# Localization of states and elementary vs composite in QM

• TrickyDicky
In summary: 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.Inconsistency is hard to find in a field that has been developing for almost a century, but I don't think there would be any serious consequences from this assumption if it turned out to be wrong.

#### 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?

TrickyDicky said:
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.

Thanks, kith.
kith said:
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.

## 1. What is localization of states in quantum mechanics?

Localization of states in quantum mechanics refers to the ability to determine the position and momentum of a particle with a certain degree of precision. It is a fundamental concept in quantum mechanics and is related to the uncertainty principle, which states that the more precisely we know the position of a particle, the less we know about its momentum and vice versa.

## 2. How are elementary particles different from composite particles in quantum mechanics?

Elementary particles are particles that cannot be broken down into smaller components and are considered to be the building blocks of matter. Composite particles, on the other hand, are made up of smaller particles and are not considered to be fundamental. In quantum mechanics, elementary particles follow different rules and interactions compared to composite particles.

## 3. What is the significance of studying the localization of states in quantum mechanics?

The study of localization of states in quantum mechanics is important because it allows us to understand the behavior and properties of particles at the atomic and subatomic level. This understanding is crucial for many practical applications, such as in the development of new technologies like transistors and lasers, as well as in the field of quantum computing.

## 4. How does the concept of localization of states relate to the wave-particle duality in quantum mechanics?

The concept of localization of states is closely related to the wave-particle duality in quantum mechanics. According to this principle, particles can exhibit both wave-like and particle-like behavior. The localization of states refers to the particle-like behavior of particles, while the wave-like behavior is described by the wave function. These two aspects are interconnected and must be taken into account when studying the behavior of particles in quantum mechanics.

## 5. Can localization of states be observed experimentally?

Yes, localization of states can be observed experimentally. One way to do this is through the double-slit experiment, where particles are fired at a barrier with two narrow slits. The resulting pattern on a screen behind the barrier shows the particle's localization, as it passes through one of the slits and creates an interference pattern. This experiment demonstrates the wave-like and particle-like behavior of particles in quantum mechanics, including their localization.