379 research outputs found
Heisenberg-type higher order symmetries of superintegrable systems separable in cartesian coordinates
Heisenberg-type higher order symmetries are studied for both classical and
quantum mechanical systems separable in cartesian coordinates. A few particular
cases of this type of superintegrable systems were already considered in the
literature, but here they are characterized in full generality together with
their integrability properties. Some of these systems are defined only in a
region of , and in general they do not include bounded solutions.
The quantum symmetries and potentials are shown to reduce to their
superintegrable classical analogs in the limit.Comment: 23 Pages, 3 figures, To appear in Nonlinearit
Superintegrability of the Fock-Darwin system
The Fock-Darwin system is analysed from the point of view of its symmetry
properties in the quantum and classical frameworks. The quantum Fock-Darwin
system is known to have two sets of ladder operators, a fact which guarantees
its solvability. We show that for rational values of the quotient of two
relevant frequencies, this system is superintegrable, the quantum symmetries
being responsible for the degeneracy of the energy levels. These symmetries are
of higher order and close a polynomial algebra. In the classical case, the
ladder operators are replaced by ladder functions and the symmetries by
constants of motion. We also prove that the rational classical system is
superintegrable and its trajectories are closed. The constants of motion are
also generators of symmetry transformations in the phase space that have been
integrated for some special cases. These transformations connect different
trajectories with the same energy. The coherent states of the quantum
superintegrable system are found and they reproduce the closed trajectories of
the classical one.Comment: 21 pages,16 figure
Classical ladder functions for Rosen-Morse and curved Kepler-Coulomb systems
Producción CientíficaLadder functions in classical mechanics are defined in a similar way as ladder operators in the context of quantum mechanics. In the present paper, we develop a new method for obtaining ladder functions of one dimensional systems by means of a product of two ‘factor functions’. We apply this method to the curved Kepler–Coulomb and Rosen–Morse II systems whose ladder functions were not found yet. The ladder functions here obtained are applied to get the motion of the systems.Ministerio de Economía, Industria y Competitividad (project MTM2014-57129-C2-1-P)Junta de Castilla y León-FEDER (projects BU229P18 / VA057U16 / VA137G18)
SUSY approach to Pauli Hamiltonians with an axial symmetry
A two-dimensional Pauli Hamiltonian describing the interaction of a neutral
spin-1/2 particle with a magnetic field having axial and second order
symmetries, is considered. After separation of variables, the one-dimensional
matrix Hamiltonian is analyzed from the point of view of supersymmetric quantum
mechanics. Attention is paid to the discrete symmetries of the Hamiltonian and
also to the Hamiltonian hierarchies generated by intertwining operators. The
spectrum is studied by means of the associated matrix shape-invariance. The
relation between the intertwining operators and the second order symmetries is
established and the full set of ladder operators that complete the dynamical
algebra is constructed.Comment: 18 pages, 3 figure
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