A photon strikes a stationary electron and scatters exactly backwards along the line of its incoming path. Treating the system as in Newtonian mechanics, apply the conservation of energy and momentum to show that the increase in wavelength of the photon is 2h/mc, where m is the rest mass of the electron. [Hint: Use E^(2)=p^(2)c^(2)+m^(2)c^(4) for the energy.]

Question

anonymous · Answer

You can solve this by using the conservation laws (as stated). This means you just equate the quantities before and after the scatter.

For energy:$$E_{\gamma i}+E_{ei}=E_{\gamma f}+E_{ef}$$For momentum:$$p_{\gamma i}+p_{ei}=p_{\gamma f}+p_{ef}$$
You're given what to use for energy:$$E=\sqrt{(pc)^{2}+(mc^{2})^{2}}$$
Note: a photon has no mass (that term will drop out) and an electron will not have any momentum (that term drops out).

Your goal is to obtain an expression for energy and for momentum, then solve the system of equations.

anonymous · Answer

I've done that and I get $$\lambda \prime-\lambda=\frac{ h }{ m_{e}c }(1-\cos \theta)$$, which I then plug 90 degrees into cos which brings that term to zero. So I'm left with$$\lambda \prime-\lambda=\frac{ h }{ m _{e}c }$$ not $$\frac{ 2h }{ m _{e}c }$$.

anonymous · Answer

I'm so silly, just realized the incident angle is 180 degrees not 90. It all works out now! Thanks!

anonymous · Answer

Haha, yes, it's what supplies the negative sign from the cosine to allow the terms to add. Nice work!