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

arXiv:quant-ph/9702010 · Modern Physics Letters A

TRAJECTORY-COHERENT STATES AND THE HEISENBERG'S UNCERTAINTY RELATION

A. G. Karavaev

Published Sep 28, 1994• 0 citations
arXivHTMLPDF

Abstract

In this paper we investigate the problem of minimizing the Heisenberg's uncertainty relation with the trajectory-coherent states. The conditions of minimization for Hamiltonian and trajectory are obtained. We show that the trajectory-coherent states minimize the Heisenberg's uncertainty relation for special Cauchy problem of the Schrödinger equation only.

DOI
10.1142/s0217732394002677

Related papers

COHERENT STATES OF QUANTUM NONLINEAR 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 [Formula: see text].

arXiv:quant-ph/9512005•Modern Physics Letters B•Dec 20, 1995•1 citation

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

Schroedinger uncertainty relation and its minimization states

D. A. Trifonov

An introductory survey on the Schroedinger uncertainty relation and its minimization states is presented with minimal number of formulas and some historical points. The case of the two canonical observables, position and momentum, is discussed in greater detail: basic properties of the two subsets of minimization states (canonical squeezed and coherent states) are reviewed and compared. The case of two non-canonical observables is breafly outlined. Stanfard SU(1,1) and SU(2) group-related coherent states can be defined as states that minimize Schroedinger inequality for the three pairs of generators simultaneously. The symmetry of the Heisenberg and Schroedinger relations is also discussed, and two natural generalizations to the cases of several observables and several states are noted.

arXiv:physics/0105035•May 12, 2001•9 citations

GENERALIZED HEISENBERG RELATION AND QUANTUM HARMONIC OSCILLATORS

P. NARAYANA SWAMY

We study the consequences of the generalized Heisenberg uncertainty relation which admits a minimal uncertainty in length such as the case in a theory of quantum gravity. In particular, the theory of quantum harmonic oscillators arising from such a generalized uncertainty relation is examined. We demonstrate that all the standard properties of the quantum harmonic oscillators prevail when we employ a generalized momentum. We also show that quantum electrodynamics and coherent photon states can be described in the familiar standard manner despite the generalized uncertainty principle.

arXiv:quant-ph/0610079•International Journal of Modern Physics A•Dec 10, 2006•1 citation

On some states minimizing uncertainty relations: A new look at these relations

K. Urbanowski

The Heisenberg–Robertson (HR) and Robertson–Schrödinger (RS) uncertainty relations are studied. Analyzing the RS uncertainty relation we found, that there can exist a large set of states of the quantum system under considerations, for which the lower bound of the product of the standard deviations of a pair of noncommuting observables, A and B, is zero, and which differ from those described in the literature for the HR uncertainty relation. These states are not eigenstates of either the observable A or B. The correlation function for these observables in such states is equal to zero. We have also shown that the so-called “sum uncertainty relations” also do not provide any information about lower bounds on the standard deviations calculated for these states. We additionally show that the uncertainty principle in its most general form has two faces: One is that it is a lower bound on the product of standard deviations, and the other is that the product of standard deviations is an upper bound on the modulus of the correlation function of a pair of the noncommuting observables in the state under consideration.

arXiv:2411.08131•Modern Physics Letters A•Mar 25, 2026•0 citations

Coherent states for the anharmonic oscillator and classical phase space trajectories

H. S. Sharatchandra

Unique set of coherent states for the anharmonic oscillator is obtained by requiring i. under the quantum mechanical time evolution a coherent state evolves into another, governed by trajectory in the classical phase space (of a related hamiltonian); ii. the resolution of identity involves exactly the classical phase space measure.The rules are invariant under unitary transformations of the quantum theory and canonical transformations of the classical theory. The states are almost, but not quite, minimal uncertainty wave packets. The construction can be generalized to quantum versions of integrable classical theories.

arXiv:quant-ph/9707032•Jul 17, 1997•0 citations
In this paper we investigate the problem of minimizing the Heisenberg's uncertainty relation with the trajectory-coherent states. The conditions of minimization for Hamiltonian and trajectory are obtained. We show that the trajectory-coherent states minimize the Heisenberg's uncertainty relation for special Cauchy problem of the Schrödinger equation only.
An introductory survey on the Schroedinger uncertainty relation and its minimization states is presented with minimal number of formulas and some historical points. The case of the two canonical observables, position and momentum, is discussed in greater detail: basic properties of the two subsets of minimization states (canonical squeezed and coherent states) are reviewed and compared. The case of two non-canonical observables is breafly outlined. Stanfard SU(1,1) and SU(2) group-related coherent states can be defined as states that minimize Schroedinger inequality for the three pairs of generators simultaneously. The symmetry of the Heisenberg and Schroedinger relations is also discussed, and two natural generalizations to the cases of several observables and several states are noted.
We study the consequences of the generalized Heisenberg uncertainty relation which admits a minimal uncertainty in length such as the case in a theory of quantum gravity. In particular, the theory of quantum harmonic oscillators arising from such a generalized uncertainty relation is examined. We demonstrate that all the standard properties of the quantum harmonic oscillators prevail when we employ a generalized momentum. We also show that quantum electrodynamics and coherent photon states can be described in the familiar standard manner despite the generalized uncertainty principle.
The Heisenberg–Robertson (HR) and Robertson–Schrödinger (RS) uncertainty relations are studied. Analyzing the RS uncertainty relation we found, that there can exist a large set of states of the quantum system under considerations, for which the lower bound of the product of the standard deviations of a pair of noncommuting observables, A and B, is zero, and which differ from those described in the literature for the HR uncertainty relation. These states are not eigenstates of either the observable A or B. The correlation function for these observables in such states is equal to zero. We have also shown that the so-called “sum uncertainty relations” also do not provide any information about lower bounds on the standard deviations calculated for these states. We additionally show that the uncertainty principle in its most general form has two faces: One is that it is a lower bound on the product of standard deviations, and the other is that the product of standard deviations is an upper bound on the modulus of the correlation function of a pair of the noncommuting observables in the state under consideration.
Unique set of coherent states for the anharmonic oscillator is obtained by requiring i. under the quantum mechanical time evolution a coherent state evolves into another, governed by trajectory in the classical phase space (of a related hamiltonian); ii. the resolution of identity involves exactly the classical phase space measure.The rules are invariant under unitary transformations of the quantum theory and canonical transformations of the classical theory. The states are almost, but not quite, minimal uncertainty wave packets. The construction can be generalized to quantum versions of integrable classical theories.