The Relativistic Harmonic Oscillator in a Uniform Gravitational Field

We present the relativistic generalization of the classical harmonic oscillator suspended within a uniform gravitational field measured by an observer in a laboratory in which the suspension point of the spring is fixed. The starting point of this analysis is a variational approach based on the Euler–Lagrange formalism. Due to the conceptual differences of mass in the framework of special relativity compared with the classical model, the correct treatment of the relativistic gravitational potential requires special attention. It is proved that the corresponding relativistic equation of motion has unique periodic solutions. Some approximate analytical results including the next-to-leading-order term in the non-relativistic limit are also examined. The discussion is rounded up with a numerical simulation of the full relativistic results in the case of a strong gravity field. Finally, the dynamics of the model is further explored by investigating phase space and its quantitative relativistic features.

How the spin-orbit interaction appears with the same order of a non-commutative change of framework for relativistic fermions

In this research, in a difficult but absolutely precise way of calculation, we show how a very tiny amount of a non-commutative change of quantum space would appear almost as big as a normal physical interaction, namely the Rashba spin-orbit interaction, for relativistic fermions. Hence, in order to show that, we firstly solve a relativistic equation of motion of a Dirac particle, influenced by a typical harmonic energy-dependent interaction for commutative and non-commutative frameworks via the Nikiforov–Uvarov exact approach. Then to study perturbation effects of a spin-orbit interaction, we apply it for both mentioned frameworks, obtaining their energy polynomial relations and discriminant formula to precisely extract all physical-admissible roots of their quartic equations. In this step, we analyze the behaviors of their quartic eigenvalue polynomials in four sections and accurately compare them one by one. Finally, we distinctly show that the magnitude of the physical spin-orbit perturbation appears, almost of the same order of imposing a non-commutative geometry change of framework, as an outstanding result.

Influences of the coordinate dependent noncommutative space on charged and spin currents

We study the charged and spin currents on a coordinate dependent noncommutative space. Starting from the noncommutative extended relativistic equation of motion, the nonrelativistic approximation is obtained by using the Foldy–Wouthuysen transformation, and then the charged and spin currents are derived by using the extended Drude model. We find that the charged current is twisted by modifying the off-diagonal elements of the Hall conductivity, however, the spin current is not affected up to leading order of the noncommutative parameter.