The College of Education for Pure Sciences, Department of Physics, discussed a master’s thesis on the propagation of wave packets in single layer and double layer graphene.
The thesis presented by the researcher (Munira Waheed Mahan) included the study of quantum particle dynamics expressed by the caustic wave packet and the vibrational motion phenomenon known as (Zitterbewegung ZB) in monolayer and bilayer graphene analytically and numerically. It is clear that the kinetics of the wave beam at an arbitrary time directly depends on the relationship between the initial values ​​of the spin polarizations of the borrowed electron amplitudes on the A and B sublattices. An analytical formula for the expected value of the site effect in monolayer and bilayer graphene is derived from the equations of motion in Heisenberg representation . On the other hand, the probability density relationship in monolayer graphene expressed by the wave function was derived using the Schrödinger representation. Several parameters of the wave beam such as the width d, the initial value of the wave vector in the longitudinal direction kxo and the initial spin-shield polarization of the wave beam in the A and B sub-lattices have been taken into account. Heisenberg representation was used for the physical interpretation of the results. Our results revealed the possibility of transient oscillation of the site-effect-rate oscillation in monolayer graphene and non-transient oscillations. Moreover, if the initial wave packet is split into two parts moving in opposite direction along the graphene web, the phenomenon of zitterbewegung can be clearly achieved. If the initial direction of the metastable twist coincides with the wave vector, wave packet splitting does not occur and the wave packet center is shifted at t > 0 along the same direction. However, the direction of wave packet motion can be estimated by the direction of the wave vector, mainly by the phase difference between the components of the α and β metastable spin polarizations. It is important to discover from the presented results the possibility of controlling the nature of the fluctuation of the site effect values ​​in monolayer and bi-layer graphene and thus controlling the vibrational motion or the zitterbewegung.

message goal
The aim of this study is to provide a theoretical model to study the dynamics of charged quantum particles represented by caustic wave packets and to study the vibrational motion known as the Zitterbewegung phenomenon or the ZBW in single-layer and two-layer graphene, analytically and numerically, through the use of long-wave approximation, where the study included calculating the average value expected. In this study, an analytical formula for the location effect was derived using Heisenberg's representation. Several values ​​of the initial borrowed spin polarizations of the wave beams were taken into consideration.
In this study, a general time frame was developed for calculating the time growth of wave packets by describing the Hamiltonian effect of graphene for two-dimensional electronic gas, as this allows the possibility of verifying the dynamic effects of wave packets and the potential periodic oscillations of charge carriers. The effect of the wiggle spin polarizations and the phase difference between the wave function components on the ZBW oscillatory phenomenon, and the patterns formed for that movement as a result of changing the polarizations of the wiggle, were studied. The results obtained from the study contribute to predicting future access to nanoelectronic devices that operate at very high frequencies, which may reach the order of 1021 Hz.