Wolfram Alpha:

```
Compton Effect
--------------
Photons scattered from electrons had longer wavelength than those incident
upon the target.  Gave clear and independent evidence of particle-like behavior
in support of the photoelectric effect.

recoiled electron pe= √(E2 - (mec2)2)/c
^
/
/
/
photon ------->o.....
pi = h/λi       \θ
\
\
v
scattered photon
pf = h/λf

Conservation f energy:
hνi + mec2 = hνf + √(pe2c2 + me2c4)

Conservation of momentum:
_    _    _
pi = pe + pf
_    _    _   _
pe2 = (pi - pf).(pi - pf)
= pi2 + pf2 - 2pipfcosθ

multiply by c2 and substitute E = pc = hν

(pec)2 = (hνi)2 + (hνf)2 - 2h2νiνfcosθ

square energy conservation expression to get

(pec)2 = (hνi)2 + (hνf)2 - 2h2νiνf + 2mec2(hνi - hνf)

-2h2νiνfcosθ = -2h2νiνf + 2mec2(hνi - hνf)

λf - λi = (h/mec)(1 - cosθ)

Compton Wavelength
--------------------

The Compton wavelength of a particle is equivalent to the wavelength
of a photon whose energy is the same as the rest-mass energy of the
particle.

The Compton wavelength, λ, is derived as follows:

E = hf = hc/λ = mc2

∴ λ = h/mc```