We consider the quantum mechanics on the noncommutative plane with the generalized uncertainty relations Δx1Δx2≥θ2,Δp1Δp2≥θ̄2,ΔxiΔpi≥ℏ2,Δx1Δp2≥η2\documentclass[12pt]{minimal}
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\begin{document}${\Delta } x_{1} {\Delta } x_{2} \ge \frac {\theta }{2}, {\Delta } p_{1} {\Delta } p_{2} \ge \frac {\bar {\theta }}{2}, {\Delta } x_{i} {\Delta } p_{i} \ge \frac {\hbar }{2}, {\Delta } x_{1} {\Delta } p_{2} \ge \frac {\eta }{2}$\end{document}. We show that the model has two essentially different phases which is determined by κ=1+1ℏ2(η2−θθ̄)\documentclass[12pt]{minimal}
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\begin{document}$\kappa = 1 + \frac {1}{\hbar ^{2} } (\eta ^{2} - \theta \bar {\theta })$\end{document}. We construct a operator π̂i\documentclass[12pt]{minimal}
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\begin{document}$\hat {\pi }_{i}$\end{document} commuting with x̂j\documentclass[12pt]{minimal}
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\begin{document}$\hat {x}_{j} $\end{document} 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.
