Question

Consider two photons, one with energy 1 = 2 MeV traveling to the right, and the other with energy 2 = 3 MeV moving to the left. The two photons collide head-on and produce a positron-electron pair. In this problem you will calculate the velocities of the two particles after the collision. Use the Doppler Shift equation to find the velocity of a frame of reference S' such that the two photons have the same energy. Which direct is S

Answer 1

In the lab frame S, we have E1 and E2 for the two photons.

When we take a frame S' which is moving at a velocity v = u along photon 1 or in the right direction,

it changes the observed frequencues and wavelengths of both photons.

Since energy is related to frequency or wavelength of a photon, S' will observe different E1' and E2' than frame S.

Use doppler formula to find shift for each photon in S' frame using relative speed v = u.

Shift in frequency will shift energy since E = hf.

z = f'/f - 1 = shift

f' = (z+1)f1

where B = v/c

So, using E = hf,

E1 = hf1, E1' = hf1'

E2 = hf2, E2' = hf2'

So, if E1' = E2',

f1' = f2'

So, f1(z1+1) = f2(z2+1)

So, E1(z1+1) = E2(z2+1)

So,

2(z1+1) = 3(z2+1)

Here, z1 will use v = u, z2 will use v = -u since both are moving in opposite directions.

now, put these values in z1 and z2 formula to get u!

2z1 + 2 = 3z2 + 3

2z1 = 3z2 + 1

assuming x = u/c,

So,

So, S' is moving at a speed of (0.384615)c along photon 1, ie. towards right.