Photons Have Momentum — Question 9

Question 9

Two Compton deflections can restore direction without restoring energy

A photon initially traveling along +x has wavelength $\lambda_C$. It scatters by $90^\circ$ from one resting free electron into +y, then by $90^\circ$ from another resting free electron back into +x. For scattering from an initially resting free electron, use $E\prime=E/[1+(E/m_ec^2)(1-\cos\theta)]$, $\lambda\prime-\lambda=\lambda_C(1-\cos\theta)$, and $\lambda_C=h/(m_ec)$.

1. Find the final wavelength and photon energy.

2. Find the kinetic energy delivered to each electron, in units of $m_ec^2$.

Question 9 - Solution

Two Compton deflections can restore direction without restoring energy

(a) Method and calculation

$$\lambda_1=\lambda_C+\lambda_C=2\lambda_C,\quad\boxed{\lambda_2=3\lambda_C}.$$$$(E_0,E_1,E_2)=m_ec^2(1,1/2,1/3).$$

(b) Interpretation and further calculation

$$\boxed{K_1=E_0-E_1=\tfrac12m_ec^2},\qquad\boxed{K_2=E_1-E_2=\tfrac16m_ec^2}.$$$$\boxed{K_1+K_2=\tfrac23m_ec^2=E_0-E_2}.$$

Physical check

Net photon direction is not enough to identify the number of interactions. Wavelength shifts add across distinct resting targets.