Quantum wavepackets expanding in free space

[fog37]

Hello,

A quantum free particle (no forces acting on it) is usually introduced as a plane wave. A plane wave has a very specific momentum and energy but the function is not normalizable and the particle has a change to be everywhere in space. Mathematically, this plane wave cannot be represented by a real-valued sine wave like sin(kx-wt) which does not solve SE. The plane must be expressed as a complex wave e^i(kx-wt).

Another more realistic way to represent a free particle is as a wavepacket (summation of plane waves). Why bother with the plane wave scenario at all if it represents such an unrealistic case and a pure idealization?

Interestingly, a wavepacket made of light and traveling in a vacuum does not spread (no dispersion): the composing plane waves all travel at the same phase speed, hence the packet does not change shape and continues traveling at the group velocity. However, a quantum wavepacket, like a free electron, spreads in free space, i.e. its probability density spreads, as it travels at its group speed. I can see why that happens mathematically from the nature of the dispersion relation. But what is the overarching concept behind this natural expansion of the probability density? The quantum particle seems to behave like a free gas that continues to expands in free space. What are the consequences behind this? Most problems we deal with seem to involve forces (potentials) that influence them.
So a single electron, or proton or neutron, etc. would spread in free space while a photon does (which I infer from a light pulse not spreading).

Thanks!

 

[Nugatory]

Another more realistic way to represent a free particle is as a wavepacket (summation of plane waves). Why bother with the plane wave scenario at all if it represents such an unrealistic case and a pure idealization?

There are two reasons. First, you can't claim that a superposition of plane waves is a solution to Schrödinger's equation until you've satisfied yourself that the plane waves are themselves a solution of Schrödinger's equation. Second, although it is not physically realizable the plane wave is a very good approximation to a particle of known momentum in many scattering/reflection/transmission problems - and these are some of the most important problems we encounter.