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Probability of finding a particle in a solid angle

  1. Nov 9, 2013 #1
    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.

    Imagine that a [itex]\pi[/itex]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 [itex]\pi[/itex]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 [itex]\int[/itex]P(θ)dΩ=1.

    My ideas:
    I know the Lorentz transformations, so switching between frames is no biggie. I know Ω[itex]\equiv[/itex]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. jcsd
  3. Nov 9, 2013 #2
    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').
  4. Nov 9, 2013 #3
    P(θ) is known? Am I missing it? I'm not exactly sure what it would be, or how to get it.
  5. Nov 9, 2013 #4
    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
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