Tip:
Highlight text to annotate it
X
[MUSIC PLAYING]
For people who have to work in detail with waves, such as
people who work in optics, the small dispersion, for example,
that we have in glass, a very small change in refractive
index with frequency or wavelength, does give a
significant effect in long fibers.
And if we go to interesting situations that physicists
often look at, near atomic absorption lines, for example,
there's some very large changes in refractive index
with frequency near there.
And they can give all sorts of interesting scientific
effects, like very slow propagation
of light, for example.
And we often work technologically with wave
guides in optical fibers or in other kinds of integrated
optical circuits.
And in those different spatial shapes of waves, sometimes
called modes, do propagate at different velocities.
That could be called a kind of dispersion
from geometry or structure.
And any structure whose physical properties such as
refractive index are changing on a scale comparable with a
wavelength, they're going to show very strong dispersive
affects as well.
So in research, there's quite a lot of work in looking at
structures such as photonic crystals, where we pattern
these structures on a sub-wavelength scales, and we
get very strong dispersive effects.
And also, there have been layered optical structures in
existence for a long time in special kinds of so-called
dielectric stack structures that show quite strong and
controllable dispersion.
But in general, just ordinary propagation of light that we
see every day is not really showing very strong group
velocity effects.
But we can see them if we look for them.
Up till now, we've been looking at
waves rather generally.
And we've been noting that we can certainly find this
phenomenon of the group velocity being different from
the phase velocity.
But in perhaps most situations with classical waves, this is
typically a small effect or one only important under
relatively extreme conditions.
But with quantum mechanical waves, the
situation is quite different.
In fact, for the quantum mechanical waves associated
with something like an electron, the phase velocity
and the group velocity are almost never the same.
So group velocity is a very important
concept for quantum mechanics.
Without understanding it, we cannot even begin to
understand the movement of quantum mechanical particles
like an electron.
Let's see why this is the case.
For a free electron, the frequency omega is not
proportional to the wavevector magnitude k.
For a free electron, that is one for which the potential V
is essentially constant, basically 0 everywhere, then
we know what our Schroedinger equation would look like.
It becomes very simple.
And we know the solutions to this equation.
They are basically e to the plus or minus ikz if we're
looking at the spatial part.
The key point is that the energy here is
proportional to k squared.
And that means that if we look at the frequency associated
with that energy, which would be the energy divided by h
bar, we see that frequency is proportional to k squared and
not proportional to k.
So we can calculate now the free electron group velocity
from this expression.
Remember, the group velocity is d omega by dk.
So we can differentiate this expression
here, d by dk of this.
And we could also write it out as 1 over h bar dE by dk.
But this expression here differentiated with respect to
k, well, we're going to get 2k from the differentiation of
the k squared.
So the net result of that differentiation is
h bar k over m.
Rather trivially, that's the square root of h bar squared k
squared over m squared.
And we can rearrange that a little bit.
We can put a 2 in the top line and the bottom line and
separate out one of the m's.
This h bar squared k squared over 2m, of course, is just
the energy, E.
So if we take this group velocity, which we've just
calculated as the square root of 2E over m, it does give us
that the energy is 1/2m times the group velocity squared.
And that does correspond with our classical ideas of
velocity and kinetic energy.
This suggests to us that it is meaningful to think of the
electron as moving at the group velocity.
If we look at the phase velocity, however, the phase
velocity is simply the ratio omega over k.
It does not give us that 1/2mv squared kind of relation.
With energy equivalent to h bar omega as before, and
that's h bar squared k squared over 2m for our free electron,
then omega over k, which is our phase velocity, is
har bar k over 2m.
And if we square that up, this all becomes 1 over 2m h bar
squared k squared over 2m.
1 over 2m times E is what we get as the result.
And that would give us the energy would be twice the mass
times the square of the phase velocity.
It's not the 1/2mv squared if we take v as
being the phase velocity.
So the electron simply does not move at the phase velocity
of the wave.
It's not right.
The electron moves at the group velocity.
[MUSIC PLAYING]