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 + ​