Consider an apparatus as in Figure 3.3
consisting of two mirrors a distance L apart and containing a photon
of wavelength 
. The wave function for this
standing wave pattern looks something like
where 
. If we are to measure the momentum of the photon (say
by allowing it to collide with an electron), the de Broglie relations
tells us that we will measure 
.
If we do the actual experiment, however, we will find something like in Figure 3.3
This is because we do not have a pure traveling wave, but a standing wave consisting of waves moving in two directions, +x and -x.
We can see this mathematically by expanding 
,
The state of the photon is now the superposition of two pure states,
one of momentum 
and one of momentum 
, each with
weight equal in magnitude.
In our abstract language we would write the superposition as
so that when we measure p we will get one of the two pure states
in the composition of 
with probability somehow related to
the weights in the composition. In this case, the weights are equal so
we
expect equal probabilities as observed in the experiment.
