APPLICATION DIRECT METHOD CALCULUS OF VARIATION FOR KLEIN-GORDON FIELD

2015 ◽  
Vol 77 (23) ◽  
Author(s):  
Saiman Saiman ◽  
Rinto Agustino ◽  
Hamdani Hamdani
Keyword(s):  
Hydrogen Atom ◽  
Functional Form ◽  
Gordon Equation ◽  
Direct Method ◽  
Ritz Method ◽  
Field Equations ◽  
De Broglie Waves ◽  
Klein Gordon ◽  
Quantum Wave ◽  
Quantum Wave Equation

Klein-Gordon field is often used to study the dynamics of elementary particles. The Klein–Gordon equation was first considered as a quantum wave equation by Schrödinger in his search for an equation describing de Broglie waves. The equation was found in his notebooks from late 1925, and he appears to have prepared a manuscript applying it to the hydrogen atom. Yet, because it fails to take into account the electron's spin, the equation failed to predict the fine structure of the hydrogen atom, and overestimated the overall magnitude of the splitting pattern energy. This paper will describe in detail using the Direct Method of Calculus Variation as an alternative to solve the Klien-Gordon field equations. The Direct Method simplified the calculation because the variables are calculated and expressed in functional form of energy. The result of the calculation of Klien-Gordon Feld provided the existence of the minimizer, i.e.  with  and . Explicit form of the minimizer was calculated by the Ritz method through rows of convergent density

scholarly journals Virtual beams and the Klein paradox for the Klein-Gordon equation

2018 ◽  
Vol 64 (1) ◽  
pp. 1 ◽  
Author(s):  
A. Molgado ◽  
O. Morales ◽  
J.A. Vallejo
Keyword(s):  
Gordon Equation ◽  
Relativistic Quantum ◽  
Quantum Field ◽  
Method Of Images ◽  
Klein Paradox ◽  
Virtual Particles ◽  
Klein Gordon ◽  
Quantum Wave ◽  
Well Posed ◽  
Quantum Wave Equation

Whenever we consider any relativistic quantum wave equation we are confronted with the Klein paradox, which asserts that incident particles will suffer a surplus of reflection when dispersed by a discontinuous potential. Following recent results on the Dirac equation, we propose a solution to this paradox for the Klein-Gordon case by introducing virtual beams in a natural well-posed generalization of the method of images in the theory of partial differential equations. Thus, our solution considers a global reflection coefficient obtained from the two contributions, the reflected particles plus the incident virtual particles. Despite its simplicity, this method allows a reasonable understanding of the paradox within the context of the quantum relativistic theory of particles (according to the original setup for the Klein paradox) and without resorting to any quantum field theoretic issues.

scholarly journals Solitons for the Nonlinear Klein-Gordon Equation

2010 ◽  
Vol 10 (2) ◽  
Author(s):  
J. Bellazzini ◽  
V. Benci ◽  
C. Bonanno ◽  
A.M. Micheletti
Keyword(s):  
Solitary Waves ◽  
Gordon Equation ◽  
Energy Charge ◽  
Sufficient Conditions ◽  
Orbital Stability ◽  
Field Equations ◽  
Klein Gordon ◽  
Ratio Energy ◽  
Klein Gordon Equation

AbstractIn this paper we study existence and orbital stability for solitary waves of the nonlinear Klein-Gordon equation. The energy of these solutions travels as a localized packet, hence they are a particular type of solitons. In particular we are interested in sufficient conditions on the potential for the existence of solitons. Our proof is based on the study of the ratio energy/charge of a function, which turns out to be a useful approach for many field equations.

OCTONIC GRAVITATIONAL FIELD EQUATIONS

2013 ◽  
Vol 28 (21) ◽  
pp. 1350112 ◽  
Author(s):  
SÜLEYMAN DEMİR ◽  
MURAT TANIŞLI ◽  
TÜLAY TOLAN
Keyword(s):  
Wave Equation ◽  
Gordon Equation ◽  
The Other ◽  
Massive Graviton ◽  
Field Equations ◽  
Gravitational Field Equations ◽  
Number Systems ◽  
Klein Gordon ◽  
Linear Gravity ◽  
Klein Gordon Equation

Generalized field equations of linear gravity are formulated on the basis of octons. When compared to the other eight-component noncommutative hypercomplex number systems, it is demonstrated that associative octons with scalar, pseudoscalar, pseudovector and vector values present a convenient and capable tool to describe the Maxwell–Proca-like field equations of gravitoelectromagnetism in a compact and simple way. Introducing massive graviton and gravitomagnetic monopole terms, the generalized gravitational wave equation and Klein–Gordon equation for linear gravity are also developed.

scholarly journals Derivation of a Relativistic Wave Equation more Profound than Dirac’s Relativistic Wave Equation

2018 ◽  
Vol 10 (6) ◽  
pp. 102
Author(s):  
Koshun Suto
Keyword(s):  
Wave Equation ◽  
Hydrogen Atom ◽  
Dirac Equation ◽  
Potential Energy ◽  
Gordon Equation ◽  
Relativistic Wave Equation ◽  
Relativistic Wave ◽  
Departure Point ◽  
Klein Gordon ◽  
Klein Gordon Equation

The author has previously derived an energy-momentum relationship applicable in a hydrogen atom. Since this relationship is taken as a departure point, there is a similarity with the Dirac’s relativistic wave equation, but an equation more profound than the Dirac equation is derived. When determining the coefficients  and β of the Dirac equation, Dirac assumed that the equation satisfies the Klein-Gordon equation. The Klein-Gordon equation is an equation which quantizes Einstein's energy-momentum relationship. This paper derives an equation similar to the Klein-Gordon equation by quantizing the relationship between energy and momentum of the electron in a hydrogen atom. By looking to the Dirac equation, it is predicted that there is a relativistic wave equation which satisfies that equation, and its coefficients are determined. With the Dirac equation it is necessary to insert a term for potential energy into the equation when describing the state of the electron in a hydrogen atom. However, in this paper, a potential energy term is not introduced into the relativistic wave equation. Instead, potential energy is incorporated into the equation by changing the coefficient  of the Dirac equation.

scholarly journals SECOND-ORDER CORRECTIONS TO THE NONCOMMUTATIVE KLEIN–GORDON EQUATION WITH A COULOMB POTENTIAL

2011 ◽  
Vol 26 (23) ◽  
pp. 4133-4144 ◽  
Author(s):  
SLIMANE ZAIM ◽  
LAMINE KHODJA ◽  
YAZID DELENDA
Keyword(s):  
Hydrogen Atom ◽  
Coulomb Potential ◽  
Lamb Shift ◽  
Gordon Equation ◽  
Energy Levels ◽  
Space Time ◽  
Second Order ◽  
Noncommutative Space ◽  
Klein Gordon ◽  
Klein Gordon Equation

We improve the previous study of the Klein–Gordon equation in a noncommutative space–time as applied to the hydrogen atom to extract the energy levels, by considering the second-order corrections in the noncommutativity parameter. Phenomenologically we show that noncommutativity is the source of Lamb shift corrections.

scholarly journals The Euler-Lagrange expression and degenerate lagrange densities

1972 ◽  
Vol 14 (4) ◽  
pp. 482-495 ◽  
Author(s):  
D. Lovelock
Keyword(s):  
Variational Principle ◽  
Fourth Order ◽  
Gordon Equation ◽  
Einstein Equations ◽  
Theoretical Physics ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Lagrange Equations ◽  
Klein Gordon ◽  
Klein Gordon Equation

It is well known that many of the field equations from theoretical physics (e.g. Einstein field equations, Maxwell's equations, Klein-Gordon equation) can be obtained from a variational principle with a suitably chosen Lagrange density. In the case of the Einstein equations the corresponding Lagrangian is degenerate (i.e., the associated Euler-Lagrange equations are of second order whereas in general these would be of fourth order), while in the cases of the Maxwell and Klein-Gordon equations the Lagrangian usually used is not degenerate.

scholarly journals Bound states of the Klein-Gordon equation with n-dimensional scalar and vect or hydrogen atom-type potentials

2003 ◽  
Vol 52 (7) ◽  
pp. 1579
Author(s):  
Chen Chang-Yuan ◽  
Liu Cheng-Lin ◽  
Lu Fa-Lin ◽  
Sun Dong-Sheng
Keyword(s):  
Hydrogen Atom ◽  
Bound States ◽  
Gordon Equation ◽  
Atom Type ◽  
Klein Gordon ◽  
Klein Gordon Equation
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