The Landau problem and noncommutative quantum mechanics

TL;DR

This paper investigates the conditions under which noncommutative quantum mechanics and the Landau problem are equivalent, identifying specific parameter values that establish their equivalence in the lowest Landau level and discussing implications for bounds on noncommutative effects.

Contribution

It demonstrates the specific potential choice that makes noncommutative quantum mechanics equivalent to the Landau problem at certain parameters, clarifying how bounds on noncommutative effects vary across systems.

Findings

Equivalence established at ${ ilde heta} = 0.22 imes 10^{-11} cm^2$ in the lowest Landau level.

Different systems imply different bounds for ${ ilde heta}$.

Possible explanation for varying bounds in literature.

Abstract

The conditions under which noncommutative quantum mechanics and the Landau problem are equivalent theories is explored. If the potential in noncommutative quantum mechanics is chosen as $V= \Omega \aleph$ with $\aleph$ defined in the text, then for the value ${\tilde \theta} = 0.22 \times 10^{-11} cm^2$ (that 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 differents values for ${\tilde \theta}$ and, therefore, the possible bounds for ${\tilde \theta}$ should be searched in a physical independent scenario. This last fact could explain the differents bounds for $\tilde \theta$ found in the literature.