How to Find the Energy of a Scattered Photon in a Moving Electron Frame?

[mathman44]

Homework Statement

Consider a photon scatter off an electron heading towards the photon. After the collision the photon is scattered at 90 degrees. Find the energy E of the scattered photon.

The Attempt at a Solution

Boosting into the electron frame the incoming photon energy increases from

$$E$$ to

$$E_{boost}=E*\gamma*(1+\beta)$$. Then I can apply the standard compton equation to find the scattered photon energy in the boosted frame, provided that the scatter angle is also 90 degrees in the boosted frame (is this a legal move?).

$$E=\frac{E_{boost}}{1+\frac{E_{boost}}{m_e*c^2}}$$

So far so good, but how do I transform the energy back into the initial frame? How will the photon energy change as I boost back to the initial frame?

 

[TSny]

Then I can apply the standard compton equation to find the scattered photon energy in the boosted frame, provided that the scatter angle is also 90 degrees in the boosted frame (is this a legal move?).

No, the angle will not be 90^o in the boosted frame. In the boosted frame, the scattered photon will have a component of momentum parallel to the boost direction as well as a perpendicular component.

how do I transform the energy back into the initial frame? How will the photon energy change as I boost back to the initial frame?

The energy and momentum of the photon together make up a 4-vector. So, you can apply the Lorentz transformation equations to this 4-vector. This is also how you can get the scattered angle in the boosted frame.