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I have heard many times that we can treat a moving particle as a:

  1. classical particle

  2. non-relativistic

  3. relativistic particle

  4. ultra-relativistic particle

While I know equations for 1, 2, & 3, I really don't know what is the difference between ultrarelativistic and relativistic particle. Can anyone explain a bit or provide some hyperrefs.

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An ultra-relativistic particle is any particle you observe to have almost all its energy stored in the form of momentum. In other words, we are talking about particles that have only a very tiny fraction of their total energy stored in (rest)mass.

The relativistic mass-energy-momentum relationship

$$E^2 - c^2 \ p^2 \ = \ c^4 \ m^2 $$

is valid for a particle with (rest)mass $m$ regardless its speed. Depending on the relative magnitude of the various terms a particle is referred to as ultra-relativistic, relativistic, or non-relativistic.

An ultra-relativistic particle speeds by with $E \approx c \ p >> m \ c^2$. Examples are neutrinos (at almost any energy), but also protons accelerated to full speed in the LHC.

In contrast, non-relativistic particles (I prefer to reserve the term classical particles for particles behaving in 'non-quantum' fashion) are characterized by $E \approx m \ c^2 >> c \ p$.

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Depending on the relative sizes of the $mc^2$ and the $pc$ sides of this right triangle, a particle is called non-relativistic, relativistic, or ultra-relativistic.

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  • $\begingroup$ Thank you! But i am to use the same equations for ultra relativistic and relativistic particles? I guess yes, but a confirmation would be nice. $\endgroup$ – 71GA Jul 22 '13 at 13:16
  • $\begingroup$ Good point. Have incorporated this in my answer. $\endgroup$ – Johannes Jul 22 '13 at 13:24
  • $\begingroup$ @71GA With ultra-relativistic particles is is common to drop the mass when working kinematics problems. It makes things really easy---often easier than Newtonian problems. $\endgroup$ – dmckee Jul 22 '13 at 21:18

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