A New Look at Energy — Question 8

Question 8

Two massless particles can form a system with nonzero invariant mass

Two photons travel in opposite directions with energies $E_1=3.00\,\mathrm{MeV}$ and $E_2=1.00\,\mathrm{MeV}$. Use $p=E/c$ for each and choose +x along photon 1.

1. Find the system total energy, momentum and invariant mass.

2. Find the center-of-momentum velocity and the energy of each photon in that frame.

Question 8 - Solution

Two massless particles can form a system with nonzero invariant mass

(a) Method and calculation

$$E=E_1+E_2=4.00\,\mathrm{MeV},\quad Pc=E_1-E_2=2.00\,\mathrm{MeV}.$$$$\boxed{Mc^2=\sqrt{E^2-P^2c^2}=2\sqrt3\,\mathrm{MeV}}.$$

(b) Interpretation and further calculation

$$\boxed{V/c=Pc/E=1/2},\quad\gamma_V=2/\sqrt3.$$$$\boxed{E_1\prime=\gamma_VE_1(1-1/2)=\sqrt3\,\mathrm{MeV}=E_2\prime}.$$

Physical check

Zero individual rest masses do not imply zero system mass; the system mass depends on the vector sum of momenta.