Gauge Invariant Degeneracies and Rotational Symmetry Eigenstates in Noncommutative Plane

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.