Progress in a trajectory interpretation of the Klein-Gordon equation

A trajectory interpretation is developed for the Klein-Gordon equation in one dimension. The development is couched in a Hamilton-Jacobi representation. Equations of motion are developed. Different trajectories for a given eigenvalue energy are shown to manifest different microstates of the eigenfunction of that particular energy.     

Photon velocity from the Klein-Gordon equation

An approximate mathematical relationship for the velocity of a photon as a non-zero rest-mass quantum particle is derived from the field-free Klein-Gordon equation in the framework of the de Broglie-Bohm theory of quantum mechanics.     

Charge-magnetic-dipole bound states using the Klein-Gordon equation

We investigate bound states of a composite system consisting of a charged particle orbiting a neutral, stationary magnetic dipole. We find all bound states are metastable and none exist with angular momentum less than eleven. Our calculations is performed in two space dimensions.     

Generalized virial theorem for the Klein-Gordon equation

The virial theorem for the Klein-Gordon equation has been generalized with respect to non-integrable scale-invariant probe functions. Ground-state energies as well as certain upper bounds on the coupling constants have been established. For definiteness several kinds of attractive power potentials have been considered.     

Trajectory length constraint and the Klein-Gordon equation

It is shown that the relativistic constraint of the length of possible trajectories in the method of integrals over the trajectories results in the Klein-Gordon equation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 63–65, December, 1984.     

Bound state solution of the Klein-Gordon equation for vector and scalar Hellmann plus modified Kratzer potentials

In this study, the analytical solutions of the Klein–Gordon equation for any l states of the scalar and vector Hellmann plus modified Kratzer potential are derived by using an approximation method to the centrifugal potential term. The analytical expressions for eigenvalues and corresponding normalized eigenfunctions of the spin-zero particle have been estimated by using the parametric Nikiforov-Uvarov method. The solution for the radial part of the Klein-Gordon equation is formulated in terms of the generalized Jacobi polynomials. The energy state equation and the wave function for special cases are in good agreement with the previous literature. In addition, we have measured the numerical results of the energy eigenvalues and also the trend of the eigenvalues concerning of different potential parameters have been plotted. Furthermore, it was shown that the energy levels E and quantum numbers n and l are inversely proportional to each other.     

Similarity solutions of the cubic nonlinear Klein-Gordon equation

Using the direct method introduced by Clarkson and Kruskal recently, we obtain the similarity reductions of the cubic nonlinear Klein-Gordon equation whenz=0.     

The Klein-Gordon equation and scattering theory

Using our previously developed theory, we prove the main conclusions of scattering theory for the Klein-Gordon equation under hypotheses weaker than presently known. We consider more general equations and obtain stronger results.     

Use of the Schwarzschild metric in the Klein-Gordon equation

It is shown that substitution of the Schwarzschild metric sensor into the Klein-Gordon equation predicts the usual perihelion advance of classical general relativity.     

The Klein-Gordon equation with light-cone data

It is shown that the characteristic Cauchy problem ·u(x,t)=0,u(x,–|x|)=f(x),x ⁿ ,n1 has a unique finite energy weak solution for allf such that dx(|f|²+|f|²)< and all finite energy weak solutions of the equation are obtained in this way.     

用克莱因-戈登方程推导两质子之间的库仑势

量子电动力学认为库仑力是电荷之间交换虚光子的结果,最有力的证据是从虚光子假说直接推出库仑势能.克莱因-戈登方程(□2-m2)Φ(x,t)=0是一个用来描述质量为m,自旋为0,电荷为0的标量场的场方程,虚光子可以用粒子静止质量为0的标量场描述,因此满足克莱因-戈登方程□2Φ(x,t)=0.本文利用克莱因-戈登方程推导两质子之间的库仑势.     

Quark potentials for mesons in the Klein-Gordon equation

Two relativistic potential models are applied to describe meson spectroscopy in a unified way, encompassing both light and heavy quark systems. A combination of linear and coulomb potentials has been investigated for Klein-Gordon equation using the WKB approximation. A power-like phenomenological potential model has also been studied in the Klein-Gordon framework. Meson masses calculated for both the potentials give a good agreement with the corresponding experimental values.     

Spectral and scattering theory for the Klein-Gordon equation

Eigenfunction expansions associated with the Klein-Gordon equation, are derived in the static external field case. By employing these, we develop spectral and scattering theory. The results are almost as strong as those obtained in the Schrödinger case.     

Causal action-at-a-distance interpretation of the aspect-Rapisarda experiments

Recent results in EPR-type experiments on singlet photon pairs which establish (a) the non-existence of local hidden variables, (b) the existence of quantum superluminal correlations between the action of independent parts of a measuring device separated by space-like intervals, are interpreted causally within the frame of the stochastic interpretation of quantum mechanics.     

The Klein-Gordon equation in some anisotropic cosmologies

The Klein-Gordon equation is studied in a family of exact solutions of the Einstein equations whose material content is a perfect fluid with stiff equation of state (p=). The field equations are solved exactly for some members of the family and for massive or massless fields.     

Klein-Gordon thermal equation for a planck gas

In this paper the quantum hyperbolic equation formulated in our earlier paper [Found. Phys. Lett. 10, 599 (1997)] is applied to the study of the propagation of the initial thermal state of the universe. It is shown that the propagation depends on the barrier height. The Planck wall potential is introduced,V _P = ħ/8t_P = 1.125 10¹⁸ GeV, wheret _P is a Planck time. For the barrier heightV <V _P , the master thermal equation isthe modified telegrapher’sequation, and for barrier heightV >V _P the master equation is theKlein- Gordon equation. The solutions of both type equations for Cauchy boundary conditions are discussed.     

Levinson's theorem for the Klein-Gordon equation in one dimension

In terms of the modified Sturm-Liouville theorem, the Levinson theorem for the one-dimensional Klein-Gordon equation with a symmetric potential V(x) is established. It is shown that the number N₊ (N_-) of bound states with even (odd) parity is related to the phase shift of the scattering states with the same parity at zero momentum as and The solution of the one-dimensional Klein-Gordon equation with the energy M or -M is called as a half bound state if it is finite but does not decay fast enough at infinity to be square integrable. Received 22 December 1999     

Solutions of nonrelativistic Schrödinger equation from relativistic Klein-Gordon equation

The relativistic one-dimensional Klein-Gordon equation can be exactly solved for a certain class of potentials. But the nonrelativistic Schrödinger equation is not necessarily solvable for the same potentials. It may be possible to obtain approximate solutions for the inexact nonrelativistic potential from the relativistic exact solutions by systematically removing relativistic portion. We search for the possibility with the harmonic oscillator potential and the Coulomb potential, both of which can be exactly solvable nonrelativistically and relativistically. Though a rigorous algebraic approach is not deduced yet, it is found that the relativistic exact solutions can be a good starting point for obtaining the nonrelativistic solutions.