WEBVTT

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PROFESSOR: Einstein's argument,
we consider again our atoms.

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And this time we're going to
use the thermal equilibrium.

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We're going to really
make use of that fact.

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So, again, we have
level b and level a.

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And this is Einstein's argument.

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And we're going to
have no populations.

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If we discuss equilibrium, we
can consider a system in a box.

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And there's a few billion atoms.

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And there's some
atoms whose electron

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is going to be in state b.

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Some atoms whose electrons
are going to be in state a.

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Let's call these
numbers Nb and Na.

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And we also have
photons at temperature T

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and have a beta parameter 1 over
the Boltzmann constant times T.

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So this is the process
we're going to consider.

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So what's going to happen?

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If we came up from the edition
that we've built from 806,

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we would think,
OK, there's going

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to be absorption process and
stimulated emission process.

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And between the
two, they're going

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to be able to reach equilibrium.

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We don't believe this is not a
process that can equilibrate.

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So this would be our intuition.

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The intuition for
people that lived

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at the beginning of last
century was rather different.

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They felt that there would
be absorption process

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and there was
spontaneous emission.

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The thing that you probably
intuitively would think,

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if you are at a high level,
you spontaneously decay.

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So the thing that
was not known to them

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was this stimulated emission.

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That is what Einstein is
credited for discovering here.

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People felt, and I think the
paper finds that makes clear

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that the intuition,
is that you have

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spontaneous emission,
in which spontaneously

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by some kind of instability,
the higher state goes

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to the bottom.

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And you have absorption.

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But then Einstein figured
out that you couldn't achieve

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equilibrium in that way.

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Well, the way we're
going to do it,

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we're going to put the
two things we know--

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the absorption
and the stimulated

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emission-- and see
that we don't get it

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to work, the equilibrium.

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But then when we add the
spontaneous emission, we will.

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And as it turns out,
the spontaneous emission

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is a little harder to calculate.

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If we were to do it
in 806, it probably

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would be a matter of
two lectures involving

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an electromagnetic field.

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So we will not do it.

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But the good thing is that
Einstein's argument tells you

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the speed of spontaneous
emission, the rate

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of spontaneous
emission, in terms

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of this rate of stimulated
emission or absorption.

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So it does the
calculation for you

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by some other
thermodynamical means.

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So we're going to
use here three facts.

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One is that the--

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three facts-- one is the
populations are in equilibrium.

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So Na dot is they stop
changing is equal to 0.

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And Nb dot is equal to 0.

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They don't stop changing
because nothing happens.

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All the time there
will be emission,

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and there will be absorption.

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But if you reach equilibrium,
the number of atoms

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remain the same on every state.

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So that's our statement
that the populations

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achieve equilibrium.

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The second statement is that the
equilibrium is thermodynamical.

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So it's thermal equilibrium.

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That is Nb over Na, for example,
is the Boltzmann factor e

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to the minus beta, Eb over
e to the minus beta Ea.

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And this is equal to e to the
minus beta h bar omega ba.

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You get e to the minus
beta, eb minus ea.

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But eb minus ea
is h bar omega ba.

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So that's thermal equilibrium.

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And the last thing
that we need is

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a statement about the photons.

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What do they do when
they have equilibrium?

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And that was known already
due the work of Planck

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and others, black
body equilibrium.

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So we need to know something
about the thermal radiation.

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And the way one describes
this is in terms of a function

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U of omega d omega.

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In the black body
radiation, there

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are at a given
temperature, there

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are photons with
very little energy.

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There are some largest
number of photons

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with some energy associated
with temperature,

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and then it decays.

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So you have photons
of all energy.

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So if you want a description of
what's going on in black body

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radiation, you can
consider the energy

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in the photons in
the frequency range.

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But you even must
be more precise.

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It is the energy per unit
volume in a frequency range.

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Because if it's different,
the energy of the black body

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cavities, this room
or it's a little box.

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So it's an energy per
volume per frequency range.

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And that's what
this quantity is.

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So let me write it.

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Energy per unit volume in
the frequency range dw.

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In other words, it's kind of a
proxy for the number of photons

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available.

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All the photos have at some
value of the frequency,

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of energy e omega.

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So if you know the
energy, you basically

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are getting here the number of
photons with frequency omega

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in that range per unit
volume, all that stuff.

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So this has a formula.

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And the formula that
was known to people

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was this quantities and then
omega cube, d omega, over e

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to the beta h bar omega minus 1.

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So this is the basis
of the calculation.

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What do we do?

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We have to consider
the possible processes.

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OK, so our processes
are absorption.

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And in this case,
we go from a to b.

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And let's try to
write a rate for them.

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So what would the
rate depend on?

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Well, here's some
little assumptions.

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Certainly, if you don't have
particles in the a state,

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you cannot have this process.

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So this process, the total
rate that we observe in the box

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will depend on an Na.

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The more particles
you have in this state

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a, the larger the probability
that you get the transitions,

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and the larger the rate.

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It will also be affected
by the number of photons

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available at that
frequency that can

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produce a transition in
proportional to that.

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And finally, the quantity
that our study of perturbation

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theory will tell us
about, but at that time,

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finds that was not known, it's
a transition coefficient, Bab,

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he called it.

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And this is the unknown one.

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And this is what we don't know.

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We know U. We assume we know Na.

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This is the transition
rate per atom

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and then multiplied by
the number of atoms.

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So this is transition
rate per atom.

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Then we have the process
of spontaneous emission.

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And then we will be
another coefficient, Bba.

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And it would depend on
the number of particles

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that are in the state b,
because spontaneous emissions

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that transition from b to a.

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We call it-- oh, not
spontaneous stimulated.

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I'm sorry-- stimulated emission.

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This is the one
we're considering.

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It's stimulated
by the radiation.

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So it's also proportional
to the number

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of photons present
and proportional

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to the number of
atoms that can be

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convinced to do the transition
times another coefficient, Bba.

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So this is, I think,
what we in 806 would do.

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We would consider
this two processes

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and attempt to make it work.

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And let's see what we get then.

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We're trying to get equilibrium.

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So we want the transitions to
equilibrate and, therefore,

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the populations not to change.

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So let's look-- for example--

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you could look at either one,
but you can look at Nb dot.

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It should be 0.

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But it's equal to the
rate of absorption

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minus the rate of
stimulated emission.

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You see because the
number of particles in b

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change because you get
some new particles in state

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b due to the absorption process.

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And it happens with this rate.

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And you lose some particle
because some atoms

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do the transition from the
higher level to the lower

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level.

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So what is the
rate of absorption?

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We have it here.

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It's this one, Bab
U of omega ba Na.

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And this one is Bba, the
same U of omega ba Nb.

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And we can factor the U out.

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And this is the wrong
calculation, I must say,

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because we're missing
that extra process that

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was intuitive to
Einstein, but to us it's

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a little less clear--

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Nb times U of omega ba.

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OK, I can do a one
more little thing.

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I can factor an Na.

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And this becomes Bab minus Bba
to the minus beta h bar omega

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Ba U of omega ba.

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OK, I use the ratio of Na
over Nb being thermodynamical.

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So Nb over Na was used
from point two to get this.

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And this should be equal to 0.

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But this equation
can't be satisfied.

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What do you have here?

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You should be able to
equilibrate at any temperature.

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On the other hand,
what is our intuition

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about these quantities,
Bab and Bba?

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They should be
temperature independent.

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These are properties of the
geometry of those states

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and the overlaps of
the wave functions.

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These are atomic
physics properties

00:15:35.350 --> 00:15:38.030
of the levels of the particles.

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We will calculate them.

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And here is the input of
how many photons there

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are, how many atoms there are.

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And the number of
photons certainly

00:15:49.300 --> 00:15:50.560
depend on the temperature.

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The number of atoms
for equilibrium

00:15:52.780 --> 00:15:54.260
depend on the temperature.

00:15:54.260 --> 00:15:57.220
But this is a factor
that says, well,

00:15:57.220 --> 00:16:01.210
how likely is the transition
once you have a photon

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and once you have an atom?

00:16:02.890 --> 00:16:07.990
And that depends like we did
for the ionization, calculated

00:16:07.990 --> 00:16:10.450
some matrix elements
that are totally

00:16:10.450 --> 00:16:12.800
independent of temperature.

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So these numbers are totally
independent of temperature.

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And we're asking
this to be 0, which

00:16:20.740 --> 00:16:23.655
requires this factor to be 0.

00:16:23.655 --> 00:16:26.300
And this depends on temperature.

00:16:26.300 --> 00:16:31.510
So you cannot attain
equilibrium with this way.

00:16:31.510 --> 00:16:43.600
So it's impossible to satisfy
for all temperatures given

00:16:43.600 --> 00:16:48.990
that Bab and Bba are constants.

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So we're missing a process.

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This is the process that
Einstein thought was intuitive,

00:17:12.700 --> 00:17:16.214
the process of
spontaneous emission.

00:17:19.569 --> 00:17:24.490
So we add one more process.

00:17:24.490 --> 00:17:29.745
It's called
spontaneous emission.

00:17:36.820 --> 00:17:40.795
And it's a process
also from b to a.

00:17:45.373 --> 00:17:48.670
And it's going to have a rate.

00:17:48.670 --> 00:17:52.120
But it's not going
to depend, that rate,

00:17:52.120 --> 00:17:55.570
on the number of photons,
because it's happening

00:17:55.570 --> 00:17:57.940
independently of the photons.

00:17:57.940 --> 00:18:01.420
So we don't have this U factor.

00:18:01.420 --> 00:18:06.790
We do have the Nb because
each of the b atoms

00:18:06.790 --> 00:18:09.160
can spontaneously decay.

00:18:09.160 --> 00:18:11.330
But we don't have the U factor.

00:18:14.880 --> 00:18:16.710
So what do we have?

00:18:16.710 --> 00:18:23.320
A rate, which is the
term by a coefficient

00:18:23.320 --> 00:18:27.940
that Einstein called it a,
that's why the name a and b

00:18:27.940 --> 00:18:32.440
coefficients of
Einstein, a times and b.

00:18:35.490 --> 00:18:38.310
So that's the
spontaneous emission rate

00:18:38.310 --> 00:18:42.510
per atom multiplied by
the number of atoms.

00:18:42.510 --> 00:18:47.630
So we go back to our equation--

00:18:47.630 --> 00:18:52.130
rate of absorption minus
rate of stimulated emission

00:18:52.130 --> 00:18:57.470
minus the rate of
spontaneous emission.

00:18:57.470 --> 00:19:00.020
So I'll write it here.

00:19:00.020 --> 00:19:07.310
0 is equal to Nb dot
equal minus A Nb--

00:19:07.310 --> 00:19:10.240
that's the spontaneous emission.

00:19:10.240 --> 00:19:11.900
We write it first.

00:19:11.900 --> 00:19:13.990
And then we'll write
the other two--

00:19:13.990 --> 00:19:30.520
Bba and Nb U omega ba
plus Bab U of omega ba Na.