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M1L2c.txt
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M1L2c.txt
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#
# File: content-mit-8422-1x-captions/M1L2c.txt
#
# Captions for 8.422x module
#
# This file has 166 caption lines.
#
# Do not add or delete any lines. If there is text missing at the end, please add it to the last line.
#
#----------------------------------------
This approach-- and I wanted to show you
that to you-- involved a lot of notation, you know,
a's, little alphas, normal modes, and such.
Let me just simply show you another much, much shorter path
way how you see that everything looks and smells
like an harmonic oscillator.
And then we do the last step, which is rather straight
forward, we quantize the electromagnetic field.
I could quantize it right now, but we
should take a short break from the appendix in atom photon
interaction and sort of do a more intuitive shortcut
to the same physics.
Any questions at this point?
If you feel it's confusing, remind yourself.
This is just classical physics.
We are just rewriting Maxwell's equations in new variables.
So what we do now is, as I said, let's repeat
some of this derivation by focusing on energy.
So if you write down energy for our systems of atoms, photons,
and coulomb fields, we have particles alpha
with mass m alpha and velocity v alpha.
This is our kinetic energy.
And then we use the well known expression
for the electromagnetic field, which
is the spatial integral over e square plus b square.
And this integral over the electromagnetic field energy
density can actually be nicely separated
into an integral over the longitudinal field, which
we have introduced, and a second integral which
involves the transverse field.
Of course, the magnetic field is only transverse
because the divergence of the magnetic field is 0
and that means the magnetic field does not have
any longitudinal component.
So this part here, as we actually have shown,
is given by simply the coulomb energy.
Well, we have charged density at position a
and a prime interacting with a coulomb interaction.
And eventually, if you want to treat that further,
this coulomb energy should be split into some divergent self
energy, which is the energy of the electron interacting
with itself and people know how to deal
with it, and the interaction energy,
for example, between the proton and the electron,
which is responsible for the atomic structure.
So in other words, we know how to deal with it,
this actually becomes atomic structure.
This is in contrast to this part here
which is the classical energy of the radiation field
which is sort of the transverse part of the energy.
So to describe now radiation, we have
to focus on this transverse part.
We are now expressing the transverse part
by the vector potential and its derivatives.
So we introduce the vector potential as before,
and I don't want to go through the re-derivation,
I just want to show you how the total energy now
appears in terms of the vector potential.
And what we need for that is, of course, the vector
potential and its derivative.
The vector potential depends on polarization,
depends on the Fourier component,
and we call this derivative of the vector potential
the conjugate momentum.
So now focusing on energy, we, of course,
reduce everything, as before, to the vector potential
and its derivative.
So we sum over polarization.
We integrate over all Fourier components.
And the integral involves now the vector potential-- well,
the square of it or in complex notation,
the complex conjugate depends on polarization.
We have c square k square.
This part, of course, comes from the magnetic field,
which depends on the vector potential.
It's a spatial derivative, the curl,
and this gives the k square.
Whereas the part of electromagnetic energy which
is related to the electric field, the electric field
is a temporal derivative of the vector potential,
and therefore, it involves the temporal derivatives which
are now the canonical momentum.
So this equation should really remind you now.
It's an energy equation of the energy
of an harmonic oscillator because the energy is now
a sum over all oscillators, but this
is sort of x square, the amplitude
of the oscillator squared, this is the potential energy.
And here we have the derivative of x, the velocity or momentum
and this should remind you of the kinetic energy
of an harmonic oscillator.
So in other words, this should tell you
that by focusing on the transverse component
of the vector potential doing sort of separating out--
I mean, yeah, it's about lots of summation,
k summation, polarization summation.
We, in the end, find that each such mode
is an harmonic oscillator.
Any questions?
Let me just show you now-- and it
may help you to go through the previous derivation--
let me now introduce what I did before the normal modes
and then show you how the energy looks
like defined in normal modes.
So we have this normal mode variables
which are defined as a superposition of a and a dot.
And I'm not telling you here with all the indices
whether it's a polarization component, the k components.
I just want to tell you the structure the normal mode is
superposition of a dot and a.
And with that, the energy of the electromagnetic field
can be written-- well, we always have
to sum over Fourier components.
We always have to sum over the polarization.
But then, we have something which
is really very, very simple and intuitive.
It's just the square of alpha, with the covert polarization,
and the prefecter is h bar omega over 2.
I mean it looks like quantum mechanics, h bar
omega times a dagger a plus one half,
but this is purely classical.
The alpha, there is no quantization.
There is no operator.
We've simply defined something new, namely alpha,
in terms of the vector potential.
So in that sense, if you now go back to the derivation,
the alphas are nothing else than some elaborate combination
of the transverse electric and magnetic field.
H bar only enters for the constant in the definition
of the normal mode parameter.
So actually, I have introduced h bar
by choosing this parameter wisely in such a way
that it connects with quantum mechanics.
But h bar has been completely-- it's a completely arbitrary
introduction here.
I could have said h bar 2 1.
OK.
So I hope you enjoyed or, at least,
did not dislike this excursion into classical physics.
So we have now two equations for describing the energy
of an harmonic oscillator.
So both of these equations will look
like an harmonic oscillator, but if something
looks like an harmonic oscillator,
it is an harmonic oscillator.
And I think I should always remind you
that, so far, everything has been purely classical.
And also let me write down that h bar enters solely
through the constant in the definition of the normal mode
parameter alpha.
If you want to know more about it,
there is a second reference.
I will actually show you the cover page in a few moments,
but there is a second book by Claude Cohen-Tannoudji
and collaborators.
Not atom photon interaction, but called Photons and Atoms.
It's a whole book on rigorously defining QED.
So a whole book has been written about the subject