Two phases of the noncommutative quantum mechanics

Abstract

We consider quantum mechanics on the noncommutative plane in the presence of magnetic field $B$. We show, that the model has two essentially different phases separated by the point $B\theta =c{\hslash }^2/e$, where $\theta$ is a parameter of noncommutativity. In this point the system reduces to exactly-solvable one-dimensional system. When $\kappa =1-eB\theta /c{\hslash }^2<0$ there is a finite number of states corresponding to the given value of the angular momentum. In another phase, i.e. when $\kappa >0$ the number of states is infinite. The perturbative spectrum near the critical point $\kappa =0$ is computed.

4 pages, presentation still improved, mistakes corrected, new sections added

References in corpus (2)

• Quantum Hall states as matrix Chern-Simons theory
• Noncommutativity in Space and Primordial Magnetic Field

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