The **Dirac delta function is zero everywhere except at the point where its argument is zero**.
At that point, it is just the right kind of infinity so that

This is the definition of the delta function. It picks of the value of the function at the point where the argument of the delta function vanishes. A simple extension of the definition gives.

The transformation of an integral allows us to compute

the effect of the argument being a function.

If we make a wave packet in p-space using the delta function,
and we transform to position space,

we just get the state of definite .

This is a state of definite momentum written in momentum space.

Its Fourier transform is

This is a state of definite position written in position space.

Jim Branson 2013-04-22