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1effect
1effect is offline
#55
Mar9-08, 06:22 PM
P: 321
Quote Quote by DrGreg View Post
Note that the reason why it makes sense to consider [tex]\Sigma E[/tex] and [tex]\Sigma \textbf{p}[/tex] is because of conservation of energy and momentum. We want to replace our multi-particle system with a single particle which behaves like the system, as far as possible.

For each particle, the four dimensional vector [tex]\left(E, \textbf{p}c \right) [/tex] is a "4-vector", which means that it obeys the Lorentz transform

[tex] E' = \gamma_u \left(E - u p_x \right) [/tex]
[tex] p_x' = \gamma_u \left(p_x - \frac{u E}{c^2} \right) [/tex]
[tex] p_y' = p_y [/tex]
[tex] p_z' = p_z [/tex]
Yes, you were right all along about [tex] p_y' = p_y [/tex]
[tex] p_z' = p_z [/tex], I had to do the calculations myself, it wasn't obvious.
This leads to:

[tex]E'^2-(p'_xc)^2=E^2-(p_xc)^2[/tex]
and ultimately to:

[tex]E'^2-(\vec{p'}c)^2=E^2-(\vec{p}c)^2[/tex]


Now I am very happy.