TL;DR
This paper analyzes the behavior and evolution of free wave packets, highlighting their spatial spread, shape stability over short times, asymptotic form, and special non-shaping Hermite-Gauss packets.
Contribution
It provides a comprehensive description of wave packet evolution, including new insights into shape stability and the identification of Hermite-Gauss packets that do not change shape.
Findings
Wave packets' spatial spread changes simply over time.
Short-term probability distribution moves with little shape change.
Long-term wave packets settle into a form related to momentum distribution.
Abstract
We discuss four general features of force-free evolution: (1) The spatial spread of any packet changes with time in a very simple way. (2) Over sufficiently short periods of time (whose duration is related to the spread in momentum of the packet) the probability distribution moves but there is little change in shape. (3) After a sufficiently long period (related to the initial spatial spread) the packet settles into a simple form simply related to the momentum distribution in the packet. In this asymptotic regime, the shape of the probability distribution no longer changes except for its scale, which 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 as they evolve.
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