Harmonic oscillator in a background magnetic field in noncommutative quantum phase-space

We solve explicitly the two-dimensional harmonic oscillator and the harmonic oscillator in a background magnetic field in noncommutative phase-space without making use of any type of representation. A key observation that we make is that for a specific choice of the noncommutative parameters, the time reversal symmetry of the systems get restored since the energy spectrum becomes degenerate. This is in contrast to the noncommutative configuration space where the time reversal symmetry of the harmonic oscillator is always broken.Comment: 7 pages Late

Variations on the Planar Landau Problem: Canonical Transformations, A Purely Linear Potential and the Half-Plane

The ordinary Landau problem of a charged particle in a plane subjected to a perpendicular homogeneous and static magnetic field is reconsidered from different points of view. The role of phase space canonical transformations and their relation to a choice of gauge in the solution of the problem is addressed. The Landau problem is then extended to different contexts, in particular the singular situation of a purely linear potential term being added as an interaction, for which a complete purely algebraic solution is presented. This solution is then exploited to solve this same singular Landau problem in the half-plane, with as motivation the potential relevance of such a geometry for quantum Hall measurements in the presence of an electric field or a gravitational quantum well

Bound state energies and phase shifts of a non-commutative well

Non-commutative quantum mechanics can be viewed as a quantum system represented in the space of Hilbert-Schmidt operators acting on non-commutative configuration space. Within this framework an unambiguous definition can be given for the non-commutative well. Using this approach we compute the bound state energies, phase shifts and scattering cross sections of the non- commutative well. As expected the results are very close to the commutative results when the well is large or the non-commutative parameter is small. However, the convergence is not uniform and phase shifts at certain energies exhibit a much stronger then expected dependence on the non-commutative parameter even at small values.Comment: 12 pages, 8 figure

Spectrum of the non-commutative spherical well

We give precise meaning to piecewise constant potentials in non-commutative quantum mechanics. In particular we discuss the infinite and finite non-commutative spherical well in two dimensions. Using this, bound-states and scattering can be discussed unambiguously. Here we focus on the infinite well and solve for the eigenvalues and eigenfunctions. We find that time reversal symmetry is broken by the non-commutativity. We show that in the commutative and thermodynamic limits the eigenstates and eigenfunctions of the commutative spherical well are recovered and time reversal symmetry is restored

Twist Deformation of Rotationally Invariant Quantum Mechanics

Non-commutative Quantum Mechanics in 3D is investigated in the framework of the abelian Drinfeld twist which deforms a given Hopf algebra while preserving its Hopf algebra structure. Composite operators (of coordinates and momenta) entering the Hamiltonian have to be reinterpreted as primitive elements of a dynamical Lie algebra which could be either finite (for the harmonic oscillator) or infinite (in the general case). The deformed brackets of the deformed angular momenta close the so(3) algebra. On the other hand, undeformed rotationally invariant operators can become, under deformation, anomalous (the anomaly vanishes when the deformation parameter goes to zero). The deformed operators, Taylor-expanded in the deformation parameter, can be selected to minimize the anomaly. We present the deformations (and their anomalies) of undeformed rotationally-invariant operators corresponding to the harmonic oscillator (quadratic potential), the anharmonic oscillator (quartic potential) and the Coulomb potential.Comment: 20 page

Formulation, Interpretation and Application of non-Commutative Quantum Mechanics

In analogy with conventional quantum mechanics, non-commutative quantum mechanics is formulated as a quantum system on the Hilbert space of Hilbert-Schmidt operators acting on non-commutative configuration space. It is argued that the standard quantum mechanical interpretation based on Positive Operator Valued Measures, provides a sufficient framework for the consistent interpretation of this quantum system. The implications of this formalism for rotational and time reversal symmetry are discussed. The formalism is applied to the free particle and harmonic oscillator in two dimensions and the physical signatures of non commutativity are identified.Comment: 11 page