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
at
.
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