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We study two quantum mechanical systems on the noncommutative plane using arepresentation independent approach. First, in the context of the Landauproblem, we obtain an explicit expression for the gauge transformation thatconnects the Landau and the symmetric gauge in noncommutative space. This leadus to conclude that the usual form of the symmetric gauge$\vec{A}=\left(-\frac{\beta}{2}\hat{Y},\frac{\beta}{2}\hat{X}\right)$, in whichthe constant $\beta$ is interpreted as the magnetic field, is not true innoncommutative space. We also be able to establish a precise definition of$\beta$ as function of the magnetic field, for which the equivalence betweenthe symmetric and Landau gauges is hold in noncommutative plane. Using thesymmetric gauge we obtain results for the spectrum of the Quantum Hall system,its transverse conductivity in the presence of an electric field and otherstatic observables. These results amend the literature on Quantum Hall Effectin noncommutative plane in which the incorrect form of the symmetric gauge, innoncommutative space, is assumed. We also study the non-equilibrium dynamics ofsimple observables for this system. On the other hand, we study the dynamics ofthe harmonic oscillator in non-commutative space and show that, in general, itexhibit quasi-periodic behavior, in striking contrast with its commutativeversion. The study of the dynamics reveals itself as a most powerful tool tocharacterize and understand the effects of non-commutativity.

Dynamic and static properties of Quantum Hall and Harmonic Oscillator systems on the non-commutative plane