Why can't a photon transfer all of its energy to an electron?

[darrenhb]

Homework Statement

Pretty straight forward, I just have to show why a photon can't transfer all of its energy to an electron. I understand this in theory but I'm stuck at how to show it.

Homework Equations

1) E_p + m_ec² = E_p' + E_e

Where E_p is the energy of the photon, E_p' is the energy of the scattered photon, and the rest is obvious.

2) E = hf

3) p = p'cos($$\theta$$) + p_ecos($$\phi$$)

4) p'sin($$\theta$$) = p_esin($$\phi$$)

p is the initial momentum of the photon, p' is the final momentum of the photon, p_e is the momentum of the electron after scattering. $$\theta$$ is the angle of the scattered photon and $$\phi$$ is the angle of the scattered electron.

The Attempt at a Solution

I figure I have to use conservation of momentum and energy to show that it's a contradiction. I was going to assume $$\theta$$ and $$\phi$$ were 0, but I'm not sure if I can do that. I've hit a roadblock, I'm not sure how to go about this. A hint in the right direction would be much appreciated!

 

[Maxim Zh]

Using the relativistic dispersion formula:
$$E^2 = m^2 c^4 + p^2 c^2$$
you can prove that the conservation laws for energy and momentum can not be satisfied simultaneously if E_p'=0.

 

[darrenhb]

Thanks! That equation was the missing link I think, I figured it out. :)