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

Reviews in Mathematical Physics

COHERENT STATES AND QUANTUM DYNAMICS OF NON-LINEAR SYSTEMS

Zbigniew Haba

Published Nov 1, 1996• 2 citations
Publisher

Abstract

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].

DOI
10.1142/s0129055x96000391

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 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

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

DYNAMICS OF GENERALIZED COHERENT STATES

S. De Martino, Silvio De Siena, Fabrizio Illuminati

We show that generalized coherent states follow Schrödinger dynamics in time-dependent potentials. The normalized wave-packets follow a classical evolution without spreading; in turn, the Schrödinger potential depends on the state through the classical trajectory. This feedback mechanism with continuous dynamical re-adjustment allows the packets to remain coherent indefinitely.

arXiv:quant-ph/9501010•Modern Physics Letters B•Jun 10, 1995•5 citations

THEORY OF CONTROLLED QUANTUM DYNAMICS

Salvatore De Martino, Silvio De Siena, Fabrizio Illuminati

We introduce a general formalism, based on the stochastic formulation of quantum mechanics, to obtain localized quasi-classical wave packets as dynamically controlled systems, for arbitrary anharmonic potentials. The control is in general linear, and it amounts to introduce additional quadratic and linear time-dependent terms to the given potential. In this way one can construct for general systems either coherent packets moving with constant dispersion, or dynamically squeezed packets whose spreading remains bounded for all times. In the standard operatorial framework our scheme corresponds to a suitable generalization of the displacement and scaling operators that generate the coherent and squeezed states of the harmonic oscillator.

•Jan 1, 1997•3 citations

Coherent states for the hydrogen atom

Pushan Majumdar, H. S. Sharatchandra

We construct wave packets for the hydrogen atom labeled by the classical action-angle variables with the following properties. (i) The time evolution is exactly given by classical evolution of the angle variables. (The angle variable corresponding to the position on the orbit is now noncompact and we do not get exactly the same state after one period; however, the gross features do not change. In particular, the wave packet remains peaked around the labels.) (ii) The resolution of the identity using this overcomplete set involves exactly the classical phase-space measure. (iii) The semiclassical limit is related to Bohr-Sommerfeld quantization. (iv) They are almost minimum uncertainty wave packets in position and momentum.

arXiv:quant-ph/9708010•Physical Review A•Nov 1, 1997•37 citations
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].
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
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.
We introduce a general formalism, based on the stochastic formulation of quantum mechanics, to obtain localized quasi-classical wave packets as dynamically controlled systems, for arbitrary anharmonic potentials. The control is in general linear, and it amounts to introduce additional quadratic and linear time-dependent terms to the given potential. In this way one can construct for general systems either coherent packets moving with constant dispersion, or dynamically squeezed packets whose spreading remains bounded for all times. In the standard operatorial framework our scheme corresponds to a suitable generalization of the displacement and scaling operators that generate the coherent and squeezed states of the harmonic oscillator.
We construct wave packets for the hydrogen atom labeled by the classical action-angle variables with the following properties. (i) The time evolution is exactly given by classical evolution of the angle variables. (The angle variable corresponding to the position on the orbit is now noncompact and we do not get exactly the same state after one period; however, the gross features do not change. In particular, the wave packet remains peaked around the labels.) (ii) The resolution of the identity using this overcomplete set involves exactly the classical phase-space measure. (iii) The semiclassical limit is related to Bohr-Sommerfeld quantization. (iv) They are almost minimum uncertainty wave packets in position and momentum.