So far, we have performed our Fourier Transforms at and looked at the result only at . We will now put time back into the wave function and look at the wave packet at later times. We will see that the behavior of photons and non-relativistic electrons is quite different.
Assume we start with our Gaussian (minimum uncertainty) wavepacket
We can do the Fourier Transform to position space, including the time dependence.
To cover the general case, lets expand
around the center of the wave packet in k-space.
Performing the Fourier Transform, we get
We see that the photon will move with the velocity of light and that the wave packet will not disperse, because .
For the NR electron, the wave packet moves with the correct group velocity, , but the wave packet spreads with time. The RMS width is .
A wave packet naturally spreads because it contains waves of different momenta and hence different velocities. Wave packets that are very localized in space spread rapidly.
Jim Branson 2013-04-22