Understanding the Compton Effect: Derivation of the Equation

[The_ArtofScience]

The Compton effect does make sense conceptually but I'm having trouble with the actual derivation.

The equation starts out from the principles of conservation of energy and mass: E_{\gamma}+E_{e}=E_{\gamma'}+E_{e'}
which strangely proceeds into the form: \ E-E'}+m=\sqrt{p^2+m^2}} The \sqrt{p^2+m^2}} actually expands into (E-E'cos\theta)^2{}+m^2 when the RHS is squared

So my question is where does the \sqrt{p^2+m^2}} come from and why does it expand to (E-E'cos\theta)^2{}+m^2?

 

[jtbell]

where does the \sqrt{p^2+m^2}} come from

The relativistic relationship between (rest) mass, momentum and energy is E^2 = (pc)^2 + (mc^2)^2. Your source must be using units that make c = 1. I assume this refers to the energy of the electron after the interaction (because the electron is usually assumed to be stationary before the interaction) so it really should read

E_{\gamma} - E^{\prime}_{\gamma} + m = \sqrt {(p^{\prime}_e)^2 + m^2}

and why does it expand to (E-E'cos\theta)^2{}+m^2?

I assume the E and the E' refer to the outgoing photon (gamma). I'm sorry, different textbooks use different routes to derive the Compton-scattering equation and I'm too tired to try to guess which route your source uses. Does your source give any more details?

 

[jtbell]

Looking at your second question again, that step probably comes from conservation of momentum. Usually in this derivation one let's the direction of the incoming photon be along the x-axis. Then the outgoing electron and photon both have x- and y-components of momentum. The x- and y-components are each conserved.

Also, for a photon, E = pc which in your units (with c = 1) becomes E = p. So that E^{\prime} \cos \theta is probably the x-component of the outgoing photon momentum: p_x^{\prime} = p^{\prime} \cos \theta = E^{\prime} \cos \theta.

 

[The_ArtofScience]

Thank you jtbell for clearing that up. I'm still curious where that radical came from, but for now I'm very satisfied with the explanations. =D