732 research outputs found
-particle sigma model: Momentum-space quantization of a particle on the sphere
We perform the momentum-space quantization of a spin-less particle moving on
the group manifold, that is, the three-dimensional sphere , by
using a non-canonical method entirely based on symmetry grounds. To achieve
this task, non-standard (contact) symmetries are required as already shown in a
previous article where the configuration-space quantization was given. The
Hilbert space in the momentum space representation turns out to be made of a
subset of (oscillatory) solutions of the Helmholtz equation in four dimensions.
The most relevant result is the fact that both the scalar product and the
generalized Fourier transform between configuration and momentum spaces deviate
notably from the naively expected expressions, the former exhibiting now a
non-trivial kernel, under a double integral, traced back to the non-trivial
topology of the phase space, even though the momentum space as such is flat. In
addition, momentum space itself appears directly as the carrier space of an
irreducible representation of the symmetry group, and the Fourier transform as
the unitary equivalence between two unitary irreducible representations.Comment: 29 pages, 3 figure
The Quantum Arnold Transformation and its applications
The Quantum Arnold Transformation, a unitary operator mapping the solutions of the Schr¨odinger equation for time-dependent quadratic Hamiltonians into free-particle solutions, is revisited. Possible applications and extensions are also outlined: the analytic construction of harmonic states for the free particle, the Quantum Arnold-Ermakov-Pinney transformation and the description of the
Release & Recapture method
Coherent States for infinite homogeneous waveguide arrays
Perelomov coherent states for equally spaced, infinite homogeneous waveguide
arrays with Euclidean E(2) symmetry are defined, and new resolutions of the identity
are constructed in Cartesian and polar coordinates. The key point to construct these
resolutions of the identity is the fact that coherent states satisfy Helmholtz equation
(in coherent states labels) an thus a non-local scalar product with a convolution
kernel can be introduced which is invariant under the Euclidean group. It is also
shown that these coherent states for the Eucliean E(2) group have a simple and
natural physical realization in these waveguide arrays.Spanish MICINN through the project PGC2018-097831-B-I00 and Junta de Andalucı́a through the project FEDER/UJA-1381026
The triple-shell structure and collimated outflows of the planetary nebula NGC 6891
Narrow-band H and [N {\sc ii}] images and high-dispersion
spatially-resolved echelle spectroscopy of the planetary nebula NGC 6891 are
presented. These observations show a great wealth of structures. The bright
central nebula is surrounded by an attached shell and a detached outer halo.
Both the inner and intermediate shells can be described as ellipsoids with
similar major to minor axial ratios, but different spatial orientations. The
kinematical ages of the intermediate shell and halo are 4,800 and 28,000 years,
respectively. The inter-shell time lapse is in good agreement with the
evolutionary inter-pulse time lapse. A highly collimated outflow is observed to
protrude from the tips of the major axis of the inner nebula and impact on the
outer edge of the intermediate shell. Kinematics and excitation of this outflow
provide conclusive evidence that it is deflected during the interaction with
the outer edge of the intermediate shell. At the same time, both the kinematics
and the morphology of the intermediate shell appear to be affected by this
interaction.Comment: To be published in MNRAS, 9 pages, 5 figure
Harmonic states for the free particle
Different families of states, which are solutions of the time-dependent free
Schr\"odinger equation, are imported from the harmonic oscillator using the
Quantum Arnold Transformation introduced in a previous paper. Among them,
infinite series of states are given that are normalizable, expand the whole
space of solutions, are spatially multi-localized and are eigenstates of a
suitably defined number operator. Associated with these states new sets of
coherent and squeezed states for the free particle are defined representing
traveling, squeezed, multi-localized wave packets. These states are also
constructed in higher dimensions, leading to the quantum mechanical version of
the Hermite-Gauss and Laguerre-Gauss states of paraxial wave optics. Some
applications of these new families of states and procedures to experimentally
realize and manipulate them are outlined.Comment: 21 pages, 3 figures. Title changed, content added, references adde
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