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M3L14e.txt
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M3L14e.txt
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#
# File: content-mit-8-421-3x-subtitles/M3L14e.txt
#
# Captions for 8.421x module
#
# This file has 156 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
We spend now the rest of today and parts of next Monday
in a microscopic derivation of spontaneous emission,
using field quantization.
But I just want to make you aware that we know already what
it is.
We have a classical derivation of the V coefficient.
And Einstein's treatment gives us the ratio of A and B.
So we know already, at this point,
what the rate of spontaneous emission is.
But it is nice, I think also important for our education,
to obtain it in a microscope way where we really
show how we have to-- you know, some, overall modes and such,
to obtain the expression.
Also, I want to ask you questions.
I want to ask you clicker questions afterwards.
And one clicker question for you is,
what happens to spontaneous emission on one
and two-dimension?
Certain things will change.
And it's much clearer what will change,
if you have a clear understanding how we sum up
all the modes, how all the possible modes contribute
to spontaneous emission.
And of course, in two-dimension and one-dimension,
you have a different density of modes.
So with that motivation, we need a quantized electromagnetic
field where we quantize the field for each mode.
And then, we go back, we do the summation overall modes,
and we have really understood in the most fundamental,
microscopic way how photons and light interact.
OK.
So our next chapter is the quantization
of the radiation field.
OK.
Field quantization.
We discussed the quantization of the electromagnetic field,
really, from first principles, from vector potential,
radiation field, Coulomb gauge, transverse vector potential
in 8422.
So we dedicate one or two classes
to just discuss all of the steps to have
a full quantization of the electromagnetic field
with all the bells and whistles.
So sometimes, when I teach this course, I sort of-- well,
you've heard about field quantization.
I can refer to that, or I can refer you to 8422.
But in the end, I thought, why don't I just give you
a 10-minute derogation, just sort of focusing
on the essential, because this makes this course more
self-contained, more complete.
So I give you now a 10-minute quantization
of the electromagnetic field, pretty much,
going straight to showing you electromagnetic field
is a harmonic oscillator.
And now, let's use the quantum description
of the harmonic oscillator.
And then we have a quantum description
of the electromagnetic field.
So this is not rigorous, but it is logically compete.
So in the discussion of the quantization
of the electromagnetic field, we focus just
on a single mode of the electromagnetic field.
Each mode will be a harmonic oscillator.
And then, we have many harmonic oscillators comprising
the electromagnetic field.
So a similar mode, we assume that we
have plain waves, with a polarization,
with an amplitude.
The electric field is the derivative
of the vector potential.
And the shortest way to show you analogy
with the harmonic oscillator is to remind you
that the total energy, which is actually,
if you are wondering about a factor of 2, the electric
and the magnetic part.
The total energy is quadratic in the amplitude of the vector
potential.
By the way, there is a factor of 1/2.
Because if you have a sinusoidal variation,
you take the time average.
Cosine square average is 1/2.
Well, if the total energy is quadratic in the amplitude,
this immediately allows us to draw analogies
to an harmonic oscillator.
And we can use the vector potential
of the similar mode of the electromagnetic field
to define two quantities, Q and P.
Let me write it in that way.
Omega Q plus iP is related to A in the following way.
And yes, I was just wondering about this.
V is the volume.
We assume everything happens in a finite volume of space.
Where you would say, I have two new quantities, Q and P,
so I need two equations.
And the two equations involve A and A complex conjugate.
So now, we had an expression for the total energy,
in terms of with amplitude of the vector potential.
So now, I can rewrite it.
The amplitude square of the vector potential
is A times A star.
And with that, I get the total energy
to be proportional to Q square plus P square.
And that should remind you-- and everything was set up
to remind you-- that this looks like an harmonic
oscillator, which, where Q is the position variable and P
is the momentum variable.
So now-- I mean, all this is classical.
All this is just clever definitions.
But now, we have to do a leap to quantum physics.
We cannot logically derive it.
We have to make a leap.
And the leap is that we postulate that this is now
a quantum-- we postulate that this should now
be described as a quantum harmonic oscillator.
And this transition is done by simply postulating
that the two quantities we have defined
have fulfilled the canonical commutator for position
and momentum.
So we've started with a vector potential, expressed the energy
as a vector potential.
And now we say, we recognize through those definitions.
that this is an harmonic oscillator with variables Q
and P, which are defined in terms of the vector potential.
So then, you know, if you have the quantized harmonic
oscillator, you immediately introduce
creation and annihilation operators,
which are linear superpositions of Q and P
in the following form.
And A dagger has a minus sign here.
And all the prefactors were cleverly set up in such a way
that the commutator of A and A dagger is 1.
And now, we can do all the substitutions.
We can express P and Q by A and A dagger.
But P and Q were related to the vector potential, A.
And the vector potential, A square, defined the energy.
So now, we have an expression for the energy, which
is no longer involving A, or P and Q.
It involves A and A dagger.
And surprise, surprise.
We find that our total energy, because we have operators
now has become an Hamiltonian, has this well-known result
with the photon number operator, A dagger A plus 1/2.
So this is the quickest way, which
takes us, in a few minutes, to the quantized electromagnetic