###

Fourier Transform of Gaussian *****

We wish to **Fourier transform the Gaussian wave packet** in (momentum) k-space
to get
in position space.
The Fourier Transform formula is

Now we will transform the integral a few times to get to the standard definite integral of
a Gaussian for which we know the answer.
First,

which does nothing really since
.

Now we want to **complete the square** in the exponent inside the integral.
We plan a term like
so we define

Again
.
Lets write out the planned exponent to see what we are missing.

We need to multiply by
to cancel the extra term in the completed square.

That term can be pulled outside the integral since it doesn't depend on
.

So now we have the standard Gaussian integral which just gives us
.

Lets **check the normalization**.

Given a normalized
, we get a normalized
.
The RMS deviation, or standard deviation of a Gaussian can be read from the distribution.

Squaring
, we get

Reading from either the coefficient or the exponential we see that

For the width in k-space,

Reading from the coefficient of the exponential, we get

We can see that as we vary the width in k-space, the width in x-space varies to keep the product constant.

Translating this into momentum, we get the limit of the **Heisenberg Uncertainty Principle**.

In fact the Uncertainty Principle states that

so the Gaussian wave packets seem to saturate the bound!

Jim Branson
2013-04-22