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

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

. 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:

First consider an electron that is initially confined to a region of a size nm (roughly atomic dimensions) in a gaussian wave packet. For simplicity we will let ; that is, the electron is (on average) at rest. If the electron is free (as we have assumed throughout this treatment) then its wave packet will expand to times its initial size in a time s.
If the same electron is confined much more loosely to a region of a size m, the time required for it to disperse until is times longer: ns.
The same electron initially confined to a 1 mm sized wave packet will take 0.0172 s to disperse to a wave packet 1.414 mm in size; and so on.
A one-gram marble localized to within 0.1 mm will delocalize spontaneously (the physical meaning of dispersion) to 0.1414 mm only after s. (That is, years!)
The centre of a wave packet with a finite k_o moves with time at the group velocity, as expected for the mean position of a particle. It simultaneously broadens just like the (on average) stationary particle; this must be so in order to preserve Galilean invariance, which is still applicable as long as the velocities are nonrelativistic.
This raises the question: What do the ``wiggles'' represent? When the particle is (on average) at rest, its wave packet is just a ``bump'' that spreads out with time; when it is moving, it acquires all these oscillations of phase with a wavelength satisfying de Broglie's formula. Is it really ``there'' at the peaks and ``not there'' at the points where the function crosses the axis? No. Except for the overall ``envelope'' it is just as ``there'' at one point as at another. This is a direct consequence of using the complex exponential form (rather than a cosine) for the travelling wave. Although the plots above show only the real part, there is an imaginary part that is a maximum when the real part is zero and vice versa so that the absolute magnitude is always (except for the overall ``envelope'') the same.
In that case, what is the point of even having these ``wiggles?'' Well, although no experiment can measure the absolute phase of a wavefunction, the relative phase of two probability amplitudes being added together is what causes interference, which is the key to all observable quantum mechanical phenomena.
It is also worth remembering that by adding together two travelling waves propagating in opposite directions it is possible to make a standing wave, whose wavefunction really is a real oscillatory function for which the particle is actually never found at positions where the amplitude is zero.