A wavepacket is a mathematical function that contains all information about the probabilistic [position] and [momentum] of a [particle] (more exactly, a "[wave-particle]" but nobody says this--see [wave/particle duality] if you are unfamiliar with this notion). Actually, the concept of a wavepacket has applications in all wave-related fields, but in this writeup I'll concentrate on the quantum mechanical meaning.

A single [sinusoidal] wave extends uniformly over all space, and thus has no [location]. A wavepacket is the sum of many sinusoidal waves. A [summation] of [sinusoid]s produces a function that is more [localized]--although it still extends over all [space], it has a peak. The position of a particle is described by its wavepacket, and the peak of the wavepacket corresponds to the region of space which has the highest probability of being found to be the particle's position upon measurement. In case you weren't aware of it, there is a finite (albeit ridiculously low but philosophically nonzero) probability that every component of your body would be measured to be on [Jupiter]. Physicists are now convinced that the wavepacket describes everything that can be known about a particle (see [Copenhagen Interpretation]).

Mathematically, a wavepacket (in one dimension; the extension to three dimensions is trivial) is described by the following [function].

ψ(x,t) = ∫φ(k)e^{i(kx - w(k)t)}dk,

where w(k) is E(k)/(h/2π). For a [free particle], E(k) is just (hk/2π)²/2m. Notice that if φ(k) is a [delta function] then the wavepacket of a [free particle] consists of a single, unlocalized and un-normalizable [sinusoid]. As [David Griffiths] puts it, There is no such thing as a free particle with a definite energy.

Let t = 0 correspond to a time when we know the structure of the wavepacket. Then it is clear that φ(k) is just the [Fourier Transform] of ψ(x,0). When φ(k) is known, the wavepacket's structure over time is fully characterized. φ(k) typically peaks at some value k₀ and has a narrow range. It can be shown that the [group velocity] of the wavepacket (as opposed to the [phase velocity]) is given by dw/dk |k₀. In the case of a free particle, the [group velocity] is twice the [phase velocity] (w/k). The [group velocity] is the more physically meaningful of the two quantities--it is the [velocity] of the particle's probabilistic peak. Notice that for a free particle, E = 1/2 mv_group², as required by the [Correspondence Principle]!

Reference: [Introduction to Quantum Mechanics] by David J. Griffiths

Wavepacket (thing)