quantumworld said:
Pat,
I think I lost u :(
I understand that the vector sum is less than the sum of the magnitudes, but doesn't mean that they are opposite in direction, it just means that there is an angle between them, but not necessarily opposite in direction.
You are right, for a general case. But the question explicitly stated that in th pion rest frame, the two photons were emitted along the axis of the pion's momentum (observed in th elab frame). Therefore, in the lab frame, the photons could either move along + {\hat k} or along - {\hat k}. That's what I used to say that they had to move in opposite directions.
And also, I thought that all photons will travel with the same velocity = c = speed of light, thus my question is how u got different energies of each one of those photons? energy = gamma*m*c^2, if these photons are traveling at the speed of light, means they have the same gamma, thus the same energy...I think I am missing something here :(
thank u
First of all, I made a typo in my last line. Sorry about that. I should have written E_1 \approx 1.5 m c^2 and E_2 \approx 0.16 m c^2 i.e. there should be no gamma factors.
But again, maybe my notation confused you. I should have written things more explicitly, that is
E_1 \approx 1.5 m_{\pi} c^2
E_1 \approx 0.16 m_{\pi} c^2
i.e. my mass in these equations is the value associated to the pion. So these are not meant tp be general formula for energies of photons, they are special cases for this specific problem. Plugging in values, you would get E_1 \approx 203 MeV and E_2 \approx 22 MeV. So the sum is 225 MeV, which is about \gamma_{\pi} m_{\pi} c^2 and the difference is 181 MeV which is about 0.8 \gamma_{\pi} m_{\pi} c^2.
So I apologize if my notation was the source of confusion. All I meant to say is that we can (of course) express the energies of the photons in terms of the energy of the pion, with the formula given above.
*But* part of your post may indicate a deeper misunderstanding. You
wrote
is how u got different energies of each one of those photons? energy = gamma*m*c^2, if these photons are traveling at the speed of light, means they have the same gamma, thus the same energy...I think I am missing something here :(
thank u
Be very careful! Of course photons may have different energies! The key point is that of course, one cannot use gamma m c^2 for a photon if one means the "mass of the photon times the gamma of the photon". That's not defined. One can may only use E = c p where p is the magnitude of the three-momentum. All photons travel indeed at the speed of light, but they may have different energies and momenta.
Again, maybe it's my notation which confused you. I was writing the energies of the photons in terms of the pion energy!
Hope this clarifies things.
Pat
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I thought they are the same, because we eat pies at pi day 
