647,603 research outputs found

    Non-Relativistic Limit of Dirac Equations in Gravitational Field and Quantum Effects of Gravity

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    Based on unified theory of electromagnetic interactions and gravitational interactions, the non-relativistic limit of the equation of motion of a charged Dirac particle in gravitational field is studied. From the Schrodinger equation obtained from this non-relativistic limit, we could see that the classical Newtonian gravitational potential appears as a part of the potential in the Schrodinger equation, which can explain the gravitational phase effects found in COW experiments. And because of this Newtonian gravitational potential, a quantum particle in earth's gravitational field may form a gravitationally bound quantized state, which had already been detected in experiments. Three different kinds of phase effects related to gravitational interactions are discussed in this paper, and these phase effects should be observable in some astrophysical processes. Besides, there exists direct coupling between gravitomagnetic field and quantum spin, radiation caused by this coupling can be used to directly determine the gravitomagnetic field on the surface of a star.Comment: 12 pages, no figur

    Perturbation of coupling matrices and its effect on the synchronizability in arrays of coupled chaotic systems

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    In a recent paper, wavelet analysis was used to perturb the coupling matrix in an array of identical chaotic systems in order to improve its synchronization. As the synchronization criterion is determined by the second smallest eigenvalue λ2\lambda_2 of the coupling matrix, the problem is equivalent to studying how λ2\lambda_2 of the coupling matrix changes with perturbation. In the aforementioned paper, a small percentage of the wavelet coefficients are modified. However, this result in a perturbed matrix where every element is modified and nonzero. The purpose of this paper is to present some results on the change of λ2\lambda_2 due to perturbation. In particular, we show that as the number of systems nn \to \infty, perturbations which only add local coupling will not change λ2\lambda_2. On the other hand, we show that there exists perturbations which affect an arbitrarily small percentage of matrix elements, each of which is changed by an arbitrarily small amount and yet can make λ2\lambda_2 arbitrarily large. These results give conditions on what the perturbation should be in order to improve the synchronizability in an array of coupled chaotic systems. This analysis allows us to prove and explain some of the synchronization phenomena observed in a recently studied network where random coupling are added to a locally connected array. Finally we classify various classes of coupling matrices such as small world networks and scale free networks according to their synchronizability in the limit.Comment: 7 pages, 2 figures, 1 tabl