Show that an isolated electron cannot absorb a photon completely.

Promise a salute

The proof for a photon cannot transfer all its energy to an isolated electron is a follows

By Conservation of Energy

hν  +  m₀C² = mC²                                ...(1)

Where m₀ = Rest mass of an electron

m = Relativistic mass of electron after complete photon absorption

C = Speed of light in vacuum

m = γm₀

Here γ² = 1/ (1 - β²) or γ = 1/ (1 - β²)^1/2

and  β = V/C

Here V = velocity of electron after complete absorption of photon

Equation (1) is written as

hν = m₀C² (γ – 1)                     ...(2)

By conservation of momentum

hν/C = mV = γm₀ βC

or hν = γm₀ βC²                       ...(3)

Equating (2) & (3) we obtain

m₀C² (γ – 1) = γm₀ βC²

or (γ – 1)= γβ

or  1/ (1 - β²)^1/2  - 1 = β/(1 - β²)^1/2

Solving which we obtain

β(1-β) = 0

Hence either  = 0, and the electron is at rest after the interaction which is not possible as conservation of momentum will not hold good otherwise.

or β = 1 or V = C, i.e. the electron is moving with the speed of light. This is again impossible.

Consequently simultaneous conservation of energy & momentum requires the presence of a third body (namely, the nucleus).