- #1
ryanwilk
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Homework Statement
Consider the decay A -> B + C (where A is not at rest). In the rest frame of A, B is emitted in a random direction (all directions have equal probability) and I need to show that in the lab frame, the energy distribution of B is uniform.
(We assume that B has negligible mass)
Homework Equations
(Let c=1)
The Attempt at a Solution
So I started by writing down the 4-momenta of A and B in the rest frame of A (choosing the momentum of B to be along the x axis):
PA' = (mA,0,0,0)
PB' = (EB',EB',0,0)
In the lab frame:
PA = (EA,(EA2-mA2)1/2,0,0)
PB = (EB,EB,0,0)
Lorentz boosting along the x-axis, I can determine the maximum and minimum energy that B can have:
[tex]
E_B^{\mathrm{min,max}} = \frac{E_A}{2} \bigg(1-\frac{m_C^2}{m_A^2}\bigg) \bigg(1 \pm \sqrt{1-\frac{m_A^2}{E_A^2}}\bigg)
[/tex]
using:
[tex]
\gamma = \frac{E_A}{m_A}\>,\gamma \beta = \frac{p_A}{m_A},\>E_A' = \frac{m_A^2-m_C^2}{2m_A}
[/tex]
I could also Lorentz boost in all other directions to get expressions for the energy. But I have no idea how to show that each of these energies in equally probable?
Any help would be appreciated.
Thanks!