Consider a meson bound in a point nucleus with in nite mass mn. Identifying the lagrangian density las the integrand of 1, to nd the conjugate momenta is just to take derivatives. Thus it is not possible to have probability interpretation for kleingordon equation. Relativistic quantum mechanics kleingordon equation dirac. This wavefunction obeys a firstorder equation the schroedinger equation. Fok as a wave equation under the conditions of cyclicity in the fifth coordinate and was shortly afterwards deduced by several authors for example, w. 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. These are some notes that i made using the klein gordon equation as a starting place for understanding imaginary dimensions in reality. In his book the investors manifesto, william berstein explains how to use the gordon equation to estimate expected real returns on stocks for stocks on the long run. Analytical solution for the kleingordon equation and action function.
Comparison among kleingordon equation, dirac equation. The klein gordon equation is the linear partial differential equation which is the equation of motion of a free scalar field of possibly nonvanishing mass m m on some possibly curved spacetime lorentzian manifold. The aim here is to show that there is no impediment to a particle interpretation of the kleingordon equation once a small extension to the usual formalism is included. Parallel spectral numerical methodsthe kleingordon. Its solutions include a quantum scalar or pseudoscalar field, a field whose. The plane waves 10 are basic solutions and the field 9 is constructed by.
The klein gordon 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 klein gordon equation as the basic field equation of the scalar field. Stationary baseband solutions of the equation come about as a balance between nonlinearity and dispersion, and thus represent solitary wave solutions to the system. Interpretation of the kleingordon probability density. The kleingordon equation is the linear partial differential equation which is the equation of motion of a free scalar field of possibly nonvanishing mass m m on some possibly curved spacetime lorentzian manifold. It is a quantized version of the relativistic energymomentum relation. Klein gordon equation f or a free relativistic particle the total energy e is no longer giv en b y the equation w e used to deriv e the sc hro edinger equation.
It is secondorder in space and time and manifestly lorentzcovariant. Pdf solving the kleingordon equation using fourier. The kg equation originally arose in an attempt to give a relativistic generalization of the schr odinger equation. Kleingordon equation particles in exponentialtype molecule. These are some notes that i made using the kleingordon equation as a starting place for understanding imaginary dimensions in reality.
The kleingordon equation in curved spacetime from within the proper time formalism of schwinger and dewitt has been treated, at least from the gaussian approximation, by beken stein 2, 3. However, the situation becomes quite di erent in the theory of special relativity. The kleingordon equation kge is the wellknown relativistic wave equation that describes spinzero particles. Feb 05, 2018 we all know that the schrodinger equation is a non relativistic wave equation. B c2 would decrease without lower bound if the charge number of nucleus zis su ciently large. Furthermore, we also give a numeric solution using a splitstep algorithm, and also present two physical applications of the sine gordon equation. The idea was to let be the complexvalued wave function describing a spinless particle of mass m. Thanks for contributing an answer to physics stack exchange. Aug 22, 2016 strictly speaking, the klein gordon equation should really be considered as the equation of motion for a classical or quantum field, not a wave equation, which is why doing all of this seems handwavy if we interpret it as the latter. The general solutions found, could be used for future explorations on the study for other speci. Two dimensional simulations of the kleingordon equation can be found in yang. Kleingordon equation relativistic quantum mechanics.
Furthermore, we also give a numeric solution using a splitstep algorithm, and also present two physical applications of the sinegordon equation. The nonrelativistic equation for the energy of a free particle is. Therefore, as long as energy and momentum follows the einsteins relation eq. Separation of variables and exact solution of the klein. Kleingordon equation with a powerlaw nonlinearity 1, kleingordon equation with a powerlaw nonlinearity 2, modi. The relativisticallyinvariant quantum equation describing spinless scalar or pseudoscalar particles, for example, and mesons. The two coupled pdes can be combined into a single. Notes on the kleingordon equation by ron poteet issuu. View kleingordon equation research papers on academia. Boundary value problems for the kleingordon equation. Separation of variables in the klein gordon equation.
These equations appear in the study of relativistic and quantum physics. See also special cases of the nonlinear kleingordon equation. The gordon equation states that the longterm expected real inflation adjusted return from the market should approximate the inflationadjusted compound yearly growth rate in dividends plus the current dividend yield. Nonperturbative calculation of qed processes involving a strong electromagnetic field, especially provided by strong laser facilities at present. Notes on the sine gordon equation david gablinger january 31, 2007 abstract in this seminar, we will introduce the sinegordon equation, and solve it using a baecklund transfomation. Chapter 11 em lorentz force derived from klein gordons.
Solving the kleingordon equation using fourier spectral methods. What is the relationship between the dirac equation and. In nonrelativistic mechanics, the energy for a free particle is. Stationary solutions of the kleingordon equation in a. This extension has already been found necessary in previous work 4,5 to accommodate lorentz invariance in the manyparticle case. Since we want the lagrangian density to be invariant under lorentz transformations a linear wave equation, the lagrangian density can contain only the. How to derive general solution to the kleingordon equation. The kleingordon equation kleinfockgordon equation or sometimes kleingordonfock equation is a relativistic wave equation, related to the schrodinger equation. A qualitative study of the damped dung equation and.
The linear kleingordon equation occurs as a modification. Equation 12 is of course the relativistic energymomentum relation all. The signs in the formula for w1are taken arbitrarily. 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.
He then tried to find solutions, but was dismayed to find that problems like nonpositivedefinite probability curre. Kleingordon equation a relativistic that is satisfying the requirements of the theory of relativity quantum. Notes on the sine gordon equation david gablinger january 31, 2007 abstract in this seminar, we will introduce the sine gordon equation, and solve it using a baecklund transfomation. The klein gordon equation 21 is completely separable in 3, therefore we look for its solution in the form. Suppose w wx,t is a solution of the nonlinear klein gordon equation. The curvilinear coordinates in which the variables are. Pdf notes on the kleingordon equation researchgate. Kleingordon equation in schrodinger form 3 which is the usual relativistic formula for the energy of a particle. All types of external electromagnetic fields containing arbitrary functions which admit of separation of variables in the kleingordon equation by using three firstorder differential symmetry operators, and stationary fields admitting separation of variables by using two first and one secondorder differential operators, are found. The gordon equation can it predict future market returns. The gordon growth model is used to determine the intrinsic value of a stock based on a future series of dividends that grow at a. Exact solutions nonlinear partial differential equations secondorder hyperbolic partial differential equations nonlinear klein gordon equation 7.
But avoid asking for help, clarification, or responding to other answers. The aim here is to show that there is no impediment to a particle interpretation of the klein gordon equation once a small extension to the usual formalism is included. New solutions of the ddimensional kleingordon equation via. Strictly speaking, the klein gordon equation should really be considered as the equation of motion for a classical or quantum field, not a wave equation, which is why doing all of this seems handwavy if we interpret it as the latter. Kleingordon equation in schrodinger form physics pages. Em lorentz force derived from klein gordons equation 11. Comparison among kleingordon equation, dirac equation and. A benchmark test for computer performance article pdf available january 2015 with 345 reads how we measure reads. Blow up of three dimensional radially symmetric real solutions to this equation have recently been numerically studied by donninger and schlag. Several analytical results found in the literature, including the socalled klein gordon oscillator, are obtained as particular cases of this unified. Solution of the coupled kleingordon schrodinger equation. Continuum 3d 32, 32 3d5 3s12, 3p12 2sl2, 12 dirac fine structure increased 3p32, 3d32 2sl2 12 lamb shift increased hyperfine structure.
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