Special Relativity: Pion Decay - Two Photons

[FaradayCage]

Homework Statement

A (pi)0 meson whose rest mass is 135 MeV/c2 is moving with a kinetic energy of 1 GeV. It decays in flight into two photons whose paths are along the direction of motion of the meson. Find the energies of the two photons.

Homework Equations

Lab Frame:

The invariant: $$E^2 = P^2c^2 + m^2c^4$$
Conservation of Energy: $$E = E_1 + E_2 = hf_1 + hf_2$$
Conservation of Momentum: $$P = P_1 + P_2 = hf_1/c + hf_2/c$$

ZMF Frame:

The invariant: $$E^2 = P'^2c^2 + m^2c^4$$
Conservation of Energy(ZMF frame): $$m_0c^2 = E'_1 + E'_2 = hf'_1 + hf'_2$$
Conservation of Momentum(ZMF frame): $$P' = P'_1 + P'_2 = 0$$

The Attempt at a Solution

Using invariant and $$E = E_1 + E_2$$, $$P = P_1 + P_2$$ (lab frame)

We have $$E_1^2 + E_2^2 + 2E_1E_2 = (P_1^2 + P_2^2 + 2P_1P_2)c^2 + m_o^2c^4$$

Not sure what other equations I can get from conditions given. Too many unknowns?? I get the feeling this should end up with a quadratic to give the two different energies...

 

[kuruman]

I find it easier to treat problems of this kind by introducing gamma in the equations. Then, for a particle,

$$E=\gamma m_0c^2$$

$$K=(\gamma-1)m_0c^2$$

$$p=\sqrt{\gamma^2-1}m_0c$$

Use these in your energy and momentum conservation equations. Pay attention to the relative direction of travel of the two photons. Is the angle between the direction of the two photons zero or 180^o?

 

[FaradayCage]

Thanks a lot for the help :) I sorted it in the end by finding the energies in the ZMF after the decay and then using the doppler shift to transform them back into the lab frame.