Rotational Symmetry and Gauge Invariant Degeneracies on 2D Noncommutative Plane
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| Format: | Preprint |
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2021
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| _version_ | 1866916497567252480 |
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| author | Rusli, M. N. N. M. Nurisya, M. S. Zainuddin, H. Umar, M. F. Jellal, A. |
| author_facet | Rusli, M. N. N. M. Nurisya, M. S. Zainuddin, H. Umar, M. F. Jellal, A. |
| contents | 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 $θ$ 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 $θ$ is proportional to the ratio between $\hbar$ and $mω$. For the scale $Bθ= \hbar$, 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 $θ$. Finally, if $0 < Bθ< \hbar$, $θ$ has to also be proportional to the ratio between $\hbar$ and $mω$ 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 $θ$ on the system are properly shown for all cases. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_08666 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Rotational Symmetry and Gauge Invariant Degeneracies on 2D Noncommutative Plane Rusli, M. N. N. M. Nurisya, M. S. Zainuddin, H. Umar, M. F. Jellal, A. Quantum Physics High Energy Physics - Theory Mathematical Physics 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 $θ$ 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 $θ$ is proportional to the ratio between $\hbar$ and $mω$. For the scale $Bθ= \hbar$, 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 $θ$. Finally, if $0 < Bθ< \hbar$, $θ$ has to also be proportional to the ratio between $\hbar$ and $mω$ 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 $θ$ on the system are properly shown for all cases. |
| title | Rotational Symmetry and Gauge Invariant Degeneracies on 2D Noncommutative Plane |
| topic | Quantum Physics High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2112.08666 |