M1L2b.txt

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When, we start out with Maxwell's equations,
we have actually six components of the electromagnetic field.
And we will see in a moment that this is redundant.
What will turn out to be very important to separate
the local fields from the radiation field
is a Fourier transformation.
And so we introduce an expansion into plane waves.
And what then happens, of course,
is that the derivative operator becomes the k vector.
So therefore, when you had Maxwell's equations
with the curl del cos b, it turns into k cos b.
And this is now important because we
have now separate equations for the component
of the electric and magnetic field
which is parallel onto k projected onto k.
And of course, the cross product takes out the component
which is perpendicular to k.

So that's something which is sort of nice that the Fourier
transform allows us to separate the fields
into longitudinal fields where the vector of the Fourier
component is parallel to the k vector
and into transpose component.
The difference between curly and non-curly vectors
is one is a Fourier, the other one
is other spatial components.
I have to actually say the [INAUDIBLE] API
sort of very-- tries to be super accurate in choosing symbols.
They don't want to-- [INAUDIBLE] is really
a master of elegance and perfection,
so they don't want to use the same [INAUDIBLE]
for a spatial component and for the Fourier component.
On the other hand, I have to tell you one thing.
For the next hour, use a little bit your intuition.
If you see e, it means electric field.
Whether it is curly whether it's italic, whether it's boldface,
just don't let those differences clutter your view.
So I would say don't ask me too often
what exactly does it mean now because you have
gone from curly to non-curly.
It is explained in the appendix.
I will tell you everything which is necessary,
but these are of some-- I mean, subtleties
which are more for rigorous [INAUDIBLE]
the mathematics in a more rigorous way.
So we have the spatial Fourier components.
And as I mentioned, what you can do only because of the Fourier
transform, you have now the distinction,
a rigorous separation, between the parallel
and the longitudinal fields.

So of course, you can also now get back
what is the longitudinal field not in Fourier space,
but in position space by just taking
the longitudinal or transverse components of the Fourier
transform and transforming them back.
Why I'm elaborating on that is we
will see in the next few minutes that the transverse field
is the field which propagates.
The longitudinal field is the Coulomb field.
It belongs to the atoms and will not
become a new degree of freedom.
And we see that immediately in the next equations.
When we look now at Maxwell's equations,
we find that the transverse and longitudinal fields decouple.
We have two kinds of equations which are completely decoupled.
The longitudinal electric field is the Coulomb field
associated with a charge entity whereas the transverse fields
are sort of creating themselves.
And this is radiation, how the electric and magnetic fields
create themselves as they propagate.
Now, the fact that the parallel electric field can be expressed
by the charge density or it's expressed
by the momentary position of charges-- and this
means immediately that the longitudinal field is not
an independent variable or an extra degree of freedom.
Canonical, of course, we describe particles
by their position.
And here, you see explicitly that the parallel field, which
we, the longitudinal field, which we defined the way we
did it, that this part of the field
is a momentary electric field, the Coulomb field
associated with the charges.
The next thing, of course, is to define the vector potential
and the scalar potential.
So we can, by using the standard substitution,
we can now express electric and magnetic field
by scalar potential and by a vector potential.
And we continue to have the separation
between longitudinal and transverse fields.
Namely, the transfers electromagnetic fields
depend only on the transverse component
of the vector potential.
And the transpose component is gauge invariant.
In other words, you know there are
different gauges we can consider when we introduce
electromagnetic potentials.
But the choice of gauge does not affect the transverse vector
potential.
Well, now we want to pick our gauge
and we pick the gauge which is most convenient.
This is typically for low energy quantum physics
for describing atoms and radiation-- the Coulomb gauge.
So we have the freedom of speech.
And the Coulomb gauge is written by-- that the divergence
of the vector potential is zero.
If you now kind of thing about Fourier transform,
this means k dot the Fourier components is zero
and that means the vector potential does not have
any longitudinal component.
So with that, we have now reduced
equations for six variables-- three electric field,
three magnetic field components.
By introducing vector and scalar potential, you go to four.
We've eliminated one more with the Coulomb gauge.
So we have three.
This scalar potential is the Coulomb potential.
It's not an extra degree of freedom.
And what is left are the two components, the two field
components, of the transverse vector potential.
And these are now, as we will see, the independent variables
of the radiation field.
So in other words, when we talk about the electromagnetic field
in the remainder of the course, the part
of the electromagnetic field we are interested in
are the fields generated by the transverse vector potential.
Now, what is next is to go into normal modes.
The reason is that we want to identify
each normal mode of the electromagnetic field
as an independent harmonic oscillator.
Now, I'm showing you those equations,
but you sort of have to sit back because now, there will be
alphas, As, lower As, upper As.
They all mean pretty much the same.
In one case, you're normalized.

Maybe something has to be factored out or not.
So the derivation in the appendix of API
is exactly [INAUDIBLE] distinction.
But let me just give you a flyover here,
and that is we are now taking the transverse components
from the electorate magnetic field
and we define new parameters which are the normal modes.
And what we actually want to see in a moment
is that those normal mode variables
are harmonic oscillators.
Those normal modes can be defined by the original field
which are transverse.
But since we have expressed the transverse field
by the transverse vector potential,
they also simply can be explained
by the transverse vector potential.
So to the mind you, we have done a Fourier transform
in the spatial coordinate but not
yet in the temporal coordinate.

OK, so what we obtain now is an equation
of motion for the normal mode variables.

And here, I have reminded you that the transverse vector
potential, how it can be expressed by the normal mode
variables.
So in other words, we do-- yes, it
looks complicated and a little bit messy,
but it's classical physics.
And all we have done is introduced normal modes
in a just more complicated notation
as you may have seen it in 803 or some other course
and for the first time we talked about normal modes
of a pendulum or a chain of sprigs.

So we have identified normally modes by the equation above
and then it's a mathematical identity
that our radiation field, our transverse vector potential,
can simply be expanded into normal modes.
So the equation of motion for the transverse fields
involve, of course, the transverse vector potential.
Here, we have our equations for the transverse fields.
So this is what people want to describe.
And if we express everything in terms of the vector potential,
then we have an equation which involves the transverse vector
potential, but in addition, because it's
a differential equation, the first and second derivative.
So that means that, well, if you have a second order
differential equation, which is the full description
of the electromagnetic field, then
you know that the solution of the second order equation
needs as an initial condition the field itself
in the first derivative.
So therefore, our classical fields
are determined by a perpendicular
and its derivative at the initial time.
So we need A and we need A dot, and they are coupled.
But what we are doing right now is--
and this is the idea behind the normal modes.
The normal modes' coordinates combine A and A dot.
It's the same when you have a harmonic oscillator
and you want to introduce coordinates
for the normal mode.
They are a combination of position and momentum.
And then, they are decoupled.
But position and momentum always couple
or oscillate back and forth.
And the same happens here between A
and A dot and the normal mode coordinates
are the linear combination.
So in other words, the equation of motion, coupled source,
and normal mode means that we have introduced
decoupled normal modes.
And whenever in classical physics we have normal modes,
normal modes means that time dependence
is e to the i omega t.
Then, we have distilled the problem of coupled components
to decoupled harmonic oscillators.
So at this point, each mode acts is an independent harmonic
oscillator.
So what I wanted to show you here
is clearly as a fly over and as an appetizer
to read more in the book, that we
can start with electromagnetic field components.
We have rigorously-- we haven't just
assumed we have now empty space and only [INAUDIBLE] radiation.
We have rigorously separated the electromagnetic field
in what belongs to the atoms and what
belongs to radiation and the trick versus spatial Fourier
component.
At that moment, we had a description
in terms of the transverse vector potential,
and by using purely classical physics,
by combining the vector potential and its derivative,
we found normal modes and they are now completely independent