The idea of a wave packet is that you superimpose a lot of plane waves to create a localized free electron. But when and why is this useful? It seems that wave packets are often used to derive transport equations inside solids. But in principle the true electronic wavefunctions given by Blochs theorem are perfectly valid solutions to the Schrodinger equation and are even normalizeable. So why do we ever bother to talk about wavepackets and what is the physical reason for using these?
$\begingroup$ Sometimes, without wave packets you cannot integrate by parts. $\endgroup$– AccidentalFourierTransformJan 9, 2017 at 18:25
You don't "use" wave packets. They simply arise due to uncertainty.
If you have a single wave of known frequency, then it is the same everywhere.
If you combine two waves of slightly different frequency, you get a "beat" effect, due to alternate cancellation and reinforcement.
If you combine many waves of slightly different frequencies, the waves almost always cancel, but they do occasionally reinforce, but these reinforcements come further and further apart.
If you combine infinitely many waves having a distribution of frequencies, you get only one region of reinforcement, and that is called a "particle" or "wave packet".
If you are uncertain of the frequency (which you are) then you can think of it as a distribution of frequencies, and there you are.
$\begingroup$ This addresses the mathematical origin of wave packets. It seems to me that OP is asking about their physical origin. $\endgroup$ Jan 9, 2017 at 18:32
2$\begingroup$ @AccidentalFourierTransform: If you (accidentally :) take a fourier transform of a photon (like with a lens), then you see the distribution, right? If that's physical or mathematical I guess I don't know. $\endgroup$ Jan 9, 2017 at 18:41
3$\begingroup$ Ah, the elusive thin line between physics and mathematics... What I meant is that I have the feeling that OPs question is "why do you need wave packets", not "how do you generate them/how are they formed"; your answer seems to address the latter question though. Of course, I might be wrong. $\endgroup$ Jan 9, 2017 at 18:47
$\begingroup$ Yes I am more interested in the physical origin. $\endgroup$ Jan 9, 2017 at 18:51
$\begingroup$ Has anyone ever physical described the wave packet or an origin? $\endgroup$ Jan 9, 2017 at 18:56
You seem to be unaware of the whole field of particle physics, with its enormous number of scattering experiments which studied have led to the standard model of particle physics. This is based on quantum field theory which works with creation and annihilation operators on plane wave wave functions of the appropriate equations ( dirac for fermions for example).
Plane waves are not a good model for an incoming electron beam or the positron beam coming to meet it , as plane waves go from +infinity to -infinity and the two beams would have a very small probability of meeting within the experimental area, which experimentally is not the case.
Usually one is taught the Feynman diagrams for the interaction, i.e. point particles meeting at vertices, but one needs, a consistent quantum mechanical model since the commutator relations will give a volume in momentum and space for the elementary particles. This is treated with the wavepacket formalism, see for example in this online QFT book at page 49 equations 5.6 and 5.7, where in order to describe a real particle a wave packet is defined.
So wave packets are a necessary conceptual mathematical tool to bring into consistency creation and annihilation operators operating on plane wave solutions, and the quantum mechanical modeling of particles within the QFT framework.