Localization of states and elementary vs composite in QM

[TrickyDicky]

While browsing Wikipedia I bumped into this sentence that seemed partially wrong to me but maybe I didn't understand what it is referring to so would like for some expert to help me elucidate it: "Even if an elementary particle has a delocalized wavepacket, the wavepacket is in fact a quantum superposition of quantum states wherein the particle is exactly localized. This is not true for a composite particle, which can never be represented as a superposition of exactly-localized quantum states."
There are two parts: If the wavepacket is a superposition of plane waves, in what sense are they "quantum states wherein the particle is exactly localized"?
And if this was so, why in the bound state wavepacket case this can never be so?

 

[kith]

There are two parts: If the wavepacket is a superposition of plane waves, in what sense are they "quantum states wherein the particle is exactly localized"?

You could write
$$e^{-ipx'} \propto \int_{-\infty}^{+\infty} \delta(x-x') e^{-ip(x-x')} dx,$$
so in this sense, a plane wave can be viewed as a formal superposition of delta functions just like a Gaussian wavepacket can be viewed as a formal superposition of plane waves. But as you probably know, delta functions are not square integrable so they are not proper quantum states but idealizations.

And if this was so, why in the bound state wavepacket case this can never be so?

This probably refers to entanglement. If the composite state is not a product state, you cannot assign definite wavefunctions to the single particles at all.

 

[TrickyDicky]

Thanks, kith.

You could write
$$e^{-ipx'} \propto \int_{-\infty}^{+\infty} \delta(x-x') e^{-ip(x-x')} dx,$$
so in this sense, a plane wave can be viewed as a formal superposition of delta functions just like a Gaussian wavepacket can be viewed as a formal superposition of plane waves. But as you probably know, delta functions are not square integrable so they are not proper quantum states but idealizations.

Yes, they are idealizations, with certain validity as approximations for many different situations(like for VP propagators) but not to undermine the very foundation of QM, the HUP. It looks like in particle physics they take seriously this idealization, the paragraph I quoted goes on to say: "It is in this sense that physicists can discuss the intrinsic "size" of a particle: The size of its internal structure, not the size of its wavepacket. The "size" of an elementary particle, in this sense, is exactly zero."
Since in particle physics this assumption about the internal structure of elementary particles is essential to their interpretation of scattering experiments of i.e.electrons as probes of the internal structure of i.e. protons I wonder if some inconsistency can be derived from this de facto dispensing with the HUP.

This probably refers to entanglement. If the composite state is not a product state, you cannot assign definite wavefunctions to the single particles at all.

I see what you mean, you cannot assign them states since when considering the isolated composite particle they are acting as virtual, so this is by definition. What I find ironic here is that the composite state itself is mathematically equivalent to elementary particle one, even if physically by definition we cannot assign it the same meaning as in the elementary particle case.