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The uncertainty principle tells us that $$\sigma_x\sigma_p \geq \frac{\hbar}{2}, $$ which means that the more precisely we measure a particle's position, the more imprecise we will know its momentum. However, the standard deviation $\sigma_x$ can't be zero, and therefore its wave function will always have some spread to it. That got me thinking, is that the reason why particles such as protons, electrons, or neutrons have size? Is their size determined by the average of the spread of their wave function's standard deviation in position space when one collapses their wave functions an infinite amount of times?

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  • $\begingroup$ Eeeeek.... "width" is a technical resonance term and not "size", which you are apparently asking about... Can you supplant the first with the second? And what do you do with pointlike particles like the μ? $\endgroup$ – Cosmas Zachos Aug 7 at 22:34
  • $\begingroup$ I'm aware of the term 'pointlike' in physics, but that is exactly why I'm asking, as if my hypothesis behind my question were true, µ would have a size, eventhough it's negligible. $\endgroup$ – Cazo Aug 7 at 22:48
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    $\begingroup$ Related: physics.stackexchange.com/questions/495576/… . Is their size determined by the average of the spread of their wave function's standard deviation in position space when one collapses their wave functions and infinite amount of times? No, wavefunction collapse isn't even a part of standard quantum mechanics. It only exists in the Copenhagen interpretation. $\endgroup$ – Ben Crowell Aug 7 at 23:07
  • $\begingroup$ What I mean is that, when a wave function is influenced by another wave function, say light hitting an electron, the wave function localizes. Is the width of that location (bell curve), related to the size of the electron? $\endgroup$ – Cazo Aug 7 at 23:46
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In the mainstream standard model of particle physics, all matter is made up of point particles with a fixed mass which we measure as best as we can within our experimental errors. There is no width in this masses at the table.

Protons and (and neutrons bound in a nucleus) are stable composite particles , made up by a great multitude of quarks antiquarks and gluons , plus some valence quarks, and are found as quantum mechanical solutions in a QCD lattice model. Experimentally proton decay has not been seen, so the intrinsic width of the proton mass is still a delta function , although there are models that allow baryon number non conservation. (The same is true for the free neutron, because its lifetime is such that the possible width in mass is not measurable).

Width due to the quantum mechanical wave function is found theoretically and measured experimentally in resonances , and decaying elementary particles, as seen here. The Heisenberg uncertainty is directly connected with this width, but the width depends on the interactions allowed by the various conservation laws for the specific decay.

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