Relativistic energy/momentum, massless particles

[zippideedoda]

Homework Statement

A pion at rest decays into a muon and an antineutrino. The mass of the antineutrino is zero, find the energies and momenta of the muon and antineutrino. Mass of the pion is 139.57 MeV/c^2 and the mass of the muon is 105.66 MeV/c^2

Homework Equations

pion -> muon + antineutrino

(1) E=mc^2
(2) E=pc
(3) E^2 = (pc)^2 + (mc^2)^2

The Attempt at a Solution

Conservation of energy: E(pion) = E(muon) + E(antineutrino)
Using equation 1 and the given masses:
E(pion) = 139.57 MeV
E(muon) = 105.66 MeV (?)
so E(antineutrino) = E(pion) - E(muon) = 33.91 MeV
Use equation 2 for the massless antineutrino and get p = 33.91 MeV/c

since momentum is conserved and p(pion) = 0 (at rest),
p(muon) = -p(antineutrino) = -33.91 MeV/c

I don't think I'm right because if the muon has momentum it is moving and thus I can't find its energy by simply plugging its mass into equation 1.

Thanks for any help.

 

[phyzguy]

You are correct in that you can't use Equation 1 for the energy of the muon. You have to use Equation 3. Otherwise you are on target.

 

[zippideedoda]

How about this:

p(muon) = p(antineutrino) = p
E(antineutrino) = pc
E(muon) = sqrt[(pc)^2+(mc^2)^2]

plug these into conservation of E and solve for p
after some math I get

p = 29.79 MeV/c
so E(antineutrino) = 29.79 MeV
E(muon) = 109.78 MeV

The two E's add up to equal E(pion), 139.57

 

[phyzguy]

Looks good to me.

 

[rwisz]

Your approach is correct, but there is a slight error in your calculation. The energy of the muon should actually be 33.91 MeV, not 105.66 MeV. This means that the momentum of the muon is also 33.91 MeV/c, and the momentum of the antineutrino is -33.91 MeV/c, as you correctly calculated.

This result may seem counterintuitive, but it is a consequence of the relativistic equations for energy and momentum. When a massive particle (such as the pion) decays into two particles, one of which is massless (such as the antineutrino), the energy is not divided equally between the two particles. Instead, the energy is distributed in such a way that the total energy and momentum are conserved.

In this case, the antineutrino, being massless, receives all of the energy (33.91 MeV) and momentum (33.91 MeV/c) from the pion, while the muon, being massive, receives a smaller portion of the energy (also 33.91 MeV) and momentum (33.91 MeV/c). This is why the energy and momentum values for the muon are lower than its mass would suggest.

Overall, your approach was correct, but just be careful with the calculations and make sure to double check your units. Keep up the good work!