Question

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

Answer 1

A photon is a quantum of light. Light travels in packets of energy known as photons. Photons travel at the speed of light. When one body gets completely absorbed by another body after a collision it is said to be a perfectly inelastic collision.

Complete step by step answer:
In a perfectly inelastic collision, one body gets stuck or absorbed by the other body and then they move together. In a perfectly inelastic collision, a fraction of their kinetic energies gets lost in the process. In this case, for a photon to be completely absorbed by a free electron, the loss in kinetic energy needs to be zero, that is
$\Delta KE = 0$
$K{E_f} - K{E_i} = 0$
Where $K{E_f} =$Final Kinetic Energy
$K{E_i} =$Initial Kinetic Energy
$K{E_i} = K{E_f}$............[Conservation of energy]
Now, energy of photon before collision $= pc$, where $p =$momentum of photon and $c =$speed of light. And energy of electron before collision $= {m_e}{c^2}$, where ${m_e} =$mass of electron.
The energy of the whole system after collision $= \sqrt {{p^2}{c^2} + {m_e}^2{c^4}}$.
So, $K{E_i} = K{E_f}$
$\Rightarrow pc + {m_e}{c^2} = \sqrt {{p^2}{c^2} + {m_e}^2{c^4}}$
squaring both sides we get;
$\Rightarrow {\left( {pc + {m_e}{c^2}} \right)^2} = {p^2}{c^2} + {m_e}^2{c^4}$
simplifying further;
$\Rightarrow {p^2}{c^2} + {m_e}^2{c^4} + 2\left( {pc} \right)\left( {{m_e}{c^2}} \right) = {p^2}{c^2} + {m_e}^2{c^4}$
thus we get;
$\Rightarrow 2\left( {pc} \right)\left( {{m_e}{c^2}} \right) = 0$
For this equation to be true, either $p$ or ${m_e}$ have to zero, which is not at all possible.
And therefore it is not possible for a free electron to completely absorb a photon.

Note: Light was considered to be made up of particles because of Newton’s corpuscular theory. He said that light is made up of infinitesimally small particles which travel at the speed of light and their collisions with different surfaces produce different effects. It was not until Huygen’s wave theory, that Newton's theory was disapproved and it was accepted that light travels as a wave. But still there were phenomena that light exhibited which could not be explained completely by either one of the theories. Then came Einstein’s theory, the theory of dual nature of light. It said that any particle with mass can exist as both a particle and a wave at different speeds, but it is never completely either one of the both.