Rotational Symmetry and Gauge Invariant Degeneracies on 2D Noncommutative Plane

• URL: http://arxiv.org/abs/2112.08666v1
• Date: Thu, 16 Dec 2021 07:03:16 GMT
• Title: Rotational Symmetry and Gauge Invariant Degeneracies on 2D Noncommutative Plane
• Authors: M.N.N.M. Rusli, M.S. Nurisya, H. Zainuddin, M.F. Umar and A. Jellal
• Abstract summary: We find that the energy levels and states of the system are unique and hence, same goes to the degeneracies as well. The degenerate energy levels can always be found if $theta$ is proportional to the ratio between $hbar$ and $momega$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: We obtain the gauge invariant energy eigenvalues and degeneracies together with rotationally symmetric wavefunctions of a particle moving on 2D noncommutative plane subjected to homogeneous magnetic field $B$ and harmonic potential. This has been done by using the phase space coordinates transformation based on 2-parameter family of unitarily equivalent irreducible representations of the nilpotent Lie group $G_{NC}$. We find that the energy levels and states of the system are unique and hence, same goes to the degeneracies as well since they are heavily reliant on the applied $B$ and the noncommutativity $\theta $ of coordinates. Without $B$, we essentially have a noncommutative planar harmonic oscillator under generalized Bopp shift or Seiberg-Witten map. The degenerate energy levels can always be found if $\theta$ is proportional to the ratio between $\hbar$ and $m\omega$. For the scale $B\theta = \hbar$, the spectrum of energy is isomorphic to Landau problem in symmetric gauge and hence, each energy level is infinitely degenerate regardless of any values of $\theta$. Finally, if $0 < B\theta < \hbar$, $\theta$ has to also be proportional to the ratio between $\hbar$ and $m\omega$ for the degeneracy to occur. These proportionality parameters are evaluated and if they are not satisfied then we will have non-degenerate energy levels. Finally, the probability densities and effects of $B$ and $\theta$ on the system are properly shown for all cases.

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