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Can the spin of a particle increase? Can it get stronger? Can we measure this?

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Spin is an intrinsic property, so the total spin can't change but the projection can on an axis can. We can measure it and we can make particles interact with one another to produce a composite particle with higher total spin.

You might want to work on your question to get a more satisfying answer.

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Not without changing what we understand as the identity of the particle.

If the particle is truly fundamental (i.e. quarks and leptons) then the spin is fixed.

For composite particles, like the proton (which is made of two up quarks and one down) one can change the way the quarks' spin combines to make a total spin of 3/2 while keeping the particle content. However, this introduces so much energy into the quarks' interactions that it dramatically increases the composite's mass, and we simply think of it as a different particle; thus, the spin-3/2 version of the proton is known as the Delta $\Delta^+$ baryon.

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  • $\begingroup$ How on Earth did your answer get a downvote? $\endgroup$ – user154997 Jul 3 '17 at 17:10
  • $\begingroup$ @LucJ.Bourhis ¯\ _(ツ)_/¯ $\endgroup$ – Emilio Pisanty Jul 3 '17 at 17:57

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