Mecánica Cuántica en el Plano no Conmutativo Quantum Mechanics in the non Conmutative Plane

We look for the way in that the Hamiltonian operators in quantum mechanics are modified when noncommutativity in the plane is implemented. For this, a brief introduction on foundations of conventional quantum mechanics is presented and bidimensional examples are solved. New commutation relations are introduced to include non commutativity between coordinates and three examples are solved. The problem of harmonic oscillator on the plane is analyzed from point of view of creation and annihilation operators, both the conventional and the non commutative cases. Finally, the results explicitly show noncommutativity by means of noncommutativity parameter and, in the harmonic oscillator case the resulting Hamiltonian can be related with quantum optics.