1. The problem statement, all variables and given/known data
Hi everyone, I'm having a bit of trouble with solving this problem:
A ∏0 meson with rest mass m∏ has a kinetic energy K. It decays in flight into two photons whose paths are along the direction of motion of the meson. Find the energies of the two photons.
2. Relevant equations
E2=p2c2+m2c4,
which for a photon reduces to
E=pc
3. The attempt at a solution
I am using 4-vectors to solve this problem.
Before the decay, the pion has the four momenta P∏=(E/c,p,0,0). After we have two photons with four momenta P1=(E1/c,p1,0,0) and P2=(E2/c,p2,0,0).
By conservation of momentum and energy, E1+E2=E and p1+p2=p.
I would like to take the following approach - equate the four momenta and square both sides, using the fact that the quantity E2/c2-p2 is invariant.
P∏=P1+P2. Squaring and using the relevant equation above gives m∏2c2=E1E2/c2-p1p2.
But Using the fact that E1=p1c and E2=p2c reduces the right hand side to zero, which is very wrong... I believe my problem may be with the signs of the momenta - from the centre of mass frame it is clear the photons have to travel in opposite directions. But I don't expect that I would have to account for this in my working - as for example when solving collision problems in classical mechanics using conservation of momentum, I would put all my unknown velocities in as symbols, and the maths would simply give me their direction (by the sign). So can anybody help please, thankyou :)
Hi everyone, I'm having a bit of trouble with solving this problem:
A ∏0 meson with rest mass m∏ has a kinetic energy K. It decays in flight into two photons whose paths are along the direction of motion of the meson. Find the energies of the two photons.
2. Relevant equations
E2=p2c2+m2c4,
which for a photon reduces to
E=pc
3. The attempt at a solution
I am using 4-vectors to solve this problem.
Before the decay, the pion has the four momenta P∏=(E/c,p,0,0). After we have two photons with four momenta P1=(E1/c,p1,0,0) and P2=(E2/c,p2,0,0).
By conservation of momentum and energy, E1+E2=E and p1+p2=p.
I would like to take the following approach - equate the four momenta and square both sides, using the fact that the quantity E2/c2-p2 is invariant.
P∏=P1+P2. Squaring and using the relevant equation above gives m∏2c2=E1E2/c2-p1p2.
But Using the fact that E1=p1c and E2=p2c reduces the right hand side to zero, which is very wrong... I believe my problem may be with the signs of the momenta - from the centre of mass frame it is clear the photons have to travel in opposite directions. But I don't expect that I would have to account for this in my working - as for example when solving collision problems in classical mechanics using conservation of momentum, I would put all my unknown velocities in as symbols, and the maths would simply give me their direction (by the sign). So can anybody help please, thankyou :)
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