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
group (physical) velocity
The picture is a lot simpler if we assume that all
waves propagate along the
the 1-dimensional version
Gaussian distribution of wavenumbers:
has a mean wavenumber
and a variance
is the standard deviation of k).
The initial wave packet:
At t = 0, we have
If we now let
so that , we have
Completing the square, , giving
The definite integral has the value
(look it up in a table of integrals!) giving
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
at t = 0.
The requirement that the particle be somewhere at t = 0
provides the numerical value of A :
Again the definite integral equals
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"
relative to their average position
[the centre of the wave packet]
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:
- 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
- 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
(That is, years!)
- The centre of a wave packet with a finite ko
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.
Jess H. Brewer -
Last modified: Mon Nov 23 17:33:28 PST 2015