The problem of the time evolution of localized quantum states for electrons in the semiconducting and metallic single wall carbon nanotubes (SWCNT) are studied analytically and numerically. In the long-wave approximation , where a is the interatomic distance of carbon atoms in the graphene lattice, the carrier wave vector of the wave packet, we consider the time evolution of localized quantum states by using a model Hamiltonian. The localized quantum states are interpreted as a superposition of the valence band and the conduction band. The dynamic of the wave packet which is represented by localized quantum states in carbon nanotubes was considered with cylindrical symmetry. The time dependent average values of the angular and axial coordinate operators and its oscillations behavior are calculated. In addition, we taken into account the effect of an external magnetic field which is applied along the axis of carbon nanotube. The evaluations of the frequency and amplitude of zitterbewegung in the carbon nanotube are presented, which can be used for the experimental prospective studies in the nanoelectronic applications.
Heisenberg representation in the present study are used to evaluate the average values of the coordinate operators , furthermore , Schrödinger representation are used for interpreting the results of the trembling motion (Zitterbewegung) in the carbon nanotubes.