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International Journal of Theoretical Physics

, Volume 55, Issue 4, pp 2174–2181 | Cite as

Two Phases of the Non-Commutative Quantum Mechanics with the Generalized Uncertainty Relations

  • Won Sang Chung
Article
  • 86 Downloads

Abstract

We consider the quantum mechanics on the noncommutative plane with the generalized uncertainty relations \({\Delta } x_{1} {\Delta } x_{2} \ge \frac {\theta }{2}, {\Delta } p_{1} {\Delta } p_{2} \ge \frac {\bar {\theta }}{2}, {\Delta } x_{i} {\Delta } p_{i} \ge \frac {\hbar }{2}, {\Delta } x_{1} {\Delta } p_{2} \ge \frac {\eta }{2}\). We show that the model has two essentially different phases which is determined by \(\kappa = 1 + \frac {1}{\hbar ^{2} } (\eta ^{2} - \theta \bar {\theta })\). We construct a operator \(\hat {\pi }_{i}\) commuting with \(\hat {x}_{j} \) and discuss the harmonic oscillator model in two dimensional non-commutative space for three case κ > 0, κ = 0, κ < 0. Finally, we discuss the thermodynamics of a particle whose hamiltonian is related to the harmonic oscillator model in two dimensional non-commutative space.

Keywords

Noncommutative plane Generalized uncertainty relation Harmonic oscillator model 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Physics and Research Institute of Natural Science, College of Natural ScienceGyeongsang National UniversityJinjuKorea

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