Show that it is not possible for a photon to be completely absorbed by a free electron

In any collision of electron and photon the collision will be elastic.

∴ Energy and momentum will be conserved.

Energy of photon = .(λ is wavelength of photon)

Momentum of photon (p)=

Rest mass energy of electron =.(is the rest mass of electron)

Energy of electron after collision =(m is relativistic mass of electron)

Where ‘c’ is speed of light.

By use of conservation of energy,

∴ Initial energy is sum of photon energy and electron energy

Initial photon energy = pc, rest mass of photon is zero therefore its rest mass energy is zero.

Initial electron energy = rest mass energy of it =

Final energy = energy of electron as it gains velocity. is its relativistic energy, when electron moves with momentum p.

By applying the conservation of energy, we get

∴ =

Squaring both side and solving.

Solving this we get vanish.

∴ it is not possible for a photon to be completely absorbed by a free electron.