Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Lobachevskii Journal of Mathematics

Generalized Coherent States Representation

R. Sh. Kalmetev, Yu. N. Orlov, Vsevolod Zhanovich Sakbaev

Published Nov 1, 2021• 5 citations
Publisher

Abstract

Abstract The properties of the system of eigenvectors of the annihilation operator are investigated for the case of non-classical commutation relations between the operators of creation and annihilation. The existence of a measure on the set of eigenvalues of the annihilation operator is proved, with respect to which coherent states form a generalized crowded system. Formulas of covariant symbols of vectors and operators in the representation of generalized coherent states are obtained. It is shown that, similarly to the classical boson case, generalized coherent states are states with minimal uncertainty.

DOI
10.1134/s1995080221110123

Related papers

Coherent states and the -representation

J. Žák

Commuting translations in the phase plane are shown to be observables that are canonically conjugate to the annihilation operator of the harmonic oscillator states. The eigenfunctions and the eigenvalues of these commuting operators form the kq-representation. The eigenfunctions of the annihilation operator are the coherent states. The kq-eigenfunctions and the coherent states are the limiting cases of quantum-mechanical and classical wavefunctions, respectively, and it is satisfying to discover that they relate to canonically conjugate operators.

Journal of Physics A Mathematical and General•Aug 7, 1997•6 citations

Eigenvalue problem for arbitrary linear combinations of a boson annihilation and creation operator

Alfred Wünsche

Abstract The eigenvalue problem for arbitrary linear combinations kα + μα† of a boson annihilation operator α and a boson creation operator α† is solved. It is shown that these operators possess nondegenerate eigenstates to arbitrary complex eigenvalues. The expansion of these eigenstates into the basic set of number states | n >, (n = 0, 1, 2, …), is found. The eigenstates are normalizable and are therefore states of a Hilbert space for | ζ | < 1 with ζ  μ/k and represent in this case squeezed coherent states of minimal uncertainty product. They can be considered as states of a rigged Hilbert space for | ζ | ⩾ 1. A completeness relation for these states is derived that generalizes the completeness relation for the coherent states | α 〉. Furthermore, it is shown that there exists a dual orthogonality in the entire set of these states and a connected dual completeness of the eigenstates on widely arbitrary paths over the complex plane of eigenvalues. This duality goes over into a selfduality of the eigenstates of the hermitian operators kα + k* α† to real eigenvalues. The usually as nonexistent considered eigenstates of the boson creation operator α† are obtained by a limiting procedure. They belong to the most singular case among the considered general class of eigenstates with ζ  μ/k as a parameter.

Annalen der Physik•Jan 1, 1992•22 citations

Coherent States

Bryan J. Dalton, John Jeffers, Stephen Mark Barnett

Abstract This chapter defines and determines the properties of the important (though non-physical) boson and fermion coherent states, which are specified in terms of c-number (bosons) or Grassmann (fermions) variables and their complex conjugates, and defined via unitary displacement operators acting on the vacuum state. Important non-orthogonality, overcompleteness and coherence properties of coherent states are obtained. Coherent states are shown to be eigenstates of mode annihilation operators, with eigenvalues given by the c-number or Grassmann variable specifying the state. Related (unnormalised) Bargmann states are defined, depending only on the variable and not its complex conjugate, and are used to obtain useful representations of physical states and canonical forms for operators in terms of Bargmann state projectors, as well as obtaining important results such as for traces of operator products in terms of phase space integrals. Mode creation operators acting on Bargmann states are shown as equivalent to differentiation.

•Nov 13, 2014•0 citations

State-Vector Space and Canonical Coherent States in Noncommutative Plane

Sicong Jing, Tao Ling-Ping, Qiuyu Liu +1 more

The structure of the state-vector space of identical bosons in noncommutative spaces is investigated. To maintain Bose–Einstein statistics the commutation relations of phase space variables should simultaneously include coordinate-coordinate non-commutativity and momentum-momentum non-commutativity, which leads to a kind of deformed Heisenberg-Weyl algebra. Although there is no ordinary number representation in this state-vector space, several set of orthogonal and complete state-vectors can be derived which are common eigenvectors of corresponding pairs of commuting Hermitian operators. As a simple application of this state-vector space, an explicit form of two-dimensional canonical coherent state is constructed and its properties are discussed.

arXiv:hep-th/0505048•Communications in Theoretical Physics•Feb 1, 2006•6 citations

Para-Bose coherent states

J. K. Sharma, C. L. Mehta, E. C. G. Sudarshan

In place of the usual commutation relation [â,â†]=1 we consider the generalized commutation relation characteristic of the para-Bose oscillators, viz, [â, Ĥ]=Ĥ where Ĥ is the Hamiltonian (1/2)(ââ†+â†â). The number states and the representation of various operators in the basis formed by these states are obtained. We then introduce the para-Bose coherent states defined as the eigenstates of â for this generalized case. We consider some of the properties of these coherent states and also show that the uncertainty product 〈 (Δq̂)2〉 〈 (Δp̂)2〉 in this case could be made arbitrarily small.

Journal of Mathematical Physics•Oct 1, 1978•73 citations

Construction of coherent states for physical algebraic systems

Y. Hassouni, Evaldo M. F. Curado, M. A. Rego-Monteiro

We construct a general state which is an eigenvector of the annihilation operator of the generalized Heisenberg algebra. We show, for several systems characterized by different energy spectra, that this general state satisfies the minimal set of conditions required to obtain Klauder's minimal coherent states.

arXiv:math-ph/0408030•Physical Review A•Feb 11, 2005•43 citations
Abstract The properties of the system of eigenvectors of the annihilation operator are investigated for the case of non-classical commutation relations between the operators of creation and annihilation. The existence of a measure on the set of eigenvalues of the annihilation operator is proved, with respect to which coherent states form a generalized crowded system. Formulas of covariant symbols of vectors and operators in the representation of generalized coherent states are obtained. It is shown that, similarly to the classical boson case, generalized coherent states are states with minimal uncertainty.
Commuting translations in the phase plane are shown to be observables that are canonically conjugate to the annihilation operator of the harmonic oscillator states. The eigenfunctions and the eigenvalues of these commuting operators form the kq-representation. The eigenfunctions of the annihilation operator are the coherent states. The kq-eigenfunctions and the coherent states are the limiting cases of quantum-mechanical and classical wavefunctions, respectively, and it is satisfying to discover that they relate to canonically conjugate operators.
Abstract The eigenvalue problem for arbitrary linear combinations kα + μα† of a boson annihilation operator α and a boson creation operator α† is solved. It is shown that these operators possess nondegenerate eigenstates to arbitrary complex eigenvalues. The expansion of these eigenstates into the basic set of number states | n >, (n = 0, 1, 2, …), is found. The eigenstates are normalizable and are therefore states of a Hilbert space for | ζ | < 1 with ζ  μ/k and represent in this case squeezed coherent states of minimal uncertainty product. They can be considered as states of a rigged Hilbert space for | ζ | ⩾ 1. A completeness relation for these states is derived that generalizes the completeness relation for the coherent states | α 〉. Furthermore, it is shown that there exists a dual orthogonality in the entire set of these states and a connected dual completeness of the eigenstates on widely arbitrary paths over the complex plane of eigenvalues. This duality goes over into a selfduality of the eigenstates of the hermitian operators kα + k* α† to real eigenvalues. The usually as nonexistent considered eigenstates of the boson creation operator α† are obtained by a limiting procedure. They belong to the most singular case among the considered general class of eigenstates with ζ  μ/k as a parameter.
Abstract This chapter defines and determines the properties of the important (though non-physical) boson and fermion coherent states, which are specified in terms of c-number (bosons) or Grassmann (fermions) variables and their complex conjugates, and defined via unitary displacement operators acting on the vacuum state. Important non-orthogonality, overcompleteness and coherence properties of coherent states are obtained. Coherent states are shown to be eigenstates of mode annihilation operators, with eigenvalues given by the c-number or Grassmann variable specifying the state. Related (unnormalised) Bargmann states are defined, depending only on the variable and not its complex conjugate, and are used to obtain useful representations of physical states and canonical forms for operators in terms of Bargmann state projectors, as well as obtaining important results such as for traces of operator products in terms of phase space integrals. Mode creation operators acting on Bargmann states are shown as equivalent to differentiation.
The structure of the state-vector space of identical bosons in noncommutative spaces is investigated. To maintain Bose–Einstein statistics the commutation relations of phase space variables should simultaneously include coordinate-coordinate non-commutativity and momentum-momentum non-commutativity, which leads to a kind of deformed Heisenberg-Weyl algebra. Although there is no ordinary number representation in this state-vector space, several set of orthogonal and complete state-vectors can be derived which are common eigenvectors of corresponding pairs of commuting Hermitian operators. As a simple application of this state-vector space, an explicit form of two-dimensional canonical coherent state is constructed and its properties are discussed.
In place of the usual commutation relation [â,â†]=1 we consider the generalized commutation relation characteristic of the para-Bose oscillators, viz, [â, Ĥ]=Ĥ where Ĥ is the Hamiltonian (1/2)(ââ†+â†â). The number states and the representation of various operators in the basis formed by these states are obtained. We then introduce the para-Bose coherent states defined as the eigenstates of â for this generalized case. We consider some of the properties of these coherent states and also show that the uncertainty product 〈 (Δq̂)2〉 〈 (Δp̂)2〉 in this case could be made arbitrarily small.