# Probability of finding a particle in a solid angle

1. Nov 9, 2013

### cjurban

I have an interesting question that I'm not sure how to go about solving. This question has a little general relativity and (maybe) a little QM, but I wasn't sure where to post it.

Question:
Imagine that a $\pi$0 meson traveling along the z-axis (velocity v=0.99c, rest mass M) decays into two photons. The angular distribution of the photons is isotropic in the rest frame of the pion. If in the lab frame the $\pi$0 meson travels with velocity v in the z direction, what is the probability P(θ)dΩ that a photon is emitted into the solid angle dΩ?
We also know $\int$P(θ)dΩ=1.

My ideas:
I know the Lorentz transformations, so switching between frames is no biggie. I know Ω$\equiv$A/r2, and I know the differential solid angle. What's confusing to me is P(θ). Do I need to get the particle's wave function, as in P(θ)=ψ2(θ)? How would one do this? Any hints would be helpful.

2. Nov 9, 2013

### dauto

The probability P(θ) dθ that the particle is between θ and (θ + dθ) will be the same probability P'(θ') dθ' that the particle is between θ' and (θ' + dθ'), where θ and θ' are the angles as measure in the two different reference frames (Aberration formula). Since P(θ) is known, it is possible to calculate P'(θ').

3. Nov 9, 2013

### cjurban

P(θ) is known? Am I missing it? I'm not exactly sure what it would be, or how to get it.

4. Nov 9, 2013

### dauto

Didn't you say the distribution is isotropic in the rest referential frame? P(θ) must be inversely proportional to sin(θ) and the constant of proportionality is found by normalizing the probability