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Journal of Modern Optics

Why do wave packets sometimes contract?

Leon W. Cohen

Published Nov 1, 2002• 5 citations
Publisher

Abstract

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.

DOI
10.1080/0950034021000011248

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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.
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.
In this paper we discuss some aspects of the theory of wave packets. We consider a popular noncovariant Gaussian model used in various applications and show that it predicts too slow a longitudinal dispersion rate for relativistic particles. We revise this approach by considering a covariant model of Gaussian wave packets, and examine our results by inspecting a wave packet of arbitrary form. A general formula for the time dependence of the dispersion of a wave packet of arbitrary form is found. Finally, we give a transparent interpretation of the disappearance of the wave function over time due to the dispersion—a feature often considered undesirable, but which is unavoidable for wave packets. We find, starting from simple examples, proceeding with their generalizations and finally by considering the continuity equation, that the integral over time of both the flux and probability densities are asymptotically proportional to the factor 1/| x | 2 in the rest frame of the wave packet, just as in the case of an ensemble of classical particles.
A physics-oriented review is first given of wave equations: examples, the Cauchy problem, solutions in various coordinates, their symmetries, how to build a wave packet. Fourier analysis and dispersion relations are then introduced, and here the key tool is the symbol of differential operators, which makes it possible to study them in terms of algebraic polynomials involving cotangent-bundle variables. Further basic material deals with geometrical optics from the wave equation, phase and group velocity (and their dual relationship in momentum space), the Helmholtz equation and the eikonal approximation for the scalar wave equation. The Schrödinger equation is then derived with emphasis on the wave packet and its relation to classical behaviour in the light of the Einstein–de Broglie relation. For this purpose, it is shown that it is possible to build a wave packet for which the wave-like properties manifest themselves for distances of the order of atomic dimensions. The Fourier transform with respect to time of the wave packet obeys the stationary Schrödinger equation, while the wave function is found to obey a partial differential equation which is of first order in time. The wave equation The wave equation occurs in several branches of classical physics, e.g. the theory of sound, electromagnetic phenomena in vacuum and in material media, and elastic vibrations of material bodies. We consider first the simplest situation, i.e. the propagation of waves in a homogeneous, isotropic and stationary medium. Propagation is characterized by the refractive index n , a quantity that is independent of the point in space-time.
The evolution of slowly varying wave pulses in strongly dispersive and absorptive media is studied by a recursive method. It is shown that the resulting envelope function may be obtained by including correction terms of arbitrary dispersive and absorptive orders. Different definitions of pulse velocity are discussed in some detail and finally, qualitative examples of a Gaussian pulse in a cold plasma and an atomic medium are given, illustrating the importance of higher-order terms in the velocity expression. Particularly, the packet velocity is shown to be dependent on pulse width besides dispersion.