Assume we start with our Gaussian (minimum uncertainty) wavepacket
We are not interested in careful normalization here so we will drop constants.
We still need to do the integral as before. Make the substitution giving . Factor out the constant exponential that has no dependence.
We now compare this integral to the one we did earlier (so we can avoid the work of completing the square again).
Dropping the constants, we had
Our new integral is the same with the substitutions , , and . We can then write down the answer
Jim Branson 2013-04-22