An Interlude — Question 5

Question 5

Boosted two-photon annihilation produces unequal laboratory energies

A particle-antiparticle pair has total invariant rest energy W=2.00 MeV and moves at beta=.600 along x. In its rest frame, two photons emerge parallel and antiparallel to x, each with energy W/2.

1. Find the two photon energies in the lab.

2. Verify both total energy and total momentum of the moving initial system.

Question 5 - Solution

Boosted two-photon annihilation produces unequal laboratory energies

(a) Method and calculation

$$\gamma=(1-.600^2)^{-1/2}=1.25,\quad E_\pm=\gamma(W/2)(1\pm\beta).$$$$\boxed{E_+=2.00\,\mathrm{MeV},\quad E_-=.500\,\mathrm{MeV}}.$$

(b) Interpretation and further calculation

$$\boxed{E_++E_-=2.50\,\mathrm{MeV}=\gamma W}.$$$$\boxed{p_xc=E_+-E_-=1.50\,\mathrm{MeV}=\gamma\beta W}.$$

Physical check

Unequal lab photon energies are compatible with an equal-energy pair in its rest frame. Invariant mass remains W/c squared.