Abstract

We study two-dimensional Hamiltonians in phase space with non-commutativity both in coordinate and momenta. We consider the generator of rotations on the noncommutative plane and the Lie algebra generated by Hermitian rotationally invariant quadratic forms of the noncommutative dynamical variables. We show that two quantum phases are possible, chareacterized by the Lie algebras sl(2,R) or su(2) according to the relation between the non-commutativity parameters, with the rotation generator related with the Casimir operator. From this algebraic perspective, we analize the spectrum of some simple models with nonrelativistic rotationally invariant Hamiltonians in this noncommutative phase space, such as the isotropic harmonic oscillator, the Landau problem and the cylindrical well potential