so we expect the dirac equation to pass this test. Lecture 22 relativistic quantum mechanics theory of condensed. Klein gordon equation the schrodinger equation only describes particles in the non relativistic limit.
The paper also shows that in classical limit the hydrodynamic klein gordon equations lead to the madelung pseudopotential 1 as well as to the quantum pseudo potential for a charged particle given by janossy 2. Deriving the klein gordon equation relativistic quantum. Heuristic derivation of the klein gordon equation 11 b. Stationary solutions of the kleingordon equation in a. The schrodingernewton equation as nonrelativistic limit of selfgravitating klein gordon and dirac fields article pdf available in classical and quantum gravity 2921 june 2012 with 251 reads. The problem of the equivalence between the relativistic schrodinger equation and the kleingordon equation is also addressed by using arguments which can be understood by undergraduate students.
The kleingordon equation is lorentz invariant for two reasons. Chapter 7 relativistic quantum mechanics in the previous chapters we have investigated the schr. That is where cryptohermitian approach can be applied. This is a case where an incautious assumption at the outset led to misinterpretation. It discusses properties of the equation and its continuity equation based on a comparison with the schrodinger equation and its continuity equation. Also we would like to have a consistent description of the spin of the electron that in the nonrelativistic theory has to be added by hand. Pdf nonrelativistic limit of kleingordonmaxwell to. We consider numerics asymptotics for the rotating nonlinear klein gordon rkg equation, an important pde in relativistic quantum physics that can model a rotating galaxy in. Introduction to relativistic quantum mechanics 250 an equation in relativistic quantum field theory. Pasquali 1universit a dellaquila 2universit a degli studi di milano. For the complete study of the nonrelativistic limit of the cauchy problem for the nonlinear kleingordon equation we will refer to machiharanakanishiozawa 20 see also 24, 26 and references.
In the limit m 0, the dirac equation reduces to the weyl equation, which describes relativistic massless spin 1. Notes on relativistic quantum mechanics module 4 of. It is obtained in the nonrelativistic limit of the dirac equation by assuming the presence of an electromagnetic. The relativistic quantum hydrodynamic representation of klein. Nonrelativistic limit an overview sciencedirect topics. In the nonrelativistic limit, the formalism reduces naturally to that of the schr\odinger equation. It is secondorder in space and time and manifestly lorentzcovariant. Nonrelativistic limit in the energy space for nonlinear klein. We develop the nonrelativistic limit of the dirac equation and action in the. Asymptotic approximating schemes for the kleingordon. Kleingordon equation nonrelativistic limit pingback. Nonrelativistic limit of scattering theory for nonlinear.
What is the role of the factor eimt used in the derivation of s equation from. We use and motivate a splitting of the scalar kleingordon field into a sum of two fields, corresponding, in the physical. In the nonrelativistic limit in the static gravitational field, with the. Nonrelativistic limit of the klein gordon equation. Schoene1 general motors research laboratories, warren, michigan 48090 submitted by peter d. To describe the particle at particle colliders we need to incorporate special theory of relativity 8 2. Comparison among kleingordon equation, dirac equation and.
Journal of mathematical analysis and applications 71, 3647 1979 on the nonrelativistic limits of the kleingordon and dirac equations andrew y. We consider numerics asymptotics for the rotating nonlinear kleingordon rkg equation, an important pde in relativistic quantum physics that can model a rotating galaxy in. About the non relativistic limit of the phase velocity of matter waves. A complex field remains a complex field as expected for nonrelativistic problems where we are all familiar with wave functions and their interpretation. For a long time, it was believed that the dirac equation was the only valid equation for massive particles. In particular, in 18 machiharanakanishiozawa proved that any nite energy solution converges to the corresponding solution of the nonlinear schr odinger equation in the energy space, after in nite oscillations in time are removed. The relativistic quantum hydrodynamic representation of. One can show that in the non relativistic limit, the known expressions for the probability density and current are recovered. Hi, i have a conceptual question about taking nonrelativistic nr limits of the kleingordon equation, inspired by zees book on qft chapter iii. However, the kleingordon equation does not lead to a positive.
Pdf the schrodingernewton equation as nonrelativistic limit of. Oct 28, 2009 hi, i have a conceptual question about taking non relativistic nr limits of the klein gordon equation, inspired by zees book on qft chapter iii. Kleingordon equation the kleingordon equation can similarly be obtained from the rstquantization of a relativistic particle. Pdf the schrodingernewton equation as nonrelativistic. For comparison, we have also discussed the dirac equation. In particle physics, the dirac equation is a relativistic wave equation derived by british physicist paul dirac in 1928. Nonrelativistic limit of klein gordon maxwell to schrodingerpoisson. 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. Kleingordon equation nonrelativistic limit link to. Nonrelativistic limit of kleingordonmaxwell to schrodingerpoisson. In 1928, paul adrien maurice dirac 19021984 discovered the relativistic equation which now bares his name while trying to overcome the diculties of negative probability densities of the kleingordon equation1. Nonrelativistic limit of klein gordon and massless. Exact bound states of the ddimensional kleingordon equation 1427 the radial and angular parts of the kleingordon equation with this type of potential are solved using the nu method.
Fourmomentum a free particle in the nonrelativistic limit obeys. On timesplitting pseudospectral discretization for. The non relativistic limit of the kleingordon equation kg is the. The nonrelativistic limit of the dirac equation one important requirement for the dirac equation is that it reproduces what we know from nonrelativistic quantum mechanics. Notes on relativistic quantum mechanics module 4 of refresher. The klein gordon equation is lorentz invariant for two reasons. Comparison among kleingordon equation, dirac equation and relativistic stationary schr. It is a quantized version of the relativistic energymomentum relation. Exact bound states of the ddimensional klein gordon equation 1427 the radial and angular parts of the klein gordon equation with this type of potential are solved using the nu method. On timesplitting pseudospectral discretization for nonlinear kleingordon equation in nonrelativistic limit regime volume 16 issue 2 xuanchun dong, zhiguo xu, xiaofei zhao. There are two such equations, one called the kleingordon and the other one called the dirac equation.
The non relativistic limit of the real kleingordon equation. About the nonrelativistic limit of the phase velocity of matter waves. The klein gordon equation klein fock gordon equation or sometimes klein gordon fock equation is a relativistic wave equation, related to the schrodinger equation. Analysis and comparison of numerical methods for the klein. The dirac equation we will try to find a relativistic quantum mechanical description of the electron. Comparison among kleingordon equation, dirac equation. Hydrodynamic limits of the nonlinear kleingordon equation. How to use the einstein energymomentum relation to derive the klein gordon wave equation. Relativistic quantum mechanics kleingordon equation dirac.
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