How Do You Derive the Compton Effect Equation from Conservation Laws?

[gotojason]

oK SO... I have been given the task of deriving the Compton effect equ. ( $$\lambda\prime-\lambda=\frac{h}{m_ec}(1-cos\theta)$$)
exactly from the 3 following:
1) $$\frac{h}{\lambda}=\frac{h}{\lambda\prime}cos\theta +\gamma m u cos\phi$$

2) $$0=\frac{h}{\lambda\prime}sin\theta-\gamma mu sin\phi$$

3) $$\frac{hc}{\lambda}=\frac{hc}{\lambda\prime}+(\gamma-1)mc^2$$

I found the link
http://en.wikipedia.org/wiki/Compton_scattering
however I am having a hard time going from my equations to momentum and energy conservation
any ideas where to go ... I have been at it for 4+hrs and its driving me nuts
Thanks

 

[Astronuc]

To get rid of term of $$\phi$$, try

$$\frac{h}{\lambda}\,-\,\frac{h}{\lambda'}cos\theta\,=\,\gamma m u cos\phi$$


$$\frac{h}{\lambda'}sin\theta\,=\,\gamma mu sin\phi$$


Square both equations and add.

Now, IIRC, $\beta$ = u/c and $\gamma$ = $$\frac{1}{\sqrt{1-\beta^2}}$$

 

[Weimin]

1 and 2 are from momentum conservation and 3 from energy conservation.

First you need to eliminate Phi from the two first equations using the indentity cos^2(phi)+sin^2(phi)=1. Yes, you need to square up these equations after moving all other terms to one side except one with Phi.