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skrat
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Homework Statement
Negative pion with kinetic energy 100 Mev decays to electron and antineutron. What is the kinetic energy of electron , which will move in the same direction as pion did?
Homework Equations
##p^{\mu }=(\frac{E}{c},\vec{p})##
##E=T+mc^2=\sqrt{p^2c^2+m^2c^4}##
The Attempt at a Solution
Guys, I need lots of help here... I have got lots of similar problems in my book here but I can't finish a single one of them - which worries me just a bit. Ok, not true... I am really worried and frustrated and will hit somebody and ... -.- you get it.
At the beginning: ##p_{0}^{\mu }=(\frac{T_\pi +m_\pi c^2}{c},p_\pi ,0 )##,
after pion decays: ##p_{1}^{\mu }=(\frac{T_e +m_e c^2+E}{c},p_e+\frac{E}{c}cos\varphi ,\frac{E}{c}sin\varphi )##
for any direction of antineutron (E is the energy of antineutron). BUT since ##p_{0}^{\mu }=p_{1}^{\mu }## than also ##\frac{E}{c}sin\varphi =0##, therefore ##\varphi = 0##.
This means:
##p_{0}^{\mu }=(\frac{T_\pi +m_\pi c^2}{c},p_\pi ,0 )## and
##p_{1}^{\mu }=(\frac{T_e +m_e c^2+E}{c},p_e+\frac{E}{c},0)##
Now I haven't got a single clue what to do! I tried to equate each component of ##p^\mu ## but that leads me to some horrible and probably never ending calculations.
I also tried to use invariance of scalar product:
##(T_\pi +m_\pi c^2)^2-p_{\pi }^{2}c^2=(T_e+m_ec^2+E)^2-(cp_e+E)^2##
which obviously leads nowhere since I have to get rid of E and this just gives me all sorts of things multiplied by E...
Than I also tried to got to barycentric coordinate system where ##c^2(\Sigma _{i}\vec{p_i})^2=0## for ##\vec{p_i}## after the decay. Than:
##(T_\pi +m_\pi c^2)^2-p_{\pi }^{2}c^2=(T_e+m_ec^2+E)^2##
Which in my opinion also leads me nowhere, since there is no way to get rid of E...
Really guys, I am dealing with this for two days now. I beg you, teach me, help me understand so I can solve the other 25490785545 problems.
Thanks!
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