Journal of Holography Applications in Physics
Gauge Invariant Degeneracies and Rotational Symmetry Eigenstates in Noncommutative Plane
Document Type : Regular article
Authors
1 International Islamic University Malaysia
2 Institute for Mathematical Research
3 Universiti Pendidikan Sultan Idris
4 Laboratory of Theoretical Physics Faculty of Sciences Ibn Maachou Street
Abstract
We calculate the gauge invariant energy eigenvalues and degeneracies of a spinless charged particle confined in a circular harmonic potential under the influence of a perpendicular magnetic field B on a 2D noncommutative plane. The phase space coordinates transformation based on the 2-parameter family of unitarily equivalent irreducible representations of the nilpotent Lie group GNC was used to accomplish this. We find that the energy eigenvalues and quantum states of the system are unique since they depend on the particle of interest and the applied magnetic field $B$. Without B, we essentially have a noncommutative planar harmonic oscillator under the Bopp shift formulation. The corresponding degeneracy is not unique with respect to the choice of particle, and they are only reliant on the two free integral parameters. The degeneracy is not unique for the scale Bθ = h and is in fact isomorphic to the Landau problem in symmetric gauge; thus, each energy level is infinitely degenerate for any arbitrary magnitude of magnetic field. If 0 < Bθ < h , the degeneracy is unique with respect to both the particle of interest and the applied magnetic field. The system is, in principle, highly non-degenerate and, in practice, effectively non-degenerate, as only the finely-tuned magnetic field can produce degenerate states. In addition, the degeneracy also depends on the two free integral parameters. Numerical examples are provided to present the degeneracies, probability densities, and effects of B and θ on the ground and excited states of the system for all cases using the physical constants from the numerical simulation and experiment on a single GaAs parabolic quantum dot.
Keywords
- Particle on the Noncommutative Plane
- Magnetic Field
- Harmonic Potential
- Gauge Invariance
- Degeneracy
- Wavefunction
- Quantum Dot
Main Subjects
Article PDF
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Volume 2, Issue 4
November 2022Pages 11-36
- Receive Date: 10 August 2022
- Revise Date: 24 September 2022
- Accept Date: 24 September 2022