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

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Resolution of the identity for minimum-uncertainty coherent states: An example related to charged-boson coherent states

Michael Martin Nieto

Published Aug 15, 1984• 18 citations
Publisher

Abstract

After arguing on intuitive grounds that the minimum-uncertainty coherent states for general potentials are complete, the resolution of the identity for a specific example $V\ensuremath{\propto}{(z\ensuremath{-}\frac{1}{z})}^{2}$ is explicitly derived. It is then shown that these states are a generalization to arbitrary (noninteger) charge of the charged-boson coherent states.

DOI
10.1103/physrevd.30.770

Related papers

Coherent states for general potentials. I. Formalism

Michael Martin Nieto, L. M. Simmons

We first review the properties of the harmonic-oscillator coherent states which can be equivalently defined as (a) a specific subset of the $x\ensuremath{-}p$ minimum-uncertainty states, (b) eigenstates of the annihilation operator, or (c) states created by a certain unitary exponential displacement operator. Then we present a new method for finding coherent states for particles in general potentials. Its basis is the desire to find those states which most nearly follow the classical motion, but it is most nearly a generalization of the minimum-uncertainty method. The properties of these states are discussed in detail. Next we show that the annihilation operator and displacement operator methods, as heretofore defined, cannot be applied to general potentials (whose eigenvalues are not equally spaced). We define a generalization of these methods but show that the states so defined are not, in general, equivalent to the minimum-uncertainty coherent states. We discuss a number of properties of our coherent states and the procedures we have used.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Sep 15, 1979•210 citations

Coherent states withSU(2) andSU(3) charges

Manu Mathur, Samir K Paul

We define coherent states carrying SU (2) charge by exploiting the Schwinger boson representation of the SU (2) Lie algebra. These coherent states satisfy the continuity property and provide resolution of identity on S 3 . We further generalize these techniques to construct the corresponding SU (3) charge coherent states. The SU ( N ) extension is also discussed.

arXiv:quant-ph/0303146•Journal of Physics A Mathematical and General•Aug 23, 2005•1 citation

Coherent states for general potentials. II. Confining one-dimensional examples

Michael Martin Nieto, L. M. Simmons

We apply our minimum-uncertainty coherent-states formalism, which is physically motivated by the classical motion, to two confining one-dimensional systems: the harmonic oscillator with centripetal barrier and the symmetric P\"oschl-Teller potentials. The minimum-uncertainty coherent states are discussed in great detail, and the connections to annihilation-operator coherent states and displacement-operator coherent states are given. The first system discussed provides an excellent bridge between the harmonic oscillator and more general potentials because, even though it is a nonharmonic potential, its energy eigenvalues are equally spaced. Thus, its coherent states have many, but not all, of the properties of the harmonic-oscillator coherent states.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Sep 15, 1979•136 citations

Spin minimum uncertainty states for refined uncertainty relations

Hao Dai, Yue Zhang

Minimum uncertainty states of the conventional Heisenberg uncertainty relation have been extensively studied and are often regarded as the most classical quantum states from the perspective of uncertainty, providing valuable insight into the nature of quantumness and its potential applications. In this work, we investigate the minimum uncertainty states associated with an information-theoretic refinement of the Heisenberg uncertainty relation in general spin systems. Using two different approaches, the matrix formulation and the Wick symbol representation, we derive explicit expressions for the states that saturate the uncertainty bound. We show that spin coherent states indeed achieve minimum uncertainty, consistent with their conventional identification as the classical states of spin systems. Moreover, we also identify additional classes of minimum uncertainty states beyond the coherent family. Finally, we compare the spin-system results with the previously studied bosonic case and elucidate the origin of the differences between the two settings.

quant-pharXiv:2512.17307•Journal of Physics A Mathematical and Theoretical•Dec 22, 2025•0 citations

Generalized Coherent States

David Stoler

Using an operator technique similar to one we have used in a previous paper, we study the generalized coherent states which were introduced by Titulaer and Glauber and discussed recently by Bialynicka-Birula. Using this technique, we provide an alternative derivation of the most general coherent state and then derive a number of interesting properties, some of which do not seem to be in the literature. For example, we show that the generalized coherent states are not minimum-uncertainty packets.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Oct 15, 1971•188 citations

Coherent states for general potentials. III. Nonconfining one-dimensional examples

Michael Martin Nieto, L. M. Simmons

We apply our minimum-uncertainty coherent-states (MUCS) formalism to two one-dimensional systems that have continua: the symmetric Rosen-Morse potential and the Morse potential. The coherent states are discussed analytically in great detail, and the connections to annihilation-operator and displacement-operator coherent states are given. For the Rosen-Morse system the existence of a continuum does not prevent one from obtaining the coherent states in analytic, closed form. The Morse system, with its energy-dependent natural classical variable ${X}_{c}$, has a natural quantum operator $X$ which is Hamiltonian-dependent. This Hamiltonian dependence is complicated and prevents an easy analytic solution for the MUCS. However, approximate MUCS can be obtained by analytic approximation techniques.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Sep 15, 1979•114 citations
After arguing on intuitive grounds that the minimum-uncertainty coherent states for general potentials are complete, the resolution of the identity for a specific example V\ensuremath∝(z\ensuremath−1z)2V\ensuremath{\propto}{(z\ensuremath{-}\frac{1}{z})}^{2}V\ensuremath∝(z\ensuremath−z1​)2 is explicitly derived. It is then shown that these states are a generalization to arbitrary (noninteger) charge of the charged-boson coherent states.
We first review the properties of the harmonic-oscillator coherent states which can be equivalently defined as (a) a specific subset of the $x\ensuremath{-}p$ minimum-uncertainty states, (b) eigenstates of the annihilation operator, or (c) states created by a certain unitary exponential displacement operator. Then we present a new method for finding coherent states for particles in general potentials. Its basis is the desire to find those states which most nearly follow the classical motion, but it is most nearly a generalization of the minimum-uncertainty method. The properties of these states are discussed in detail. Next we show that the annihilation operator and displacement operator methods, as heretofore defined, cannot be applied to general potentials (whose eigenvalues are not equally spaced). We define a generalization of these methods but show that the states so defined are not, in general, equivalent to the minimum-uncertainty coherent states. We discuss a number of properties of our coherent states and the procedures we have used.
We apply our minimum-uncertainty coherent-states formalism, which is physically motivated by the classical motion, to two confining one-dimensional systems: the harmonic oscillator with centripetal barrier and the symmetric P\"oschl-Teller potentials. The minimum-uncertainty coherent states are discussed in great detail, and the connections to annihilation-operator coherent states and displacement-operator coherent states are given. The first system discussed provides an excellent bridge between the harmonic oscillator and more general potentials because, even though it is a nonharmonic potential, its energy eigenvalues are equally spaced. Thus, its coherent states have many, but not all, of the properties of the harmonic-oscillator coherent states.
Minimum uncertainty states of the conventional Heisenberg uncertainty relation have been extensively studied and are often regarded as the most classical quantum states from the perspective of uncertainty, providing valuable insight into the nature of quantumness and its potential applications. In this work, we investigate the minimum uncertainty states associated with an information-theoretic refinement of the Heisenberg uncertainty relation in general spin systems. Using two different approaches, the matrix formulation and the Wick symbol representation, we derive explicit expressions for the states that saturate the uncertainty bound. We show that spin coherent states indeed achieve minimum uncertainty, consistent with their conventional identification as the classical states of spin systems. Moreover, we also identify additional classes of minimum uncertainty states beyond the coherent family. Finally, we compare the spin-system results with the previously studied bosonic case and elucidate the origin of the differences between the two settings.
We apply our minimum-uncertainty coherent-states (MUCS) formalism to two one-dimensional systems that have continua: the symmetric Rosen-Morse potential and the Morse potential. The coherent states are discussed analytically in great detail, and the connections to annihilation-operator and displacement-operator coherent states are given. For the Rosen-Morse system the existence of a continuum does not prevent one from obtaining the coherent states in analytic, closed form. The Morse system, with its energy-dependent natural classical variable Xc{X}_{c}Xc​, has a natural quantum operator XXX which is Hamiltonian-dependent. This Hamiltonian dependence is complicated and prevents an easy analytic solution for the MUCS. However, approximate MUCS can be obtained by analytic approximation techniques.