A particle called neutral pi meson (or pion) decays into two photons (and nothing else).If the pion (whose rest mass is 135Mev) is initially at rest,
What are the energies of the photons?
How are they moving with respect to each other ?
It's easier to start with the second part of the question. The momentum of the poin is zero, so the sum of the momenta of the two photons (p1 and p2) is zero:
p1 + p2 = 0 ------->
p1 = - p2
Then because the energy of a photon with momentum p = |p| c, you see that the energies of the two photons are equal. The sum of the energies of the two photons has to be 135 MeV, because of energy conservation, so each photon has half this energy.
To determine the energies of the photons, we can use the principle of conservation of energy and momentum.
1. Conservation of energy: In the rest frame of the pion, the total energy before decay is equal to the total energy after decay. Since the pion is initially at rest, its energy is solely its rest mass energy. Therefore, the total energy before decay is 2 * (mass of pion) = 2 * 135 MeV = 270 MeV.
2. Conservation of momentum: In the rest frame of the pion, the initial momentum is zero. After the decay, the momentum should still be zero because it is a two-photon decay, meaning the photons should move in opposite directions to cancel out their individual momenta.
Now, let's assign the energies of the photons as E1 and E2, where E1 is the energy of the first photon and E2 is the energy of the second photon. Additionally, let's assume the first photon moves to the left and the second photon moves to the right.
Using conservation of energy and momentum, we have:
E1 + E2 = 270 MeV (Conservation of energy)
E1/c - E2/c = 0 (Conservation of momentum, where c is the speed of light)
From the second equation, we can rewrite it as E1 = E2.
Substituting this relationship into the first equation, we get:
E1 + E1 = 270 MeV
2E1 = 270 MeV
Solving for E1, we find:
E1 = 135 MeV
Since E1 = E2, the energy of both photons is 135 MeV.
Therefore, the energies of the photons are 135 MeV each.
As for their motion with respect to each other, since their energies are equal, they should be moving directly away from each other in opposite directions, ensuring conservation of momentum.