The solution of the Dirac equation in the presence of the electromagnetic field on the onedimensional barrier potential is studied. The energy spectrum and the eigenfunction of the Dirac equation obtained by solving the Dirac equation and we introduced annihilation and creation operators for the Hamiltonian has an identical form in the harmonic oscillator. Regions I and III separated by a potential barrier characterized by the gap energy with the eigenfunctions as a sinusoidal function, and region II has the form of an exponent function. We found the eigenfunction involved positive and negative energy moves exponentially when passed through a barrier.

Understanding the missing matter problem in cosmological phenomena and scales of astrophysical is usually studied by modifying general relativity theory. In this article, we formulated the Kruskal-Szekeres coordinate of vacuum modified gravity model in f (R ) theory. The generalization of the field equation was obtained by generalizing Hilbert-Einstein’s action with gravitational Lagrangian in terms of f (r)function. By consider a special class of f(R) theory by taking R = R0 , we found the solution of static spherically symmetric spacetime that was known as de Sitter-Schwarzschild spacetime. The transformation rules were constructed from Kruskal-Szekeres coordinates in f (r)theory of modified general relativity to the Kruskal-Szekeres coordinate in general relativity theory. For λ ≈ 0, the Schwarzschild and Kruskal-Szekeres metric for static spherically symmetric on f (r)theory reduced to the standard Schwarzschild and Kruskal-Szekeres metric on general relativity. We also show the spacetime structure of de Sitter-Schwarzschild and Kruskal-Szekeres coordinate. This work could open a promising way to understand some features of a black hole in the f (r)theory of gravity.

Brownian motion theory is always challenging to describe diffusion phenomena around a black hole, with the main issue is how to extend the classical theory of Brownian motion to the general relativity framework. In this study, we extended the Brownian motion theory in a curved space-time come from a strong gravitational field on the Schwarzschild black hole. The Brownian motion theory in Schwarzschild space-time was derived by using the Fokker-Planck equation, and the stationary solution was analyzed by Ito, Stratonovich-Fisk, and Hanggi-Klimontovich Approach. The numerical result was found that the Brownian motion in Schwarzschild space-time 1   was reduced to the standard Brownian motion in Newtonian classical theory. According to the Hanggi-Klimontovich approach for 1   the result showed a consistent with the relativistic Maxwell distribution. The Fokker-Planck equation in Schwarzschild space-time was also formulated as a generalization of relativistic Brownian motion theory. This work could open a promising interpretation to formulate the diffusion phenomena around a massive object in the general relativity framework.

A Bosonic particle and zero mass Dirac acting on a one-dimensional barrier potential have been studied in relativistic quantum mechanics using the Dirac and Klein-Gordon equation. The equations are modified due to the influence of the electromagnetic fields and the matrix approach has been applied to obtain the representation of energy and the eigen spinor. Especially for Dirac particles, the Dirac Hamiltonian form is similar to the harmonic oscillator, so the representation of energy is an extension of the harmonic oscillator energy. In addition, the eigenfunctions involved positive energy and negative energy that moved exponentially when passed through a potential barrier.