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Main Question or Discussion Point
In graphene system, the velocity operator sometimes is v= ∂H/ħ∂p, and its matrix element is calculated as <ψvψ>, i.e., v_x = v_F cos(θ) and v_y = v_F sin(θ) [the results are the same with Eq. 25] for intraband velocity. Recently, I see a new way to calculate the velocity matrix (Mikhailov, Sergey A. "Quantum theory of the thirdorder nonlinear electrodynamic effects of graphene." Physical Review B 93.8 (2016): 085403.). But I cannot understand the Eq. 20 (see the following picture). Additionally, what the difference between the two ways to define the velocity operator?
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