Noncommutative Quantum Mechanics based in Representations of Exotic Galilei Group

Using elements of symmetry, we constructed the Noncommutative Schr\"odinger Equation from a representation of Exotic Galilei Group. As consequence, we derive the Ehrenfest theorem using noncommutative coordinates. We also have showed others features of quantum mechanics in such a manifold. As an important result, we find out that a linear potential in the noncommutative Schr\"odinger equation is completely analogous to the ordinary case. We also worked with harmonic and anharmonic oscillators, giving corrections in the energy for each one.