1. Fixing the problem with the de Broglie waves
Let’s start playing not with a single wave, but with
many waves. First, take just two waves:
Suppose that the wavelengths of the two waves
do not differ vey much: Therefore,
since

2. Let’s now add the two waves (or “superpose”
them – from Ph212 you surely remember the
terms “superposition” and “to superpose” –
meaning essentially the same as “to add
algebraically”):
Recall the identity:

3. The resulting function forms sort of “groups”

4. Now, let’s switch to a
PDF document titled
“A simple-minded
wave-packet
tutorial”:
The Tutorial

5. Summary of the wavepacket tutorial:
Superposition of many discrete waves – all with the same amplitude
Spectrum of discrete k values:
The result of superposition:
Major groups (“big fishes”)
plus a “school” of small fish.
Same k , more discrete waves:
The major groups shift farther
apart. The small groups do not
disappear, there is more of them
x axis

6. Superposition of many discrete waves with
a Gaussian distribution of amplitudes
Only major groups, the “small
fish” all gone
Same width of the Gaussian
“envelope”, but more waves
x axis
The groups shift
farther apart

7. So, if we add an infinite number of waves with
Gaussian distribution of amplitudes, only a
single group will remain, while the groups which
are its “nearest neighbors” will move away to
What does it mean, in prac-
tice, to be at a point, the x coordinate of which is
It’s the same as “nowhere”.
As follows from the above, a description in terms of a
“wavepacket” solves the localization problem.
“But if we take many waves with different k values, it’s no
longer the same situation, because the de Broglie equati-
ons precisly define the value of k !” – one can argue.
Right – but we are always talking about a narrow k
range, and the value given by the d.B. equations can be
thought of as the as the mean k value in the packet.

8. OK, so we solved the localization problem – but
what about the other one, the velocity problem?
It turns out that the description in terms of a
wavepacket solves this problem as well.
We will show that for the simplest case, with
just two waves – i.e., the one we discussed at
at the very beginning. But now we don’t limit
our considerations to t = 0 :

9. Let’s focus at the “modulation function” (i.e.,
cosine) – it is the one which determines the
motion of the “group”. We can rewrite it as:

10. What does it mean? It simply means that the group
moves with a velocity equal to:

11. Well, but what if have not just two, but many
constituent waves – in particular, when there
is an  number, or a “continuum” of k values?
Answer: the formula from the preceding
slide does not change very much:
Let’s accept that without proof – the proof
is not straightforward, it takes time to go
through it step-by-step. But if you insist you
want to see it, here it is, p. 65: Group velocity
The group velocity of a packet of de B.
waves is consistent with the classical par-
ticle velocity – it will be a homework task for
you to check it.