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Uncertainties in Gapped Graphene

Motivated by graphene-based quantum computer we examine the time-dependence of the position-momentum and position-velocity uncertainties in the monolayer gapped graphene. The effect of the energy gap to the uncertainties is shown to appear via the Compton-like wavelength $λ_c$. The uncertainties in the graphene are mainly contributed by two phenomena, spreading and zitterbewegung. While the former determines the uncertainties in the long-range of time, the latter gives the highly oscillation to the uncertainties in the short-range of time. The uncertainties in the graphene are compared with the corresponding values for the usual free Hamiltonian $\hat{H}_{free} = (p_1^2 + p_2^2) / 2 M$. It is shown that the uncertainties can be under control within the quantum mechanical law if one can choose the gap parameter $λ_c$ freely.