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

Journal of Mathematical Physics

Representation and properties of para-Bose oscillator operators. II. Coherent states and the minimum uncertainty states

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

Published Jan 1, 1981• 50 citations
Publisher

Abstract

The energy, position, and momentum eigenstates of a para-Bose oscillator system were considered in paper I. Here we consider the Bargmann or the analytic function description of the para-Bose system. This brings in, in a natural way, the coherent states ‖z;α〉 defined as the eigenstates of the annihilation operator ?. The transformation functions relating this description to the energy, position, and momentum eigenstates are explicitly obtained. Possible resolution of the identity operator using coherent states is examined. A particular resolution contains two integrals, one containing the diagonal basis ‖z;α〉〈z;α‖ and the other containing the pseudodiagonal basis ‖z;α〉〈−z;α‖. We briefly consider the normal and antinormal ordering of the operators and their diagonal and discrete diagonal coherent state approximations. The problem of constructing states with a minimum value of the product of the position and momentum uncertainties and the possible α dependence of this minimum value is considered.

DOI
10.1063/1.524756

Related papers

Two-mode para-Bose coherent states

G. M. Saxena, C. L. Mehta

The construction of eigenstates of the square of annihilation operators for a two-mode para-Bose system is reported. Bose coherent states can be deduced from these eigenstates as a special case. These states are termed para-Bose coherent states. These states are degenerate. The expansion of the coherent states in terms of two-mode para-Bose energy eigenstates has been obtained and their salient properties are discussed. Also discussed is the uncertainty product of the square of position and momentum operators in the para-Bose coherent states for a two-mode system.

Journal of Mathematical Physics•Dec 1, 1985•1 citation

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

Defining coherent states: why must they be eigenstates of the annihilation operator?

Juan Pablo Paz, Augusto J. Roncaglia

This is a pedagogical paper where we present a physically motivated approach to introduce the coherent states of a harmonic oscillator from which it is simple to rigorously derive their mathematical definition. We do this in two different ways that turn out to be equivalent but emphasize two related but different aspects of classicality. First, we analyze which are the quantum states that are the closest one can get to a point in phase space and demonstrate the validity of the following theorem: (i) The product of the uncertainty in position and that of momentum saturates the bound imposed by Heisenberg uncertainty relations for all times if and only if the state is an eigenstate of the annihilation operator. Second, we analyze the way in which the difference between the expectation value of the energy and the energy associated with the expectation values of position and momentum depends on the state, and show the validity of the following theorem (ii) the difference between the expectation value of the energy and the energy associated with the expectation values is minimal if and only if the state is an eigenstate of the annihilation operator. We also show that the reason why coherent states are chosen as the most classical ones by the decoherence process induced by coupling the particle to an environment in the standard Quantum Brownian motion model, is precisely due to the validity of the two above theorems.

arXiv:2410.16483•Oct 21, 2024•0 citations

Representations and properties of para-Bose oscillator operators. I. Energy position and momentum eigenstates

Narasimhaiengar Mukunda, E. C. G. Sudarshan, J. K. Sharma +1 more

Para-Bose commutation relations are related to the SL(2,R) Lie algebra. The irreducible representation 𝒟α of the para-Bose system is obtained as the direct sum Dβ⊕Dβ+1/2 of the representations of the SL(2,R) Lie algebra. The position and momentum eigenstates are then obtained in this representation 𝒟α, using the matrix mechanical method. The orthogonality, completeness, and the overlap of these eigenstates are derived. The momentum eigenstates are also derived using the wave mechanical method by specifying the domain of the definition of the momentum operator in addition to giving it a formal differential expression. By a careful consideration in this manner we find that the two apparently different solutions obtained by Ohnuki and Kamefuchi in this context are actually unitarily equivalent.

Journal of Mathematical Physics•Sep 1, 1980•101 citations

Coherent states of quantum non-linear systems

Z. Haba

Quantum dynamics of integrable systems is discussed. Localized wave packets generalizing the conventional coherent states of minimal uncertainty are constructed. The wave packet moves along a certain trajectory and does not change its shape for times of order 1 ¯h. In this Letter we suggest that the definition of a coherent state should be related to the Hamiltonian.The conventional coherent states [1] are defined as eigenstates of the annihilation operator α | z> = z | z> (1) The real and the imaginary parts of the complex variable z can be expressed by the mean values of the position ˆx and momentum ˆp operators in the coherent state. Under the oscillator Hamiltonian evolution Ut Ut|z> = |exp(−iωt)z> (2) where ω is the oscillator ′ √ s frequency. Then, because the state | z> is con-2¯h centrated around x = ω Rez it follows that Ut|z> has its support on the

•Jan 1, 1995•1 citation

A new set of coherent states for the isotropic harmonic oscillator: Coherent angular momentum states

Anthony J. Bracken, Howard I. Leemon

The Hamiltonian for the oscillator has earlier been written in the form where ν† and ν are raising and lowering operators for ν†ν, which has eigenvalues k (the ’’radial’’ quantum number), and λ† and λ are raising and lowering 3-vector operators for λ†⋅λ, which has eigenvalues l (the total angular momentum quantum number). A new set of coherent states for the oscillator is now defined by diagonalizing ν and λ. These states bear a similar relation to the commuting operators H, L2, and L3 (where L is the angular momentum of the system) as the usual coherent states do to the commuting number operators N1, N2, and N3. It is proposed to call them coherent angular momentum states. They are shown to be minimum-uncertainty states for the variables ν, ν† , λ, and λ†, and to provide a new quasiclassical description of the oscillator. This description coincides with that provided by the usual coherent states only in the special case that the corresponding classical motion is circular, rather than elliptical; and, in general, the uncertainty in the angular momentum of the system is smaller in the new description. The probabilities of obtaining particular values for k and l in one of the new states follow independent Poisson distributions. The new states are overcomplete, and lead to a new representation of the Hilbert space for the oscillator, in terms of analytic functions on C×K3, where K3 is the three-dimensional complex cone. This space is related to one introduced recently by Bargmann and Todorov, and carries a very simple realization of all the representations of the rotation group.

Journal of Mathematical Physics•Apr 1, 1981•7 citations
The energy, position, and momentum eigenstates of a para-Bose oscillator system were considered in paper I. Here we consider the Bargmann or the analytic function description of the para-Bose system. This brings in, in a natural way, the coherent states ‖z;α〉 defined as the eigenstates of the annihilation operator ?. The transformation functions relating this description to the energy, position, and momentum eigenstates are explicitly obtained. Possible resolution of the identity operator using coherent states is examined. A particular resolution contains two integrals, one containing the diagonal basis ‖z;α〉〈z;α‖ and the other containing the pseudodiagonal basis ‖z;α〉〈−z;α‖. We briefly consider the normal and antinormal ordering of the operators and their diagonal and discrete diagonal coherent state approximations. The problem of constructing states with a minimum value of the product of the position and momentum uncertainties and the possible α dependence of this minimum value is considered.
The construction of eigenstates of the square of annihilation operators for a two-mode para-Bose system is reported. Bose coherent states can be deduced from these eigenstates as a special case. These states are termed para-Bose coherent states. These states are degenerate. The expansion of the coherent states in terms of two-mode para-Bose energy eigenstates has been obtained and their salient properties are discussed. Also discussed is the uncertainty product of the square of position and momentum operators in the para-Bose coherent states for a two-mode system.
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.
This is a pedagogical paper where we present a physically motivated approach to introduce the coherent states of a harmonic oscillator from which it is simple to rigorously derive their mathematical definition. We do this in two different ways that turn out to be equivalent but emphasize two related but different aspects of classicality. First, we analyze which are the quantum states that are the closest one can get to a point in phase space and demonstrate the validity of the following theorem: (i) The product of the uncertainty in position and that of momentum saturates the bound imposed by Heisenberg uncertainty relations for all times if and only if the state is an eigenstate of the annihilation operator. Second, we analyze the way in which the difference between the expectation value of the energy and the energy associated with the expectation values of position and momentum depends on the state, and show the validity of the following theorem (ii) the difference between the expectation value of the energy and the energy associated with the expectation values is minimal if and only if the state is an eigenstate of the annihilation operator. We also show that the reason why coherent states are chosen as the most classical ones by the decoherence process induced by coupling the particle to an environment in the standard Quantum Brownian motion model, is precisely due to the validity of the two above theorems.
Para-Bose commutation relations are related to the SL(2,R) Lie algebra. The irreducible representation 𝒟α of the para-Bose system is obtained as the direct sum Dβ⊕Dβ+1/2 of the representations of the SL(2,R) Lie algebra. The position and momentum eigenstates are then obtained in this representation 𝒟α, using the matrix mechanical method. The orthogonality, completeness, and the overlap of these eigenstates are derived. The momentum eigenstates are also derived using the wave mechanical method by specifying the domain of the definition of the momentum operator in addition to giving it a formal differential expression. By a careful consideration in this manner we find that the two apparently different solutions obtained by Ohnuki and Kamefuchi in this context are actually unitarily equivalent.
Quantum dynamics of integrable systems is discussed. Localized wave packets generalizing the conventional coherent states of minimal uncertainty are constructed. The wave packet moves along a certain trajectory and does not change its shape for times of order 1 ¯h. In this Letter we suggest that the definition of a coherent state should be related to the Hamiltonian.The conventional coherent states [1] are defined as eigenstates of the annihilation operator α | z> = z | z> (1) The real and the imaginary parts of the complex variable z can be expressed by the mean values of the position ˆx and momentum ˆp operators in the coherent state. Under the oscillator Hamiltonian evolution Ut Ut|z> = |exp(−iωt)z> (2) where ω is the oscillator ′ √ s frequency. Then, because the state | z> is con-2¯h centrated around x = ω Rez it follows that Ut|z> has its support on the
The Hamiltonian for the oscillator has earlier been written in the form where ν† and ν are raising and lowering operators for ν†ν, which has eigenvalues k (the ’’radial’’ quantum number), and λ† and λ are raising and lowering 3-vector operators for λ†⋅λ, which has eigenvalues l (the total angular momentum quantum number). A new set of coherent states for the oscillator is now defined by diagonalizing ν and λ. These states bear a similar relation to the commuting operators H, L2, and L3 (where L is the angular momentum of the system) as the usual coherent states do to the commuting number operators N1, N2, and N3. It is proposed to call them coherent angular momentum states. They are shown to be minimum-uncertainty states for the variables ν, ν† , λ, and λ†, and to provide a new quasiclassical description of the oscillator. This description coincides with that provided by the usual coherent states only in the special case that the corresponding classical motion is circular, rather than elliptical; and, in general, the uncertainty in the angular momentum of the system is smaller in the new description. The probabilities of obtaining particular values for k and l in one of the new states follow independent Poisson distributions. The new states are overcomplete, and lead to a new representation of the Hilbert space for the oscillator, in terms of analytic functions on C×K3, where K3 is the three-dimensional complex cone. This space is related to one introduced recently by Bargmann and Todorov, and carries a very simple realization of all the representations of the rotation group.