M3L14d.txt

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This is sort of the textbook result,
but I want to rewrite the result that we
recognize the quantization of the electromagnetic field.
So instead of just looking at the power in Planck spectrum,
spectral density, and such, I want
to bring in the photon number.
I've already given you the Bose-Einstein distribution
for the photon number and [? emote ?].
So I take now equation a and multiply it
with the average photon number in a mode of omega.
This gives me on the left hand side--
well just-- I'm multiplying this with the photon number.
So on the left hand side, I have a times the photon door number.
On the right hand side, when I put in the photon number,
the photon number with this expression
just gives me the Planck distribution,
the spectral energy.
This gives me the spectral energy density
times the b coefficient.

Yes, and this is nothing else in stimulated emission.
So we realize that stimulated emission
is nothing else than n times the photon
number, the photon number n times spontaneous emission?
Similarly, we know that the rate for absorption
becomes now, well, the same, unless we
have degeneracy factors.
But, just for the fundamental discussion,
let's avoid the p counting how many degenerate levels a level
have.
Let's just assume we have a situation that we just
count every state individually.

Then I can summarize this result in the following,
that the total of rate for emission was proportional to n,
for stimulated emission, and then we
have the extra one for spontaneous emission.
Whereas the rate for absorption was n
times the spontaneous emission.
So we find that this important formula,
that emission has an n plus 1 factor,
absorption has an n factor.
And it is of course this extra plus
one, which was absolutely crucial to establish
thermal equilibrium.
If a had been zero, no thermal equilibrium
would not have been reached.

So in other words, what is already
in Einstein's treatment of the A and B coefficient
is that if you understand absorption, which
you can understand with the Schrodinger equation,
and you understand and you write it
in the fundamental way in photon numbers,
then spontaneous emission is just the rate of absorption
divided by n.

Spontaneous emission is like induced emission in it's rate,
by just one single photon.

So as I pointed out, this is a result, which is usually
obtained by second quantization, and it is already
included in Einstein's A and B coefficient.

So we could stop here.
We have already a major result, which is usually
obtained in field quantization.
But there is one deficiency, and we
want to fix it, and move on to the microscopic derivation,
and this is the following.
Right now, we really assume [INAUDIBLE] radiation.
And the ratio, n plus 1 over n was only
derived for average photon numbers in a spectral report
field.
And what he left for microscopic treatment, which
I want to present now is, even if you have just a single mode,
the atom can only interact with a single mode,
we be find that stimulated emission and absorption is
proportion, the number of photons already present.
And then there is plus 1 for spontaneous emission.
So in other words, we do it now sort of microscopically again.
And what we get out of it is that everything we learned
from Einstein's A and B coefficient is not
just valid in thermal equilibrium,
it's not just valid for average numbers,
it's really valid for single mode physics.

OK, so the agenda is what is next.

Is valid for n.
So if this expression is valid, not only
for an average over many modes, but for each single mode.

Questions about Einstein's A and B coefficient?