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M3L14ca.txt
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M3L14ca.txt
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#
# File: content-mit-8-421-3x-subtitles/M3L14ca.txt
#
# Captions for 8.421x module
#
# This file has 163 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
OK.
Einstein's A and B coefficients.
I will post one of Einstein's papers on the web sight.
He was also the first to actually discuss
mechanical forces of light.
He realized that if you have a gas at a temperature which
is different from the temperature in the room,
the gas has to equilibrate.
And the gas can only equilibrate,
lost excess velocity by transferring its momentum
to the photons.
So some equations of laser cooling,
the fact that light can exchange momentum with a particle--
and this is eventually what leads to equilibrium--
was already in papers at the beginning of the 20th century.
And it's just amazing if you read those papers,
how modern the language is and how clear the language is.
But here, I'm not talking about the mechanically effects.
But the mechanical effects of light,
which many people in this class like or use for living,
this is actually part of this equation.
Because equilibrium between a gas--
I discuss here in Einstein's A and B coefficient,
the equilibrium between the electronic structure,
the ground and excited state, with a photon field.
But Einstein also considered the equilibrium
between the motional degree of the atom.
And equilibrium between the motional degree
of the atom and the radiation field
requires the spontaneous force, the spontaneous radiation
force.
I'm not discussing it here.
What I'm discussing here is now an equilibrium
between ground state and excited state.
So the probability to find an atom in the excited state
is simply done by-- simply described
by the Boltzmann factor.
Now it's traditional in the discussion of Einstein's A
and B coefficient to allow for degeneracy factors
at ground and excited state, have degeneracy.
I have to say, I usually hate that.
I try not to talk about levels.
I just talk about quantum states,
non-degenerate or individual quantum states.
So in that sense, I try to characterize population
in the quantum state, not in a level.
But it is standard to follow Einstein's concept
where you have degeneracies.
I'm not emphasizing them here.
But I will just write them down where they belong to.
OK.
So this takes care.
We know what is a fraction of atoms in the excited state.
So this is the equilibrium.
The next thing we need is the light.
And Einstein assumed that it's a spectral density in black body
cavity.
So we need the energy density per frequency interval.
And this is nothing else than the occupation number over mode
times the energy of the photon times the density of states.
The photon number per mode is just
given by the Bose-Einstein factor,
Bose-Einstein statistics factor.
The mode density is, as you know in three dimensional,
omega squared pi squared over c cubed.
So therefore the spectral density of black body radiation
has-- and we need that in omega cube dependence.
And then it has this Bose-Einstein denominator
in the well known form.
So this is now Plank's black body spectrum in the units
where we need it.
So all we need is now to find the famous Einstein A and B
coefficient.
We have to write down a rate equation for the atoms.
So the fact is we knew already what equilibrium is.
Excited state versus ground state population
is the Boltzmann factor.
But now we write down a rate equation
which involves a black body field,
and then we compare the solution of the rate equation
to the solution we already know.
And from that, we get Einstein's A and B coefficient.
OK, so the change of in the population of the excited state
has three different terms.
One is the energy density of the black body radiation
can cause stimulated emission.
So therefore, it's proportional to the number
of atoms in the excited state.
The energy density of the black body radiation
can cause absorption.
This is proportional to the number
of atoms in the ground state.
And then this equation as it stands
would lead to contradiction when I compare
the solution of this equation to the Boltzmann factor we already
know.
And the only way to fix it is to add an extra term, which
is spontaneous emission.
If spontaneous emission were not necessarily, this A efficient
could in the end turn out to be zero,
or it can be undetermined.
But as we see, it is necessary for consistency.
So this is pretty much the famous rate equation.
And we are interested in the equilibrium solution.
In equilibrium, all derivatives vanish.
And then, by setting the derivatives to zero,
I have one equation.
And I will rewrite the equation by putting the spectral density
of the light on one site and everything else
on the other side.
And what we have here is the A coefficient, the ground state
population, the excited state population, the B coefficient.
So this is spectral density.
It's just an expression for the spectral density.
We want to put in now that the excited state fraction is
given by a Boltzmann factor.
So therefore, Ne over Ng becomes the Boltzmann factor.
And yes, there are these degeneracy factors.
So I've pretty much divided the denominator and the numerator
by their population in the excited state.
And here I get Beg.
OK.
So this is the result for the spectral density.
But we know already that the spectral density
has to be of the Plank form.
So now we can simply compare what
we know to what we obtained from the rate equation
and make sure that it matches.
It's good that we have the exponential factor.
And by bringing this expression to the form
of the other expression, we actually
have to fulfill two conditions.
One is, in order to make sure that kind
of the total expression is OK, it gives us of ratio.
The Plank body spectrum is normalized.
There is no unknown prefactor.
So this determines the ratio of A and B.
And also seems we have this functional
form of the Bose-Einstein statistics, which
is this exponential factor minus 1,
it gives us also a relation between the two B coefficient.
That one is the B coefficient for stimulated emission,
and the other one is a B coefficient for absorption.
OK so with that, we have the relation
between the A coefficient and the B coefficient.
And we find that the B coefficient for absorption
and emission are the same.
Well, we know it's the same coupling matrix.
And when the Hamiltonian, which connects ground
to excited state, excited state to ground state.
But if you really want to deal with degenerate states
and not formulate it for states, you have degeneracy factors.