Deriving the Compton Effect Equation: A System of Equations Approach

[eje5758]

Homework Statement

I need to derive the equation related to the Compton Effect from the equations for momentum and energy conservation.

Homework Equations

(1) Compton Effect: λ' -λ = h/m_e(1-cosθ)
(2) Conservation of Momentum (x-direction): h/λ= (h/λ')cosθ+γ_um_eucosΦ
(3) Conservation of Momentum (y-direction): 0= (h/λ')sinθ+γ_um_eusinΦ
(4) Conservation of Energy: h(c/λ)+m_ec² = h(c/λ')+γ_um_ec²

Where,
λ=initial wavelength of a photon
λ'= final wavelength of a photon
θ= the angle in which an electron scatters
Φ= the angle in which the photon scatters
u= speed at which electron scatters
c= speed of light in a vacuum

The Attempt at a Solution

This is a system of equations problem. So I approached the problem by eliminating squaring equations (2) and (3) and eventually eliminating Φ. This gave me [(h/λ-hcosθ/λ')²+ (h²sin²θ)/(λ')²]/γ_um²u²=1

Things started to get realllly messy from here. I successfully eliminated u, but by that time, the algebra was too far gone to get back to the Compton Effect equation. Any help would be appreciated.

 

[gleem]

The derivation should not be too messy if you kept track of all the constant factors correctly. you might help yourself by for example combining quantities as h/m₀cλ into one symbol say α. Just use care in writing down each expression and persist. Leave the cosines alone and rely on identity relationships to simplify the expressions.