For harmonic perturbations, we have derived a probability to be in the final state
proportional to the following.
The Delta Function of Energy Conservation
For simplicity of analysis lets consider the characteristics of the function
for values of
(Note that we have divided our function to be investigated by
while for all other values for
approaches zero for large
This is clearly some form of a delta function.
To find out exactly what delta function it is, we need to integrate over
We have made the substitution that
The definite integral over
(consult your table of integrals),
so the result is simple.