Wavepacket

Let's say you've got a (slightly)ditzy boy/girlfriend, and you're at the mall. You stop at a Subway to get a sandwich or something, and when you turn around, your boy/girlfriend is gone. Now, you have two options at this point:

1. Run around knocking people over, screaming his/her name, until you get desperate and call Security,
OR:
2. Reflect on your SO's basic nature. Realize that he/she likes shoes/jewelry. Construct a wave packet and superimpose it on a map of the mall, making sure high-probability areas are centered on Foot Locker and Zales.

Congratulations! You've just used a wave packet to find your boy/girlfriend! Now, leave him/her there, and look at Everything Personals.

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 sinusoids 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