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PROFESSOR: Let's turn to
our next subject then.

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So this is still time
dependent perturbation theory.

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We will be doing time dependent
perturbation theory a little

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longer, all of today's lecture
and a little of next time's

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lecture.

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So we're going to be talking
now about light and atoms.

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So this is a classic
application of what time

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dependent perturbation theory
does, the problem of radiation

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interacting with atoms is more
and more detailed applications

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and their ionization.

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So light and atoms,
and we think one

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of the situations we
will be considering

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is a collection of atoms and
they're interacting with light.

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But the way they
interact with light

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is maybe not as controlled as
we had in hydrogen ionization,

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in which you send in a wave.

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But rather, it's
interacting with light

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in thermal equilibrium.

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So there's black body radiation.

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There's all these photons,
unpolarized, in all directions,

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coming at the atoms in all ways.

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And they can produce
transitions in the atoms.

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And for that purpose, we will
consider a collection of atoms.

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And we will assume that
relevant to our situation,

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we have two energy levels.

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This state will be called b.

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Still we'll call a, Ea and Eb.

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So this is interacting
with light or with photons

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at a temperature T.

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So a little bit of
statistical mechanics today.

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In order to discover
what Einstein did,

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basically, we're going to go
through Einstein's argument

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of balance of radiation
rates and emission rates

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and things like that.

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And we need to use some
statistical physics for that.

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Einstein was very good at
using statistical physics.

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And the frequency associated
with this transition

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is Eb minus Ea over h-bar.

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And we'll take it to
be greater than 0.

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The state b will be considered
to be higher than the state a.

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So the question we ask is,
what happens to these atoms

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when light shines?

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And in particular, suppose
the light contains photons

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with a frequency omega b-a.

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That's the frequency
associated with the transition.

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So what will happen?

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And basically, a couple
of things can happen.

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These are things we
already know from time

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dependent perturbation theory.

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And surprisingly,
as we will see,

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Einstein, when he went
through his argument

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that we're going
to go through soon,

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came with a different
angle and saw things

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a little differently.

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But let's follow
up the direction

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that is based on the evidence
we've presented so far.

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So there will be two
possibilities, two cases.

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In the first case,
the atom is initially

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with its electron on
level a, electron in a.

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So what happens?

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Well, we've coupled, a system
with two levels, two harmonic

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perturbations.

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And we saw that sometimes the
particle or the system goes up

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and sometimes goes down.

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And now we're going to see
the source a little bit more

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realistically.

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Before it was a perturbation.

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Now we're going to
identify the perturbation.

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The electron is in state a.

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And here comes photons.

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And what's going to
happen is a process

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of absorption of the photon.

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And the electron
goes to state b.

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So we'll draw it here.

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So this is before.

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And here is after.

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So before, you had this
thing and the electron

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was on state a.

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There was a photon coming
here with energy h omega b-a.

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And then after, the
electron has been pushed up.

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And there's no photon anymore.

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The photon was absorbed.

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That's physically
what we would expect.

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The other possibility is
associated to the process

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of stimulated emission.

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This is absorption.

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And here we have
electron in b initially.

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And we have stimulated emission.

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And in this case of
the electron goes to a.

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And another photon is released.

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It's a very surprising
thing if you think about it.

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And here, this is the
absorption process.

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Now this stimulated
emission process,

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you have the electron
originally in b.

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And here comes the photon.

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And the end result is an
electron in a and two photons.

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We notice that, when
we're doing perturbation

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theory between the
two energy levels

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and we have the
harmonic perturbation,

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we even notice the probability
to go from the bottom

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to the top or going from
the top to the bottom

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was the same in
perturbation theory.

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And you would have
said, look, I understand

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that when you have an
electron in the bottom level,

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you need some energy from
the source to kick it up.

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But why do you need this
source to make it go down?

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It's just there.

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It should go down by itself.

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Well, you still need to couple
the system to something.

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Because if you are in
an energy eigenstate,

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you stay there forever.

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The reason it goes
down is because it

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couples to something.

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And somehow, the coupling
through the electromagnetic

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field helps this go down.

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That's why it's called
stimulated emission,

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because it's stimulated
by the radiation itself.

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And so this doesn't seem
to play too much of a role,

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but actually does, in
which just by the presence

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of a photon at the
right energy difference,

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it stimulates this to go down.

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And now you have two photons.

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So this is, of
course, something that

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is used in technology a lot.

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This is the principle of lasers.

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And therefore, this discussion
of stimulated emission,

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which was a lot due to Einstein,
is the idea of a laser.

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So laser is Light Amplification
by Stimulated Emission

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of Radiation.

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So what do you need for a laser?

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You need something that is
kind of a little strange

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to begin with.

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Because from the viewpoint
of statistical physics,

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if you have a two-level
system, particles

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will tend to be in
the lower level.

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So why would they go here?

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Well, they won't go
there by themselves.

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That's the term called
population inversion.

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You need to get
the electrons that

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are in the bottom to
go up to the next level

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and put them there.

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That's a technical
difficulty, of course.

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And it's usually solved
by optical pumping.

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We are not going to be doing
engineering here today.

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But there is a third level
here in a typical laser.

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And you can send
the electrons, that

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are sitting originally
here, with light

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again into the third level.

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And then it turns out
that that third level,

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if you have the right atom, has
a transition to this level that

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is rather quick.

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So the electrons fall
here, fall very quickly.

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And the lifetime of this
state here is very long.

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So they pretty much stay there.

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So this is the principle.

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This extra level is sometimes
called the pump level.

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And so this is done by optical
pumping, so optical pumping

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to create population inversion.

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That you get more states
in the upper state

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than you would get in
thermodynamic equilibrium.

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And then, you now have,
say, the levels here--

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I'm going to draw them this way.

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It makes it a little easier.

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I think of one electron
here, maybe two here,

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maybe four here.

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And then comes a photon and
stimulates this electron

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to go down.

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And now you have two photons.

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And they stimulate these
two electrons to go down.

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And now they have four photons.

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They can stimulate these
four electrons to go down.

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Have now eight photons.

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And that's the amplification
process of stimulation.

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Each time you stimulate
one down, you achieve this.

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In fact, technically speaking,
apparently in good lasers,

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this transition
doesn't produce light.

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You would say, oh, it
produces more photons.

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But there's some
energy levels that

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radiate by communicating
vibrational energies

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to the atoms.

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So apparently these transitions
are not electromagnetic.

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And you don't mix more light.

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So basically, stimulated
emission of radiation

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is the thing that gets
this process going.

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And that's what we're going
to try to understand now.

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The whole discussion
that follows

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and what we have to understand
about light and atoms

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will make all this
quantitative and allow

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us to produce numbers
and calculate rates.