20 research outputs found
Geometrization of some quantum mechanics formalism
There were many attempts to geometrize electromagnetic field and find out new
interpretation for quantum mechanics formalism. The distinctive feature of this
work is that it combines geometrization of electromagnetic field and
geometrization of material field within the unique topological idea. According
to the suggested topological interpretation, the Dirac equations for a free
particle and for a hydrogen atom prove to be the group--theoretical relations
that account for the symmetry properties of localized microscopic deviations of
the space--time geometry from the pseudoeuclidean one (closed topological
4-manifolds). These equations happen to be written in universal covering spaces
of the above manifolds. It is shown that "long derivatives" in Dirac equation
for a hydrogen atom can be considered as covariant derivatives of spinors in
the Weyl noneuclidean 4-space and that electromagnetic potentials can be
considered as connectivities in this space. The gauge invariance of
electromagnetic field proves to be a natural consequence of the basic
principles of the proposed geometrical interpretation. Within the suggested
concept, atoms have no inside any point-like particles (electrons) and this can
give an opportunity to overcome the difficulties of atomic physics connected
with the many-body problem.Comment: 9 page
Higher Order Quantum Superintegrability: a new "Painlev\'e conjecture"
We review recent results on superintegrable quantum systems in a
two-dimensional Euclidean space with the following properties. They are
integrable because they allow the separation of variables in Cartesian
coordinates and hence allow a specific integral of motion that is a second
order polynomial in the momenta. Moreover, they are superintegrable because
they allow an additional integral of order . Two types of such
superintegrable potentials exist. The first type consists of "standard
potentials" that satisfy linear differential equations. The second type
consists of "exotic potentials" that satisfy nonlinear equations. For , 4
and 5 these equations have the Painlev\'e property. We conjecture that this is
true for all . The two integrals X and Y commute with the Hamiltonian,
but not with each other. Together they generate a polynomial algebra (for any
) of integrals of motion. We show how this algebra can be used to calculate
the energy spectrum and the wave functions.Comment: 23 pages, submitted as a contribution to the monographic volume
"Integrability, Supersymmetry and Coherent States", a volume in honour of
Professor V\'eronique Hussin. arXiv admin note: text overlap with
arXiv:1703.0975
Isoperiodic classical systems and their quantum counterparts
One-dimensional isoperiodic classical systems have been first analyzed by Abel. Abel's characterization can be extended for singular potentials and potentials which are not defined on the whole real line. The standard shear equivalence of isoperiodic potentials can also be extended by using reflection and inversion transformations. We provide a full characterization of isoperiodic rational potentials showing that they are connected by translations, reflections or Joukowski transformations. Upon
quantization many of these isoperiodic systems fail to exhibit identical quantum energy spectra. This anomaly occurs at order O(h(2)) because semiclassical corrections of energy levels of order O(h(-2)) are identical for all isoperiodic systems. We analyze families of systems where this quantum anomaly occurs and some special systems where the spectral identity is preserved by quantization. Conversely, we point out the
existence of isospectral quantum systems which do not correspond to isoperiodic classical systems