Real and virtual photons via neutral pion decay

[padraux3]

I wasn't sure whether this question should go in advanced or introductory physics but I decided to post here since it doesn't involve any complex maths.

Homework Statement

Part 1
Consider a neutral pion at rest. On the basis of conservation of energy and momentum alone, show that it is possible for the pion to decay into two real photons (ie: two photons that obey the classical energy momentum relation for photons).

Part 2
Show that if a neutral pion decays into a single photon then the photon is virtual.
Hint: Consider the decay in the rest frame of the pion.

Homework Equations

Photon energy = |p|c
E = mc^2
Conservation laws

The Attempt at a Solution

I'm not sure if I've been going about this the right way but I've been trying all day.

Since the pion is at rest its energy is E(pion) = mc^2.
Then by the conservation of energy, the sum of the energies of the two photons is equal to E(pion). Also, the pion has zero momentum so the momentum of the two photons is also zero (thus they are moving at the same speed in opposite directions). From here I've tried to show that the energy of each photon is given by the first equation I listed. I know that the pion's energy should be evenly distributed between the two photons but I'm not sure how to show it.

Thanks in advance.

 

[tiny-tim]

Hi padraux3!

I think you're making this too complicated …

why derive the energy of a photon from its momentum?

just use energy of photon = h times … ?

(and I suppose techinically you have to show that all the other quantum numbers, like baryon number, are conserved )

 

[padraux3]

Thanks for your reply.

If I use E = hf then I end up with the equation mc^2 = h(f1+f2) where fi is the frequency of the ith photon and m is the mass of the pion. I don't really understand what I need to show. From the first hint I assumed I needed to show E = |p|c for each photon so that's why I was trying to derive the energy from momentum. I can't use conservation of quantum numbers, only energy and momentum.

 

[diazona]

I don't think you don't need to prove the energy-momentum relation (unless this is an exercise in deriving special relativity from scratch ).

I know that the pion's energy should be evenly distributed between the two photons but I'm not sure how to show it.

To show that, you can use momentum conservation in the pion's rest frame. For example, say that photon #1 has energy E₁ and momentum p₁, and photon #2 has energy E₂ and momentum p₂. What is p₁ + p₂? Then use that to show how E₁ relates to E₂.