1,004 research outputs found
Generalized coherent and intelligent states for exact solvable quantum systems
The so-called Gazeau-Klauder and Perelomov coherent states are introduced for
an arbitrary quantum system. We give also the general framework to construct
the generalized intelligent states which minimize the Robertson-Schr\"odinger
uncertainty relation. As illustration, the P\"oschl-Teller potentials of
trigonometric type will be chosen. We show the advantage of the analytical
representations of Gazeau-Klauder and Perelomov coherent states in obtaining
the generalized intelligent states in analytical way
Analytic representations based on SU(1,1) coherent states and their applications
We consider two analytic representations of the SU(1,1) Lie group: the
representation in the unit disk based on the SU(1,1) Perelomov coherent states
and the Barut-Girardello representation based on the eigenstates of the SU(1,1)
lowering generator. We show that these representations are related through a
Laplace transform. A ``weak'' resolution of the identity in terms of the
Perelomov SU(1,1) coherent states is presented which is valid even when the
Bargmann index is smaller than one half. Various applications of these
results in the context of the two-photon realization of SU(1,1) in quantum
optics are also discussed.Comment: LaTeX, 15 pages, no figures, to appear in J. Phys. A. More
information on http://www.technion.ac.il/~brif/science.htm
A few remarks on integral representation for zonal spherical functions on the symmetric space
The integral representation on the orthogonal groups for zonal spherical
functions on the symmetric space is used to obtain a
generating function for such functions. For the case N=3 the three-dimensional
integral representation reduces to a one-dimensional one.Comment: Latex file, 10 pages, amssymb.sty require
and Perelomov number coherent states: algebraic approach for general systems
We study some properties of the Perelomov number coherent states.
The Schr\"odinger's uncertainty relationship is evaluated for a position and
momentum-like operators (constructed from the Lie algebra generators) in these
number coherent states. It is shown that this relationship is minimized for the
standard coherent states. We obtain the time evolution of the number coherent
states by supposing that the Hamiltonian is proportional to the third generator
of the Lie algebra. Analogous results for the Perelomov
number coherent states are found. As examples, we compute the Perelomov
coherent states for the pseudoharmonic oscillator and the two-dimensional
isotropic harmonic oscillator
SU(2) and SU(1,1) algebra eigenstates: A unified analytic approach to coherent and intelligent states
We introduce the concept of algebra eigenstates which are defined for an
arbitrary Lie group as eigenstates of elements of the corresponding complex Lie
algebra. We show that this concept unifies different definitions of coherent
states associated with a dynamical symmetry group. On the one hand, algebra
eigenstates include different sets of Perelomov's generalized coherent states.
On the other hand, intelligent states (which are squeezed states for a system
of general symmetry) also form a subset of algebra eigenstates. We develop the
general formalism and apply it to the SU(2) and SU(1,1) simple Lie groups.
Complete solutions to the general eigenvalue problem are found in the both
cases, by a method that employs analytic representations of the algebra
eigenstates. This analytic method also enables us to obtain exact closed
expressions for quantum statistical properties of an arbitrary algebra
eigenstate. Important special cases such as standard coherent states and
intelligent states are examined and relations between them are studied by using
their analytic representations.Comment: LaTeX, 24 pages, 1 figure (compressed PostScript, available at
http://www.technion.ac.il/~brif/abstracts/AES.html ). More information on
http://www.technion.ac.il/~brif/science.htm
Darboux transformations of coherent states of the time-dependent singular oscillator
Darboux transformation of both Barut-Girardello and Perelomov coherent states
for the time-dependent singular oscillator is studied. In both cases the
measure that realizes the resolution of the identity operator in terms of
coherent states is found and corresponding holomorphic representation is
constructed. For the particular case of a free particle moving with a fixed
value of the angular momentum equal to two it is shown that Barut-Giriardello
coherent states are more localized at the initial time moment while the
Perelomov coherent states are more stable with respect to time evolution. It is
also illustrated that Darboux transformation may keep unchanged this different
time behavior.Comment: 13 page
Temporally stable coherent states in energy degenerate systems: The hydrogen atom
Klauder's recent generalization of the harmonic oscillator coherent states
[J. Phys. A 29, L293 (1996)] is applicable only in non-degenerate systems,
requiring some additional structure if applied to systems with degeneracies.
The author suggests how this structure could be added, and applies the complete
method to the hydrogen atom problem. To illustrate how a certain degree of
freedom in the construction may be exercised, states are constructed which are
initially localized and evolve semi-classically, and whose long time evolution
exhibits "fractional revivals."Comment: 9 pages, 3 figure
Representations and Properties of Generalized Statistics, Coherent States and Robertson Uncertainty Relations
The generalization of statistics, including bosonic and fermionic
sectors, is performed by means of the so-called Jacobson generators. The
corresponding Fock spaces are constructed. The Bargmann representations are
also considered. For the bosonic statistics, two inequivalent Bargmann
realizations are developed. The first (resp. second) realization induces, in a
natural way, coherent states recognized as Gazeau-Klauder (resp.
Klauder-Perelomov) ones. In the fermionic case, the Bargamnn realization leads
to the Klauder-Perelomov coherent states. For each considered realization, the
inner product of two analytic functions is defined in respect to a measure
explicitly computed. The Jacobson generators are realized as differential
operators. It is shown that the obtained coherent states minimize the
Robertson-Schr\"odinger uncertainty relation.Comment: 16 pages, published in JP
The Moyal Bracket in the Coherent States framework
The star product and Moyal bracket are introduced using the coherent states
corresponding to quantum systems with non-linear spectra. Two kinds of coherent
state are considered. The first kind is the set of Gazeau-Klauder coherent
states and the second kind are constructed following the Perelomov-Klauder
approach. The particular case of the harmonic oscillator is also discussed.Comment: 13 page
Poisson-Jacobi reduction of homogeneous tensors
The notion of homogeneous tensors is discussed. We show that there is a
one-to-one correspondence between multivector fields on a manifold ,
homogeneous with respect to a vector field on , and first-order
polydifferential operators on a closed submanifold of codimension 1 such
that is transversal to . This correspondence relates the
Schouten-Nijenhuis bracket of multivector fields on to the Schouten-Jacobi
bracket of first-order polydifferential operators on and generalizes the
Poissonization of Jacobi manifolds. Actually, it can be viewed as a
super-Poissonization. This procedure of passing from a homogeneous multivector
field to a first-order polydifferential operator can be also understood as a
sort of reduction; in the standard case -- a half of a Poisson reduction. A
dual version of the above correspondence yields in particular the
correspondence between -homogeneous symplectic structures on and
contact structures on .Comment: 19 pages, minor corrections, final version to appear in J. Phys. A:
Math. Ge
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