###

The Delta Function of Energy Conservation

For harmonic perturbations, we have derived a probability to be in the final state
proportional to the following.

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
.
For
,
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
just gives
(consult your table of integrals),
so the result is simple.

Q.E.D.

Jim Branson
2013-04-22