Compton scattering when E mc^2

[Upallnight]

Homework Statement

Consider Compton scattering of a photon by a moving electron. Before the collision the photon has a wavelength λ and is moving in the positive x direction. The electron is moving in the negative x direction with a total energy E ( including rest energy mc^2). The photon and electron collide head on. After the collision both are moving in the negative x direction.
Derive an expression for the wavelength λ' of the scattered photon. Show that if E>>mc^2, where m is the rest mass of the electron, your result reduces to
λ'=(hc\λ)(1+m^2c^4/4hcE)

Homework Equations

λ'-λ=h/mc(1-cosθ)

The Attempt at a Solution

I've tried many things to manipulate the Compton equation and conservation of energy but I can't seem to get anywhere.

 

[TSny]

Hello, Upallnight.

Show that if E>>mc^2, where m is the rest mass of the electron, your result reduces to
λ'=(hc\λ)(1+m^2c^4/4hcE)

That doesn't look right dimensionally. Is there a typo here?

Relevant equations

λ'-λ=h/mc(1-cosθ)

This Compton formula assumes that the electron is initially at rest. So, it doesn't apply to your situation (unless you want to first Lorentz transform to the rest frame of the electron, apply the Compton formula, and then transform back to the original frame.)

Or, you could just start from scratch and set up conservation of momentum and energy for the collision and do the messy algebra.