773 research outputs found
On the Hamilton-Jacobi equation for second class constrained systems
We discuss a general procedure for arriving at the Hamilton-Jacobi equation
of second-class constrained systems, and illustrate it in terms of a number of
examples by explicitely obtaining the respective Hamilton principal function,
and verifying that it leads to the correct solution to the Euler-Lagrange
equations.Comment: 17 pages, to appear in Ann. Phy
Die afrikaner en sy pers*
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Observation of Exceptional Points in Electronic Circuits
Two damped coupled oscillators have been used to demonstrate the occurrence
of exceptional points in a purely classical system. The implementation was
achieved with electronic circuits in the kHz-range. The experimental results
perfectly match the mathematical predictions at the exceptional point. A
discussion about the universal occurrence of exceptional points -- connecting
dissipation with spatial orientation -- concludes the paper.Comment: 4 pages, latex, 3 postscript figures, submitted for publicatio
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
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
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
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
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