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.

We write explicitly that depends on . For our free particle, this just means that the energy depends on the momentum. For a photon, , so , and hence . For an non-relativistic electron, , so , and hence .

To cover the general case, lets expand
around the center of the wave packet in k-space.

We anticipate the outcome a bit and name the coefficients.

For the photon, and . For the NR electron, and .

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