The most natural test of this hypothesis is to analyze a collision
event where we assume that the photon carries a momentum 
and
to see whether our expectations are in line with the experimental
results. The collision event which we shall analyze is diagrammed in
Figure 2.
Here we have an incoming photon of frequency 
colliding with a
stationary particle of mass m (in the original experiment, the
particles were electrons). After the collision, the photon transfers
some of its momentum to the particle which recoils with a momentum
p. With its momentum reduced, the photon then emerges with a new,
lower frequency 
.
Note that this effect is difficult to
understand within the classical picture given by Maxwell's equations.
Classically, we would expect the response of the electron to an
incoming electromagnetic wave of frequency 
to be an oscillatory
motion of frequency 
. With the electron oscillating at this
frequency, we would then expect it to radiate electromagnetic energy
in all directions but with the same frequency as its motion, 
.
The shift in frequency we are about to predict is a purely quantum
effect arising from the discrete nature of light.
To analyze the collision, we will use conservation of momentum and energy. To keep our results general, we shall anticipate the possibility of using very high energy photons that may leave the particle recoiling at relativistic speeds. We will use the relativistic form for the energy-momentum relationship of a particle,
Note that when the particle is at rest (p=0) we have 
, the
rest mass energy of the particle. Further, when the electron has
relatively little momentum
which is just the rest mass energy plus the usual non-relativistic form for the kinetic energy.
For energy to be conserved in our collision, the sum of the initial and final energies must be equal,
(Recall that we suppose are particle to be at rest initially.) Because we will soon apply conservation of momentum, it proves convenient to solve this equation for the momentum of the electron,
To apply conservation of momentum, we rearrange the momentum vector diagram as in Figure 2.
We now apply the law of cosines,
To ease combination with Equation (1), we first multiply
both sides by 
,
Now, we may set the right hand sides of (1) and (2) equal,
Dividing both sides by 
and using the wavelength
frequency relationship 
we get our final result,
where we have defined the quantity 
which has the dimensions of length and is
known as the Compton wavelength. For the electron it has the numerical value
.
