Why is a moving particle represented by a wave packet instead of a plane wave?
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In one sense quantum particles are particle-like simply because they are countable, and because the wavefunction is normalized. But how can we understand them moving in classical trajectories? How do we describe a particle that is behaving in a sense that is in-between the classical and quantum limits?
We describe such particles in terms of wave packets. A wave packet is a form of wave function that has a well-defined position as well as momentum. Thus wave packets tend to behave classically and are easy (and fun) to visualize. Naturally, neither the momentum nor the position is precisely defined, as is governed by the uncertainty principle.
A classic application of the uncertainty principle is in understanding wave packets. A wave packet with a very well-defined position will have a very uncertain momentum, and thus will quickly disperse as the faster components move on ahead of the slower ones. Conversely, if you construct a wave packet with a very definite momentum it will travel a long distance without dispersing, but it starts out being very broad already in position space.
The simples (and most commonly used) wave packet is a Gaussian wave packet. We can construct such a wave packet most easily in reciprocal space:
ψ~(k)=e−a2(k−k0)2ψ~(k)=e−a2(k−k0)2