General Fourier expansion in plane waves:

where we must remember that is a function of , not just a constant; the dispersion relation determines all the key physical properties of the wave such as phase velocity and group (physical) velocity .

Scalar version:

The picture is a lot simpler if we assume that all waves propagate along the direction, giving the 1-dimensional version with and .

Gaussian distribution of wavenumbers:

has a mean wavenumber and a variance (so that is the standard deviation of k).

The initial wave packet:

At t = 0, we have or where

If we now let so that , we have = Completing the square, , giving where The definite integral has the value (look it up in a table of integrals!) giving or where That is, the rms width of the wave packet about its initial mean of is and the product of the x and k widths obeys the uncertainty relation at t = 0.


The requirement that the particle be somewhere at t = 0 provides the numerical value of A : = = where . Again the definite integral equals , giving or = . We have now fully described .


What happens at later times? Each plane-wave component of has a different k and therefore progapates at a different velocity = = . Thus they all move away from x = 0 at a different rate and become spread out or dispersed [hence the name "dispersion relation" for ] relative to their average position [the centre of the wave packet] at .

The width of the wave packet, , therefore increases with time from its minimum value at t = 0. The time dependence can be calculated with some effort (not shown here); the result is

The normalization constant A will decrease with time (as the spatial extent of the wave packet increases) in order to maintain . Thus the probability of finding the particle within dx of its mean position steadily decreases with time as the wave packet disperses.


It is instructive to estimate the rate of dispersion (how fast the wave packet spreads out) for a few simple cases:

Jess H. Brewer - Last modified: Mon Nov 23 17:33:28 PST 2015