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

Generalized coherent states and the uncertainty principle

Shasanka Mohan Roy, Virendra Singh

Published Jun 15, 1982• 86 citations
Publisher

Abstract

We derive from a dynamical symmetry property that the linear and nonlinear Schr\"odinger equations with harmonic potential possess an infinite string of shape-preserving coherent wave-packet states with classical motion. Unlike the Schr\"odinger state with $\ensuremath{\Delta}x\ensuremath{\Delta}p=\frac{\ensuremath{\hbar}}{2}$, the uncertainty product can be arbitrarily large for these states showing that classical motion is not necessarily linked with minimum uncertainty. We obtain a generalization of Sudarshan's diagonal coherent-state representation in terms of these states.

DOI
10.1103/physrevd.25.3413

Related papers

Generalized Coherent States of a Particle in a Time-Dependent Linear Potential

L. Krache, Mustapha Maamache, Y. Saadi +1 more

We derive, with an invariant operator method and unitary transformation approach, that the Schrödinger equation with a time-dependent linear potential possesses an infinite string of shape-preseving wave-packet states |ϕα,λ( t )⟩ having classical motion. The qualitative properties of the invariant eigenvalue spectrum (discrete or continuous) are described separately for the different values of the frequency ω of a harmonic oscillator. It is also shown that, for a discrete eigenvalue spectrum, the states |ϕα, n ( t )⟩ could be obtained from the coherent state |ϕα,0( t )⟩.

Chinese Physics Letters•Jul 1, 2009•3 citations

Generalized coherent states and generalized squeezed coherent states

M. Venkata Satyanarayana

Roy and Virendra Singh showed that the harmonic oscillator possesses an infinite string of exact shape-preserving coherent wave-packet states \ensuremath{\Vert}n,\ensuremath{\alpha}〉 having classical motion. In this paper it is shown that the states \ensuremath{\Vert}n,\ensuremath{\alpha}〉 could be obtained from the coherent state \ensuremath{\Vert}\ensuremath{\alpha}〉 and it is also shown how a coherent state \ensuremath{\Vert}\ensuremath{\alpha}〉 could be expanded in the basis of \ensuremath{\Vert}n,\ensuremath{\alpha}〉's. Further, the possibility of ``squeezing'' the state \ensuremath{\Vert}n〉 is investigated and the ``generalized squeezed coherent states'' are obtained. The squeezed coherent states for the displaced oscillator are also defined. The physical meaning of squeezing is also pointed out.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Jul 15, 1985•121 citations

- Coherent and Squeezed States

S. Rajasekar, R. Velusamy

Certain superposition (linear combination) of energy eigenfunctions of linear harmonic oscillators are called coherent states because they are of significance in optics for the representation of coherent light waves. A hallmark of coherent states is that the variances (squares of the uncertainties) of x and px are constant in time and further their product becomes the minimum value allowed by the Heisenberg uncertainty principle. On the other hand, some linear combinations of harmonic oscillator energy functions give rise to squeezed states . A curious property of squeezed states is that the variances of x and px oscillate in time 180o out of phase with one another with the frequency twice of the oscillator. The wave packet of a coherent state possesses a minimum uncertainty. 〈x〉 and 〈px〉 have the same oscillatory forms as in the classical case. The coherent states have also been called minimum uncertainty coherent states, the Schrödinger coherent states or the Glauber coherent states [1-6]. The squeezed states are sometimes referred to in the literature as two-photon coherent states and generalized coherent states.

CRC Press eBooks•Dec 10, 2014•0 citations

New coherent states with applications to time-dependent systems

A. K. Rajagopal, John T. Marshall

A new class of coherent states is defined, having all the properties of the usual coherent states except that the uncertainty product of position and momentum need not be minimum. From the stationary-action principle, the dynamical equations of motion for the expectation values and uncertainties for position and momentum are deduced. For the time-dependent harmonic oscillator, the equation for the uncertainty in the position is found to be the well-known Lewis constraint.

Physical Review A•Nov 1, 1982•60 citations

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 AND QUANTUM DYNAMICS OF NON-LINEAR SYSTEMS

Zbigniew Haba

We discuss a quantum-mechanical analog of the classical angle variable. A localized wave packet is constructed generalizing the conventional coherent state of minimal uncertainty. For small ħ the wave packet moves along a certain trajectory close to the classical torus. In a class of integrable Hamiltonian systems we estimate that the wave packet preserves its shape at least for times of order [Formula: see text].

Reviews in Mathematical Physics•Nov 1, 1996•2 citations
We derive from a dynamical symmetry property that the linear and nonlinear Schr"odinger equations with harmonic potential possess an infinite string of shape-preserving coherent wave-packet states with classical motion. Unlike the Schr"odinger state with \ensuremathΔx\ensuremathΔp=\ensuremathℏ2\ensuremath{\Delta}x\ensuremath{\Delta}p=\frac{\ensuremath{\hbar}}{2}\ensuremathΔx\ensuremathΔp=2\ensuremathℏ​, the uncertainty product can be arbitrarily large for these states showing that classical motion is not necessarily linked with minimum uncertainty. We obtain a generalization of Sudarshan's diagonal coherent-state representation in terms of these states.
We derive, with an invariant operator method and unitary transformation approach, that the Schrödinger equation with a time-dependent linear potential possesses an infinite string of shape-preseving wave-packet states |ϕα,λ( t )⟩ having classical motion. The qualitative properties of the invariant eigenvalue spectrum (discrete or continuous) are described separately for the different values of the frequency ω of a harmonic oscillator. It is also shown that, for a discrete eigenvalue spectrum, the states |ϕα, n ( t )⟩ could be obtained from the coherent state |ϕα,0( t )⟩.
Roy and Virendra Singh showed that the harmonic oscillator possesses an infinite string of exact shape-preserving coherent wave-packet states \ensuremath{\Vert}n,\ensuremath{\alpha}〉 having classical motion. In this paper it is shown that the states \ensuremath{\Vert}n,\ensuremath{\alpha}〉 could be obtained from the coherent state \ensuremath{\Vert}\ensuremath{\alpha}〉 and it is also shown how a coherent state \ensuremath{\Vert}\ensuremath{\alpha}〉 could be expanded in the basis of \ensuremath{\Vert}n,\ensuremath{\alpha}〉's. Further, the possibility of ``squeezing'' the state \ensuremath{\Vert}n〉 is investigated and the ``generalized squeezed coherent states'' are obtained. The squeezed coherent states for the displaced oscillator are also defined. The physical meaning of squeezing is also pointed out.
Certain superposition (linear combination) of energy eigenfunctions of linear harmonic oscillators are called coherent states because they are of significance in optics for the representation of coherent light waves. A hallmark of coherent states is that the variances (squares of the uncertainties) of x and px are constant in time and further their product becomes the minimum value allowed by the Heisenberg uncertainty principle. On the other hand, some linear combinations of harmonic oscillator energy functions give rise to squeezed states . A curious property of squeezed states is that the variances of x and px oscillate in time 180o out of phase with one another with the frequency twice of the oscillator. The wave packet of a coherent state possesses a minimum uncertainty. 〈x〉 and 〈px〉 have the same oscillatory forms as in the classical case. The coherent states have also been called minimum uncertainty coherent states, the Schrödinger coherent states or the Glauber coherent states [1-6]. The squeezed states are sometimes referred to in the literature as two-photon coherent states and generalized 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.