How Do You Derive the Compton Equation?

[latitude]

Homework Statement

Derive the compton equation.

Homework Equations

$$\lambda$$` - $$\lambda$$ = h/ mc (1 - cos$$\theta$$)
E = hf = hc/$$\lambda$$

The Attempt at a Solution

Okay, I'm sorry this is so long, I'll try and make it as concise as it is possible for a whole blather of random crap to be :]

Conservation of momentum components:
h/$$\lambda$$ = h/$$\lambda$$`(cos$$\theta$$) + Pe(cos$$\psi$$)
0 = h/$$\lambda$$`(sin$$\theta$$) - Pe(sin$$\psi$$)

After some combining, squaring, and the like (getting rid of $$\psi$$):
Pe² = (h/$$\lambda$$)² - (h/$$\lambda$$`)<sup>2</sup>cos<sup>2</sup>$$\theta$$ + (h/$$\lambda$$`)²sin²$$\theta$$ - (h/$$\lambda$$)(h/$$\lambda$$`)cos$$\theta$$


E² = p²c² + E_R²
So
P² = (E² - E_R²)/c²

So I plug that into my momentum (I'm not going to write the righthand side of the equation while i show what I did w/ that)

(E² - E_R²)/c² = ...
((hc/$$\lambda$$)² - (mc²)²)/c² = .
I tried to get rid of the denominator 'c'...
(h/$$\lambda$$)² - m²c² = ...

(m²c²$$\lambda$$)/h = $$\lambda$$/$$\lambda$$` - h/$$\lambda$$`cos$$\theta$$

After some more fiddling I get to this:

$$\lambda$$` = h/m²c² - (h/$$\lambda$$)(h/m²c²)cos$$\theta$$

It's kind of close but not really... I can write out all the steps I made if that is necessary, but I'm kind of hoping I made one nice, simple-to-fix error that is glaringly obvious to the more experienced :)

Thank you :)

 

[tiny-tim]

Hi latitude!

(have a lambda: λ and a theta: θ and a psi: ψ )

Pe² = (h/$$\lambda$$)² - (h/$$\lambda$$`)<sup>2</sup>cos<sup>2</sup>$$\theta$$ + (h/$$\lambda$$`)²sin²$$\theta$$ - (h/$$\lambda$$)(h/$$\lambda$$`)cos$$\theta$$

eugh …

the first - should be a + ​