### 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