749 research outputs found
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
Scattering in three-dimensional fuzzy space
We develop scattering theory in a non-commutative space defined by a
coordinate algebra. By introducing a positive operator valued measure as a
replacement for strong position measurements, we are able to derive explicit
expressions for the probability current, differential and total cross-sections.
We show that at low incident energies the kinematics of these expressions is
identical to that of commutative scattering theory. The consequences of spacial
non-commutativity are found to be more pronounced at the dynamical level where,
even at low incident energies, the phase shifts of the partial waves can
deviate strongly from commutative results. This is demonstrated for scattering
from a spherical well. The impact of non-commutativity on the well's spectrum
and on the properties of its bound and scattering states are considered in
detail. It is found that for sufficiently large well-depths the potential
effectively becomes repulsive and that the cross-section tends towards that of
hard sphere scattering. This can occur even at low incident energies when the
particle's wave-length inside the well becomes comparable to the
non-commutative length-scale.Comment: 12 pages, 6 figure
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
Spectrum of the three dimensional fuzzy well
We develop the formalism of quantum mechanics on three dimensional fuzzy
space and solve the Schr\"odinger equation for a free particle, finite and
infinite fuzzy wells. We show that all results reduce to the appropriate
commutative limits. A high energy cut-off is found for the free particle
spectrum, which also results in the modification of the high energy dispersion
relation. An ultra-violet/infra-red duality is manifest in the free particle
spectrum. The finite well also has an upper bound on the possible energy
eigenvalues. The phase shifts due to scattering around the finite fuzzy
potential well have been calculated
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
Duality constructions from quantum state manifolds
The formalism of quantum state space geometry on manifolds of generalised
coherent states is proposed as a natural setting for the construction of
geometric dual descriptions of non-relativistic quantum systems. These state
manifolds are equipped with natural Riemannian and symplectic structures
derived from the Hilbert space inner product. This approach allows for the
systematic construction of geometries which reflect the dynamical symmetries of
the quantum system under consideration. We analyse here in detail the two
dimensional case and demonstrate how existing results in the AdS_2/CFT_1
context can be understood within this framework. We show how the radial/bulk
coordinate emerges as an energy scale associated with a regularisation
procedure and find that, under quite general conditions, these state manifolds
are asymptotically anti-de Sitter solutions of a class of classical dilaton
gravity models. For the model of conformal quantum mechanics proposed by de
Alfaro et. al. the corresponding state manifold is seen to be exactly AdS_2
with a scalar curvature determined by the representation of the symmetry
algebra. It is also shown that the dilaton field itself is given by the quantum
mechanical expectation values of the dynamical symmetry generators and as a
result exhibits dynamics equivalent to that of a conformal mechanical system.Comment: 25 Pages, References Adde
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
Non-commutative Quantum Mechanics in Three Dimensions and Rotational Symmetry
We generalize the formulation of non-commutative quantum mechanics to three
dimensional non-commutative space. Particular attention is paid to the
identification of the quantum Hilbert space in which the physical states of the
system are to be represented, the construction of the representation of the
rotation group on this space, the deformation of the Leibnitz rule accompanying
this representation and the implied necessity of deforming the co-product to
restore the rotation symmetry automorphism. This also implies the breaking of
rotational invariance on the level of the Schroedinger action and equation as
well as the Hamiltonian, even for rotational invariant potentials. For
rotational invariant potentials the symmetry breaking results purely from the
deformation in the sense that the commutator of the Hamiltonian and angular
momentum is proportional to the deformation.Comment: 21 page
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