1,685 research outputs found
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
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