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American Journal of Physics

Spreading of a free wave packet

Hai-Woong Lee

Published May 1, 1982• 18 citations
Publisher

Abstract

The spreading of a free wave packet and its relation to classical and quantum dynamics are examined. The phase-space formulation using the Wigner distribution function provides a convenient representation in which each phase point of the distribution follows a classical trajectory and the spreading occurs as a result of classical dynamics. A different formulation using a different phase-space distribution function, however, does not allow this simple interpretation of the spreading.

DOI
10.1119/1.13075

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The spreading of a free wave packet and its relation to classical and quantum dynamics are examined. The phase-space formulation using the Wigner distribution function provides a convenient representation in which each phase point of the distribution follows a classical trajectory and the spreading occurs as a result of classical dynamics. A different formulation using a different phase-space distribution function, however, does not allow this simple interpretation of the spreading.
We discuss four general features of the force-free evolution of wave packets: (1) The spatial spread of a packet changes with time in a simple way. (2) For sufficiently short durations (related to the spread in the momentum of the packet) the probability distribution will move with uniform speed and little change in shape. (3) After a sufficiently long time (related to the initial spatial spread) the wave function converges to a simple form that is simply related to the momentum distribution of the packet. In this asymptotic regime the shape of the probability distribution no longer changes, and its scale increases linearly with the time. (4) There is an infinite denumerable set of simple wave packets (the Hermite-Gauss packets) that do not change shape at any time during their evolution.
A useful and intuitive phase space picture of common semiclassical approximations and procedures is proposed and developed. Underlying the intuitive pictures is the semiclassical, C→0 limit of the Wigner distribution, which is discussed in an Appendix. Simple classical analog phase space diagrams are suggested which represent quantum amplitudes, completeness relations, stationary phase points and integrations, caustic singularities, uniformizations, and the interesting short and long time behavior of the semiclassical propagator. Semiclassical wave packet amplitudes, now known to be on equal footing with the more usual amplitudes between delocalized eigenstates of Hermitian operators, are included in the phase space pictures. It becomes apparent why the primitive semiclassical wave packet amplitudes are to some extent uniformizing, and numerical results are presented to support this conclusion. The power of the phase space picture as an aid in formulating new approximations is illustrated.
The spreading or nonspreading of wave packets in one-dimensional linear and quadratic potentials can readily be determined by solving Ehrenfest’s equations. The results for quadratic potentials can be understood in simple intuitive terms by using quantum distribution functions in phase space which, like classical distributions, rotate clocklike in time once per motional cycle. For more complicated situations, such as potentials with quantum energy spacings which are integral multiples of a basic unit, much can be learned instead by studying the quantum recurrence of their time-dependent wave functions or probability densities.
This chapter presents some of the features of the phase space picture. For quantum mechanical systems Eugene Paul Wigner introduced a quantum analogue of a phase space probability distribution called Wigner distribution function or phase space picture. Using the Wigner distribution function it is possible to perform a canonical transformation in quantum mechanics. The phase space picture is a useful candidate to illustrate the transition from classical to quantum mechanics. The Poisson brackets and canonical transformations are part of the effective formulation of classical mechanics. In the infinite height potential video sequences of the evolution of Wigner functions for a wave function of a wave packet and its mirror wave packet is constructed. The chapter also presents the application of the canonical transformations and phase space distribution function in the study of wave packet spread and coherent and squeezed states.
The relationship between the diverging of classical trajectories in chaotic many-body systems, the spreading of quantum wave packets, and the validity and use of classical molecular dynamics is explored. This analysis, which is based on the semiclassical description of wave function propagation in terms of a weighted integration over a traveling fixed width coherent state basis, suggests that the exponential divergence of nearby classical trajectories in chaotic many-body systems should result in the rapid delocalization of an initially localized quantum wave packet describing the state of the system. Thus the justification for the use of classical molecular dynamics procedures for these supposedly classical systems cannot be based on the picture of the system wave function remaining localized as its center follows a nearly classical trajectory. The quantum evolution of the system density, on the other hand, requires two propagators, and each of these propagators is represented as an integration over trajectories in the semiclassical picture. The interference between the contributions from these two integrations over classical trajectories focuses the analysis on the most important points in this trajectory pair space, which are shown to occur when both trajectories in the pair are the same. Given reasonable assumptions for the initial density for a system that is expected to be well described by classical molecular dynamics, and given an appropriate choice for the width of the coherent state basis which is employed in the semiclassical description, it is shown that the semiclassical expressions for time dependent observables and correlation functions reduce the purely classical expressions, despite the fact that an initially localized wave packet would rapidly delocalize for the same system.