No abstract

We explore the conditions under which noncommutative quantum mechanics and the Landau problem are equivalent theories. If the potential in noncommutative quantum mechanics is chosen as V=Ωℵ with ℵ defined in the text, then for the value [Formula: see text] (which measures the noncommutative effects of the space), the Landau problem and noncommutative quantum mechanics are equivalent theories in the lowest Landau level. For other systems one can find different values for [Formula: see text] and, therefore, the possible bounds for [Formula: see text] should be searched in a physical independent scenario. This last fact could explain the different bounds for [Formula: see text] found in the literature.

The possibility of detecting noncommutative space relics is analyzed using the Aharonov-Bohm effect. We show that, if space is noncommutative, the holonomy receives non-trivial kinematical corrections that will produce a diffraction pattern even when the magnetic flux is quantized. The scattering problem is also formulated, and the differential cross section is calculated. Our results can be extrapolated to high energy physics and the bound θ ∼ [10 TeV] −2 is found. If this bound holds, then noncommutative effects could be explored in scattering experiments measuring differential cross sections for small angles. The bound state Aharonov-Bohm effect is also discussed.

Quantum mechanics in a noncommutative plane is considered. For a general two dimensional central field, we find that the theory can be perturbatively solved for large values of the noncommutative parameter (θ) and explicit expressions for the eigenstates and eigenvalues are given. The Green function is explicitly obtained and we show that it can be expressed as an infinite series.For polynomial type potentials, we found a smooth limit for small values of θ and for non-polynomial ones this limit is necessarily abrupt. The Landau problem, as a limit case of a noncommutative system, is also considered.