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

Axioms

Covariance and Uncertainty Principle for Dispersive Pulse Propagation

Leon W. Cohen

Published Apr 8, 2024• 0 citations
Publisher

Abstract

We develop the concept of covariance for waves and show that it plays a fundamental role in understanding the evolution of a propagating pulse. The concept clarifies several issues regarding the spread of a pulse and the motion of the mean. Exact results are obtained for the time dependence of the covariance between position and wavenumber and the covariance between position and group velocity. We also derive relevant uncertainty principles for waves.

DOI
10.3390/axioms13040242

Related papers

The uncertainty principle in signal analysis

Leon W. Cohen

We give a simple derivation for the covariance of time and frequency. The simplicity remains when applied to arbitrary variables. We show that the covariance enters into the uncertainty principle in a fundamental way. Also we present a seeming paradox: the uncertainty principle depends only on the marginals, but the marginals carry no information about covariance. Then how is it possible that the uncertainty principle involves the covariance? The resolution of the paradox clarifies a number of issues regarding the existence of well behaved manifestly positive joint distributions of time and frequency.>

IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis•Dec 17, 2002•31 citations

Why do wave packets sometimes contract?

Leon W. Cohen

We obtain explicit expressions for the spread of a pulse in a dispersive medium. Using these results we present a simple physical picture as to why pulses sometimes contract before expanding. Explicit formulas are given for the contraction time and the duration of the contraction. We show that a fundamental quantity that enters is the covariance between position and group velocity. Both quantum and classical wave equations are considered.

Journal of Modern Optics•Nov 1, 2002•5 citations

Pulse propagation and windowed wave functions

J. Ben-Benjamin, Leon W. Cohen

We apply quasi-distribution methods developed for quantum mechanics to the propagation of pulses in dispersive media with attenuation. We show that a Schrödinger type equation follows for propagation of the pulse for each mode. One then transforms the equation to obtain an equation of evolution in the phase space of position and wavenumber. In this paper we emphasize windowed wave functions and their corresponding phase space quasi-distributions. We obtain the time evolution equation, discuss possible approximations, and compare to the Wigner distribution approximation previously derived by Loughlin and Cohen by different methods.

Journal of Modern Optics•Sep 2, 2013•7 citations

Propagation and spreading of a pulse or wave packet

Henry M. Bradford

The usual textbook discussions of the velocity and spreading of a wave packet are based on special cases and approximations which have a limited range of validity. In this paper more general relations are obtained by considering the time dependence of the moments of a wave packet. Both a packet of real sinusoidal plane waves (a real pulse, e.g., sound) and a packet of complex exponential plane waves (e.g., the quantum-mechanical wave function of a free particle) are considered. The results are similar if the real wave packet contains no standing wave components. It is shown that the centroid of a wave packet moves at a constant velocity which is a weighted average of the group velocity. The spreading of the wave packet depends on the product of the variance of the group velocity and the square of the time interval before or after the time of minimum packet width. The packet moves and spreads like a collection of particles in free flight, in which the velocities of the particles are the group velocities of the wave components of the packet.

American Journal of Physics•Nov 1, 1976•19 citations

Propagation of a step in the amplitude or envelope of a pulse or wave packet

Henry M. Bradford

The propagation of a steplike change in the amplitude of a dc pulse and a step in the envelope of a plane sinusoidal wave are investigated. It is shown that a step (e.g., the beginning or end of a pulse or wave) propagates like a pulse in the medium; i.e., its velocity is a weighted average of the group velocity, and its rate of spread is determined by the distribution of group velocities within the spectrum of the step. The results apply to a real pulse (e.g., sound) in an infinite, homogeneous, dispersive, lossless medium, and to the quantum-mechanical wave function of a free particle.

American Journal of Physics•Aug 1, 1979•2 citations

Bohm trajectories for wave propagation

Leon W. Cohen

We obtain the equations of motion for the density and phase of a propagating pulse in analogy to the Bohm formulation. We define the group velocity operator and show how it can be used to obtain the fluid-like velocity. We consider propagation with dispersion and attenuation.

Journal of Modern Optics•Aug 17, 2009•0 citations
We develop the concept of covariance for waves and show that it plays a fundamental role in understanding the evolution of a propagating pulse. The concept clarifies several issues regarding the spread of a pulse and the motion of the mean. Exact results are obtained for the time dependence of the covariance between position and wavenumber and the covariance between position and group velocity. We also derive relevant uncertainty principles for waves.
We give a simple derivation for the covariance of time and frequency. The simplicity remains when applied to arbitrary variables. We show that the covariance enters into the uncertainty principle in a fundamental way. Also we present a seeming paradox: the uncertainty principle depends only on the marginals, but the marginals carry no information about covariance. Then how is it possible that the uncertainty principle involves the covariance? The resolution of the paradox clarifies a number of issues regarding the existence of well behaved manifestly positive joint distributions of time and frequency.>
We apply quasi-distribution methods developed for quantum mechanics to the propagation of pulses in dispersive media with attenuation. We show that a Schrödinger type equation follows for propagation of the pulse for each mode. One then transforms the equation to obtain an equation of evolution in the phase space of position and wavenumber. In this paper we emphasize windowed wave functions and their corresponding phase space quasi-distributions. We obtain the time evolution equation, discuss possible approximations, and compare to the Wigner distribution approximation previously derived by Loughlin and Cohen by different methods.
The usual textbook discussions of the velocity and spreading of a wave packet are based on special cases and approximations which have a limited range of validity. In this paper more general relations are obtained by considering the time dependence of the moments of a wave packet. Both a packet of real sinusoidal plane waves (a real pulse, e.g., sound) and a packet of complex exponential plane waves (e.g., the quantum-mechanical wave function of a free particle) are considered. The results are similar if the real wave packet contains no standing wave components. It is shown that the centroid of a wave packet moves at a constant velocity which is a weighted average of the group velocity. The spreading of the wave packet depends on the product of the variance of the group velocity and the square of the time interval before or after the time of minimum packet width. The packet moves and spreads like a collection of particles in free flight, in which the velocities of the particles are the group velocities of the wave components of the packet.
The propagation of a steplike change in the amplitude of a dc pulse and a step in the envelope of a plane sinusoidal wave are investigated. It is shown that a step (e.g., the beginning or end of a pulse or wave) propagates like a pulse in the medium; i.e., its velocity is a weighted average of the group velocity, and its rate of spread is determined by the distribution of group velocities within the spectrum of the step. The results apply to a real pulse (e.g., sound) in an infinite, homogeneous, dispersive, lossless medium, and to the quantum-mechanical wave function of a free particle.