The non-commutative oscillator, symmetry and the Landau problem

Abstract

Isotropic oscillator on a plane is discussed where both the coordinate and momentum space are considered to be noncommutative. We also discuss the symmetry properties of the oscillator for three separate cases when both the noncommutative parameters Θ and Θ satisfy specific relations. We compare the Landau problem with the isotropic oscillator on noncommutative space and obtain a relation between the two noncommutative parameters with the magnetic field of the Landau problem.

Constrains of Charge-to-Mass Ratios on Noncommutative Phase Space

“…One can see that this Hamiltonian mimics the ordinary Landau problem, apart from a correction on the Larmor frequency. The noncommutative extension of the Landau system has been studied extensively [46][47][48][49][50][51][52][53][54]. However, so far the charge-tomass ratio related physics have not been studied.…”

Equivalence of curvature and noncommutativity in a physical space: Harmonic oscillator on sphere

“…3,4 Moreover, the relation between noncommutativity and the magnetic field has been shown in various other works. 4,17,18 There is a well-known relation between a constant noncommutativity parameter and constant magnetic field that is mentioned in most of these references which is θ = 1/B. But it is important to notice that this relation is not always true, specially when one deals with non-unitary transformations and nonconstant noncommutativity parameter, this parameter and magnetic field can be related differently.…”

Harmonic Oscillator Chain in Noncommutative Phase Space with Rotational Symmetry

“…Harmonic oscillator was intensively studied in the frame of noncommutative algebra [34,35,36,37,38,39,40,41,42,43,44,45,46,47,48]. Recently experiments with micro-and nano-oscillators were implemented for probing minimal length [49].…”