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Coherent states of su(1,1): correlations, fluctuations, and the pseudoharmonic oscillator

John Schliemann

Published Jan 1, 2016• 7 citations
OpenAlex

Abstract

We extend recent results on expectation values of coherent oscillator states and SU(2) coherent states to the case of the discrete representations of su(1,1). Systematic semiclassical expansions of products of arbitrary operators are derived. In particular, the leading order of the energy uncertainty of an arbitrary Hamiltonian is found to be given purely in terms of the time dependence of the classical variables. The coherent states considered here include the Perelomov-Gilmore (PG) coherent states. As an important application we discuss the pseudoharmonic oscillator and compare the PG states with the states introduced by Barut and Girardello. The latter ones turn out to be closer to the classical limit as their relative energy variance decays with the inverse square root of energy, while in the former case a constant is approached.

Related papers

Coherent states of su(1,1): correlations, fluctuations, and the pseudoharmonic oscillator

John Schliemann

Abstract We extend recent results on expectation values of coherent oscillator states and SU(2) coherent states to the case of the discrete representations of su(1,1). Systematic semiclassical expansions of products of arbitrary operators are derived. In particular, the leading order of the energy uncertainty of an arbitrary Hamiltonian is found to be given purely in terms of the time dependence of the classical variables. The coherent states considered here include the Perelomov–Gilmore (PG) coherent states. As an important application we discuss the pseudoharmonic oscillator and compare the PG states with the states introduced by Barut and Girardello. The latter ones turn out to be closer to the classical limit as their relative energy variance decays with the inverse square root of energy, while in the former case a constant is approached.

arXiv:1508.04549•Journal of Physics A Mathematical and Theoretical•Feb 19, 2016•9 citations

Barut-Girardello and Gilmore-Perelomov coherent states for pseudoharmonic oscillator and their nonclassical properties: factorization method

Mohammad Kazem Tavassoly, H R Jalali

In this paper we try to introduce the ladder operators associated with the pseudoharmonic oscillator, after solving the corresponding Schrödinger equation by using the factorization method. The obtained generalized raising and lowering operators naturally lead us to the Dirac representation space of the system which is very easier to work with, in comparison to the functional Hilbert space. The SU(1,1) dynamical symmetry group associated with the considered system is exactly established through investigating the fact that the deduced operators satisfy appropriate commutation relations. This result enables us to construct two important and distinct classes of Barut-Girardello and Gilmore-Perelomov coherent states associated with the system. Finally, their identities as the most important task are exactly resolved and some of their nonclassical properties are illustrated, numerically.

arXiv:1303.4105•Mar 17, 2013•1 citation

PERELOMOV AND BARUT–GIRARDELLO SU(1, 1) COHERENT STATES FOR HARMONIC OSCILLATOR IN ONE-DIMENSIONAL HALF SPACE

Quanhui Liu, H. ZHUO

The Perelomov and the Barut–Girardello SU(1, 1) coherent states for harmonic oscillator in one-dimensional half space are constructed. Results show that the uncertainty products ΔxΔp for these two coherent states are bound from below [Formula: see text] that is the uncertainty for the ground state, and the mean values for position x and momentum p in classical limit go over to their classical quantities respectively. In classical limit, the uncertainty given by Perelomov coherent does not vanish, and the Barut–Girardello coherent state reveals a node structure when positioning closest to the boundary x = 0 which has not been observed in coherent states for other systems.

International Journal of Modern Physics A•May 10, 2006•1 citation

Barut-Girardello coherent states of the pseudoharmonic oscillator

Dušan Popov

In the paper we have investigated some properties of the Barut-Giraradello coherent states (BGCS) (the eigenstates of the SU(1,1) lowering generator K-) for the Hamiltonian of the pseudoharmonic oscillator (PHO). By using these states, the diagonal P-representation of the density operator is constructed as a new result for this potential. In addition, we have calculated some thermal expectation values for the quantum canonical gas of the PHOs. The calculations using the BGCS representation seem to be much simpler and easier to program than the corresponding calculations in other representations (e.g. the position representation). At the end of the paper the time dependence of these states is presented.

Journal of Physics A Mathematical and Theoretical•Jun 15, 2001•48 citations

Barut-Girardello and Gilmore-Perelomov coherent states for pseudoharmonic oscillators and their nonclassical properties： Factorization method

Tavassoly, RJalali

In this paper we try to introduce the ladder operators associated with the pseudoharmonic oscillator, after solving the corresponding Schro¨dinger equation by using the factorization method. The obtained generalized raising and lowering operators naturally lead us to the Dirac representation space of the system which is much easier to work with, in comparison to the functional Hilbert space. The SU(1, 1) dynamical symmetry group associated with the considered system is exactly established through investigating the fact that the deduced operators satisfy appropriate commutation relations. This result enables us to construct two important and distinct classes of Barut-Girardello and Gilmore-Perelomov coherent states associated with the system. Finally, their identities as the most important task are exactly resolved and some of their nonclassical properties are illustrated, numerically.

中国物理B：英文版•Jan 1, 2013•0 citations

Barut—Girardello and Gilmore—Perelomov coherent states for pseudoharmonic oscillators and their nonclassical properties: Factorization method

Mohammad Kazem Tavassoly, H R Jalali

In this paper we try to introduce the ladder operators associated with the pseudoharmonic oscillator, after solving the corresponding Schrödinger equation by using the factorization method. The obtained generalized raising and lowering operators naturally lead us to the Dirac representation space of the system which is much easier to work with, in comparison to the functional Hilbert space. The SU (1, 1) dynamical symmetry group associated with the considered system is exactly established through investigating the fact that the deduced operators satisfy appropriate commutation relations. This result enables us to construct two important and distinct classes of Barut—Girardello and Gilmore—Perelomov coherent states associated with the system. Finally, their identities as the most important task are exactly resolved and some of their nonclassical properties are illustrated, numerically.

Chinese Physics B•Aug 1, 2013•11 citations
We extend recent results on expectation values of coherent oscillator states and SU(2) coherent states to the case of the discrete representations of su(1,1). Systematic semiclassical expansions of products of arbitrary operators are derived. In particular, the leading order of the energy uncertainty of an arbitrary Hamiltonian is found to be given purely in terms of the time dependence of the classical variables. The coherent states considered here include the Perelomov-Gilmore (PG) coherent states. As an important application we discuss the pseudoharmonic oscillator and compare the PG states with the states introduced by Barut and Girardello. The latter ones turn out to be closer to the classical limit as their relative energy variance decays with the inverse square root of energy, while in the former case a constant is approached.
Abstract We extend recent results on expectation values of coherent oscillator states and SU(2) coherent states to the case of the discrete representations of su(1,1). Systematic semiclassical expansions of products of arbitrary operators are derived. In particular, the leading order of the energy uncertainty of an arbitrary Hamiltonian is found to be given purely in terms of the time dependence of the classical variables. The coherent states considered here include the Perelomov–Gilmore (PG) coherent states. As an important application we discuss the pseudoharmonic oscillator and compare the PG states with the states introduced by Barut and Girardello. The latter ones turn out to be closer to the classical limit as their relative energy variance decays with the inverse square root of energy, while in the former case a constant is approached.
In this paper we try to introduce the ladder operators associated with the pseudoharmonic oscillator, after solving the corresponding Schrödinger equation by using the factorization method. The obtained generalized raising and lowering operators naturally lead us to the Dirac representation space of the system which is very easier to work with, in comparison to the functional Hilbert space. The SU(1,1) dynamical symmetry group associated with the considered system is exactly established through investigating the fact that the deduced operators satisfy appropriate commutation relations. This result enables us to construct two important and distinct classes of Barut-Girardello and Gilmore-Perelomov coherent states associated with the system. Finally, their identities as the most important task are exactly resolved and some of their nonclassical properties are illustrated, numerically.
The Perelomov and the Barut–Girardello SU(1, 1) coherent states for harmonic oscillator in one-dimensional half space are constructed. Results show that the uncertainty products ΔxΔp for these two coherent states are bound from below [Formula: see text] that is the uncertainty for the ground state, and the mean values for position x and momentum p in classical limit go over to their classical quantities respectively. In classical limit, the uncertainty given by Perelomov coherent does not vanish, and the Barut–Girardello coherent state reveals a node structure when positioning closest to the boundary x = 0 which has not been observed in coherent states for other systems.
In the paper we have investigated some properties of the Barut-Giraradello coherent states (BGCS) (the eigenstates of the SU(1,1) lowering generator K-) for the Hamiltonian of the pseudoharmonic oscillator (PHO). By using these states, the diagonal P-representation of the density operator is constructed as a new result for this potential. In addition, we have calculated some thermal expectation values for the quantum canonical gas of the PHOs. The calculations using the BGCS representation seem to be much simpler and easier to program than the corresponding calculations in other representations (e.g. the position representation). At the end of the paper the time dependence of these states is presented.
In this paper we try to introduce the ladder operators associated with the pseudoharmonic oscillator, after solving the corresponding Schro¨dinger equation by using the factorization method. The obtained generalized raising and lowering operators naturally lead us to the Dirac representation space of the system which is much easier to work with, in comparison to the functional Hilbert space. The SU(1, 1) dynamical symmetry group associated with the considered system is exactly established through investigating the fact that the deduced operators satisfy appropriate commutation relations. This result enables us to construct two important and distinct classes of Barut-Girardello and Gilmore-Perelomov coherent states associated with the system. Finally, their identities as the most important task are exactly resolved and some of their nonclassical properties are illustrated, numerically.
In this paper we try to introduce the ladder operators associated with the pseudoharmonic oscillator, after solving the corresponding Schrödinger equation by using the factorization method. The obtained generalized raising and lowering operators naturally lead us to the Dirac representation space of the system which is much easier to work with, in comparison to the functional Hilbert space. The SU (1, 1) dynamical symmetry group associated with the considered system is exactly established through investigating the fact that the deduced operators satisfy appropriate commutation relations. This result enables us to construct two important and distinct classes of Barut—Girardello and Gilmore—Perelomov coherent states associated with the system. Finally, their identities as the most important task are exactly resolved and some of their nonclassical properties are illustrated, numerically.