Relativistic momentum of two photons from a decay

[zehkari]

Homework Statement

Homework Equations

(1) E² = p²c² + m₀²c⁴

(2) E = γm₀c²

(3) E = E_γ1 - E_γ2

(4) p = E / c

(5) E = hf

(6) λ = c / f

The Attempt at a Solution

a) Using eqn (1), rearranged p = (E - m₀c²) / c , I obtained 2.9 MeV c^-1. Not sure if I have the right answer here as I converted the 1.000 GeV to 1*10³ MeV and kept both the rest energy and total energy in mega electron volts.

b) Using eqn (2), rearranged γ = E / m₀c² , γ = 7.63. Then by approximation of the lorentz factor I obtained a difference in velocity to be 2.57*10⁶ ms^-1

c) Is where I am stuck. From the observers reference frame or the labs reference frame, the two photons would be in different directions. So by using eqn (3) some how find the two different energies?
And then I guess by using eqns (5) & (6) you can find the wavelength? Bit stuck on the mathematics here.

Thanks for any help and happy holidays :).

 

[Andrew Mason]

c) Is where I am stuck. From the observers reference frame or the labs reference frame, the two photons would be in different directions. So by using eqn (3) some how find the two different energies?
And then I guess by using eqns (5) & (6) you can find the wavelength? Bit stuck on the mathematics here.

The energies of the photons have to be divided in a way that also conserves momentum. Think of the decay in the pi meson frame. Then in the lab frame.

AM

 

[zehkari]

Hello,

Thanks for replying.

In the pi meson frame then distribution of momentum would be 1/2 for each photon. Does this imply the energy of one of the photons in the lab reference frame would be E^' = γE ?

And then with de Broglie's wave-particle duality: λ = h / p , so, λ = hc/E for each wavelegth?

 

[Andrew Mason]

What would the total energy of the photons be in the rest frame of the pi meson? Then assume that the direction of one of the photons is in the direction of motion of the pi meson. Then translate the photon energies to the lab frame.

AM

 

[Andrew Mason]

You can determine the total momentum in the lab frame from the energy equation. Since energy is conserved $h\nu_1 + h\nu_2 = E_{total}$. You also know that momentum is conserved so for photons traveling in the +- direction of the meson $h\nu_1/c - h\nu_2/c = p$.

AM