GAUSSIAN WAVE PACKETS


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.

Normalization:

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 .

Dispersion:

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.

Examples:

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