Change in wavelength, photon hits a free electron.

[AbigailM]

Homework Statement

A photon with initial momentum p collides with a free electron having
a mass m that is initially at rest. If the electron and photon recoil in opposite
directions, what will be the change in the photon’ wavelength? (Hint: use
relativistic forms for energy and momentum.)

Homework Equations

Conservation of Energy:
$hf_{i}+m_{e}c^{2}=hf_{f}+\sqrt{p_{e}^{2}c^{2}+m_{e}^{2}c^{4}}$

Conservation of Momentum:
$\boldsymbol{p_{i}}=\boldsymbol{p_{f}}+\boldsymbol{p_{e}}$

The Attempt at a Solution

I won't go through the whole derivation as it's quite a bit of latex but:
If you square both equations above and introduce hf into the conservation of momentum equation, you can equate the two equations and rearrange. This will give you the compton scattering equation:

$\lambda_{2}-\lambda_{1}=\frac{h}{m_{e}c}(1-cos\theta)$

If the recoiling electron and photon are to be in opposite directions this is an angle of 180°.
Plugging this into the compton scattering equation gives:

$Δ\lambda=\frac{2h}{m_{e}c}$

,which is the change in wavelength.

Does this look ok? As always everyone, thanks for the help!

 

[TSny]

Looks ok. But it seems to me that the intent of the problem was to set up the conservation equations for the specific case of 180 degree recoil. That makes the messy algebra simpler than the general case.

However, if you can go through the algebra for the general case and then substitute the specific value of theta at the end, then that should certainly count as a solution!