In this paper we introduce a notion of quantum Hamiltonian (co)action of Hopf algebras endowed with Drinfel’d twist structure (resp., 2-cocycles). First, we define a classical Hamiltonian action in the setting of Poisson Lie groups compatible with the 2-cocycle structure and we discuss a concrete example. This allows us to construct, out of the classical momentum map, a quantum momentum map in the setting of Hopf coactions and to quantize it by using rinfel’d approach.