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I have read this question:

Elementary particle (electron) and non-elementary (proton) spagettification

and the comments where it says:

But no real elementary particle can be confined in a region smaller than its compton wavelength.

An these questions:

What is the physical significance of Compton wavelength?

Confining a particle into a region shorter than its Compton wavelength

https://en.wikipedia.org/wiki/Compton_wavelength

Where Veritas' answer says:

Yes, this will happen. But you cannot confine particle in the vacuum. To confine a particle, you must have some potential. The energy to produce pairs must come exactly from this binding potential. For example, you can confine electron using a very strong electric field. To confine an electron in a region smaller than its Compton wavelength you need a field with enough energy to create electron position pairs. Particle in a vacuum will never be confined.

So which one is right?

Question:

  1. Can elementary particles be confined into a smaller region then their Compton wavelength?
Árpád Szendrei
  • 30,008

1 Answers1

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I think it is possible to confine an electron mostly into a region smaller than its Compton wavelength ($\lambda_C=2.4\mathrm{pm}$).

First consider Bohr's model for a single electron in the field of a nucleus of charge Ze. The ground state has a normalized probability density proportional to $\exp\left(-{Z r}/{\mathrm{a}_1}\right)$. Here $\mathrm{a}_1$ is the hydrogen Bohr radius of 53pm. So that Bohr diameter is 44 times $\lambda_C$.

Next consider a fully ionized uranium nucleus an add one electron. Its Bohr radius will be 92 times smaller and hence $\mathrm{a}_{92}=0.58\mathrm{pm}$. The diameter is 1.06pm, which is still considerably smaller than $\lambda_C$.

When we add a second electron to fill the s-shell, the orbital increases in diameter. For example in helium the radius of the full s-shell 31pm instead of the scaled Bohr radius of 53pm/2; ie a 17% increase.

If this approximation holds inside a uranium atom, we can estimate that the s-shell is confined mostly to a sphere smaller than $\lambda_C$.

This can be extended to heavier nuclei as long as they are stable enough to allow for the formation of an electron shell before they decay.

Michael Momayezi
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