Photons Have Momentum — Question 8

Question 8

A free electron cannot absorb a lone photon while keeping its rest mass

A free electron initially at rest attempts to absorb a photon of positive energy E completely. No other particle, field or body participates. The final state is assumed to be one electron with unchanged rest mass $m_e$.

1. Use energy and momentum conservation to write the proposed final four-momentum.

2. Check its invariant mass and decide whether the process is possible.

Question 8 - Solution

A free electron cannot absorb a lone photon while keeping its rest mass

(a) Method and calculation

$$E_f=m_ec^2+E,\qquad p_fc=E.$$$$E_f^2-p_f^2c^2=(m_ec^2+E)^2-E^2.$$

(b) Interpretation and further calculation

$$\boxed{E_f^2-p_f^2c^2=m_e^2c^4+2m_ec^2E>m_e^2c^4}.$$$$\boxed{\text{No solution for }E>0\text{ with an unchanged electron mass}}.$$

Physical check

Photoelectric absorption involves a material system that can share momentum and change its internal energy; it is not absorption by an isolated free electron.