I'm following a derivation of the Klein-Nishina formula from scratch and this is what I have so far:
$P_{e,i} = m_0\gamma_u[1,u]$ and $P_{\gamma,i} = [\frac{\hbar\omega_i}{c^2}, \frac{\hbar\omega_i}{c}n_{\gamma,i}]$,

where $i=1,2$ is before and after and I'm assuming $n_\gamma$ is the number density.

Firstly, four momentum is $[E/c, p]$, so how has $P_{\gamma,i}$ got a $c^2$ and $n_\gamma$ in it?

So, if we accept all that. We can conserve the momenta before and after, square it and reduce it to $$P_{e1}.P_{\gamma1} = P_{e2}.P_{\gamma2}$$

Secondly, multiplying by $P_{\gamma2}$ gives $$ P_{e1}.P_{\gamma2} +P_{\gamma1}.P_{\gamma2} = P_{e2}.P_{\gamma2} + P_{\gamma2}.P_{\gamma2}$$. How does that work?

Then finally, how do I get $$ P_{e1}.P_{\gamma2} = \gamma_1m_e\hbar\omega_2(1-\frac{v_1.n_{\gamma2}}{c})$$ from the definitions on the second line?

So confused!

This post imported from StackExchange Physics at 2014-05-04 11:10 (UCT), posted by SE-user Lucidnonsense