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

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

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

The N=1 Supersymmetric Landau Problem and its Supersymmetric Landau Level Projections: the N=1 Supersymmetric Moyal-Voros Superplane

The N=1 supersymmetric invariant Landau problem is constructed and solved. By considering Landau level projections remaining non trivial under N=1 supersymmetry transformations, the algebraic structures of the N=1 supersymmetric covariant non(anti)commutative superplane analogue of the ordinary N=0 noncommutative Moyal-Voros plane are identified

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

On asymptotically flat solutions of Einstein's equations periodic in time I. Vacuum and electrovacuum solutions

By an argument similar to that of Gibbons and Stewart, but in a different coordinate system and less restrictive gauge, we show that any weakly-asymptotically-simple, analytic vacuum or electrovacuum solutions of the Einstein equations which are periodic in time are necessarily stationary.Comment: 25 pages, 2 figures, published in Class. Quant. Grav

Establishing the impact of powerful AGN on their host galaxies

Establishing the role of active galactic nuclei (AGN) during the formation of galaxies remains one of the greatest challenges of galaxy formation theory. Towards addressing this, we summarise our recent work investigating: (1) the physical drivers of ionised outflows and (2) observational signatures of the impact by jets/outflows on star formation and molecular gas content in AGN host galaxies. We confirm a connection between radio emission and extreme ionised gas kinematics in AGN hosts. Emission-line selected AGN are significantly more likely to exhibit ionised outflows (as traced by the [O III] emission line) if the projected linear extent of the radio emission is confined within the spectroscopic aperture. Follow-up high resolution radio observations and integral field spectroscopy of 10 luminous Type 2 AGN reveal moderate power, young (or frustrated) jets interacting with the interstellar medium. We find that these sources live in highly star forming and gas rich galaxies. Additionally, by combining ALMA-derived dust maps with integral field spectroscopy for eight host galaxies of z~2 X-ray AGN, we show that H-alpha emission is an unreliable tracer of star formation. For the five targets with ionised outflows we find no dramatic in-situ shut down of the star formation. Across both of these studies we find that if these AGN do have a negative impact upon their host galaxies, it must be happening on small (unresolved) spatial scales and/or an observable galaxy-wide impact has yet to occur.Comment: Invited Contribution to IAU Symposium 359 (T. Storchi-Bergmann, R. Overzier, W. Forman & R. Riffel, eds.

On asymptotically flat solutions of Einstein's equations periodic in time II. Spacetimes with scalar-field sources

We extend the work in our earlier article [4] to show that time-periodic, asymptotically-flat solutions of the Einstein equations analytic at scri, whose source is one of a range of scalar-field models, are necessarily stationary. We also show that, for some of these scalar-field sources, in stationary, asymptotically-flat solutions analytic at scri, the scalar field necessarily inherits the symmetry. To prove these results we investigate miscellaneous properties of massless and conformal scalar fields coupled to gravity, in particular Bondi mass and its loss.Comment: 29 pages, published in Class. Quant. Grav. Replaced. Typos corrected, version which appeared in Class. Quant.Gra