These are both localized in momentum about .
Check the normalization of (1).
Check the normalization of (2) using the result for a
definite integral of a Gaussian
So now we take the Fourier Transform of (1) right here.
Fourier Transform of a Gaussian
In both of these cases of (transformed from a normalized localized in momentum space) we see
We have achieved our goal of finding states that represent one free particle. We see that we can have states which are localized both in position space and momentum space. We achieved this by making wave packets which are superpositions of states with definite momentum. The wave packets, while localized, have some width in and in .
Jim Branson 2013-04-22