Coherent states of the deformed Heisenberg-Weyl algebra in non-commutative space

In two-dimensional space a subtle point that for the case of both space-space and momentum-momentum non-commuting, different from the case of only space-space non-commuting, the deformed Heisenberg-Weyl algebra in non-commutative space is not completely equivalent to the undeformed Heisenberg-Weyl algebra in commutative space is clarified. It follows that there is no well-defined procedure to construct the deformed position-position coherent state or the deformed momentum-momentum coherent state from the undeformed position-momentum coherent state. Identifications of the deformed position-position and deformed momentum-momentum coherent states with the lowest energy states of a cold Rydberg atom in special conditions and a free particle, respectively, are demonstrated.