(a) Consider a photon of angular frequency ω which is converted to a particle antiparticle pair at a height h above the ground.

Question

Pair creation is a phenomenon in which a photon is converted to a pair of particle and antiparticle, both of exactly same mass. On the other hand in pair annihilation, the particle anti-particle pair annihilate each other, are vanished, and their energy is converted into giving off a photon. 

(a) Consider a photon of angular frequency ω which is converted to a particle antiparticle pair at a height h above the ground. The particles are produced in rest with having no energy due to motion. What would be the mass of each particle in terms of ω? 

(b) Now the two particles fall down the height h. What is the total energy possessed by the particles downstairs? Down there, they annihilate each other and their energy is converted to give off a photon. What is the frequency of this photon, in terms of ω and h? 

(c) Now we send this newly created photon back to height h. In order to avoid the creation of energy, this photon must lose energy so that at height h it just have enough energy to create the pair of same mass again, without having any extra energy. What must be this energy loss for this to happen? What is the change in frequency in climbing up the height h? Does it agree with the formula we derived in class?

Answer 1

(a) As the photon has to be created two particles of equal mass and they have no energy due to motion implying their velocity is zero. Thus according to law of conservation of energy,

where m₀ is the mass of each particle.

(b) As they fall down they would gain an energy mgh, where m = m₀ =rest mass. As they were at rest, the velocity with which they fall is not relativistic.

is the total energy of the particle downstairs. Now at the time of annihilation, according to law of conservation of energy

Thus frequency of this photon is higher, because due to the attraction of gravity, energy of photon increase.

(c) To create pair of same mass again, the photon must have the same energy as the old photon and as it goes back to height h. Thus energy loss will be,

Change in frequency while climbing up the height will be,

Yes it again agrees that formula we derived in class.