We can consider photons to carry momentum. As in any collision involving particles with non zero mass, collisions involving photons and matter particles will result in the overall conservation of momentum and energy.
A photon with wavelength
can be considered as a particle with mass zero and momentum
where h = Planck's constant. When the photon collides with a stationary electron, it is scattered elastically and its wavelength increases from
to
Conservation of momentum gives
(1)
(2)
Multiplying (1) and (2) by
throughout and using
gives
(3) and
(4)
Rewriting (3) as
(5), squaring and adding (4) and (5) gives
(6)
Conservation of energy gives
Squaring both sides of the last equation gives
(7)
(7)-(6) and simplifying the result gives
Dividing by
and rearranging gives
But
and
so