Generalized Uncertainty Relations in the Non-commutative Plane

Abstract

In this paper we study two-dimensional noncommutative quantum mechanics (NCQM) with the generalized uncertainty relations \({\Delta } x_{1} {\Delta } x_{2} \ge \frac { \theta }{2}, {\Delta } p_{1} {\Delta } p_{2} \ge \frac { \bar {\theta }}{2}, {\Delta } x_{i} {\Delta } p_{i} \ge \frac {\hbar }{2}, {\Delta } x_{1} {\Delta } p_{2} \ge \frac {|{\Lambda }_{21}|}{2}\). We find the new NCQM algebra from the generalized uncertainty relations. We construct a operator \( \hat {\pi }_{i}\) commuting with \( \hat {x}_{j} \) and discuss two possibilities; One is the case that \( \hat {\pi }_{i} \) also commutes with \( \hat {p}_{j}\) and another is the case that \( \hat {\pi }_{i} \) does not commute with \( \hat {p}_{j}\). For both case we consider a motion of a charged particle in a magnetic field with a harmonic oscillator potential in the noncommutative plane.