We will now use the physical principle of superposition, the
de Broglie identification of 
as a pure state of
momentum 
, and our filter scattering example to give a
physical ``proof'' of the Fourier transform theorem.
The principle of superposition states that our state
in (2) is expressible as the superposition of pure
momentum states. As momentum is continuous we write
where the 
are the relative weights somehow related to
the probability distribution 
when measuring momentum p
or equivalently 
. Writing this in our ``concrete''
language
Now, from (5) we had 
where 
. One natural guess would be that
We may verify this by inserting our guess into (9) and seeing if we
recover 
.
where we've made the change of variables 
.