M5L25g.txt

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So now we want to perform a treatment for N particles.
We have now the individual pseudo spins 1/2.
We perform now with sum over all in particles.
We get the total spin S.
The total spin S random number has
to be smaller or equal than N/2 because we
have N spin 1/2 systems.
The incremental number is 1/2 times
the difference of the atoms which
are in the excited state minus the atoms which
are in the ground state.
And this is, of course, trivially must be smaller
than S. Because M is this z comportment of S.
And we are now describing the system by the eigenstates S
and M of the collective spin.
That means we have the following situation.
We have a manifold.
We want to show now all the energy levels.
We have a manifold, which has a maximum spin N/2.
The next manifold has N/2 minus 1.
And the last one has let's assume
we have an odd number of particles, S equals 1/2.
So here we have now N energy levels.
We can go from all the N atoms excited to all the N
atoms being de-excited.
In the following manifold, we have S is 1 less
and therefore we have a ladder of states,
which is a little bit shorter.
And eventually for S equals 1/2, we
have only 2 components plus 1/2 and minus 1/2.
So those levels interact with the electromagnetic field.
The operator of the electromagnetic field,
we have already derived that, involves
the sum of all of the little sigma pluses, sigma I pluses.
And we call the sum of all of them S plus and S minus.
And the matrix element is now for a spontaneous emission.
You have a state with SM.
S minus is the lowering operator for the N particle system.
So it goes from a state with a certain number
of atomic excitations to one excitation less.
And that means this is the act of emitting spontaneously
1 photon.
The operator S minus stays within the S manifold.
So we stay in the same letter, which
is characterized by the [? incremental ?] number S.
But we lower the incremental number by 1.
The incremental number is the measure
of the number of excitations.
And we know from general spin algebra
that this matrix element is S minus M plus 1 times S plus M.
And since I want to-- there are, of course, pre-factors
like the dipole matrix element of a similar atom.
But I always want to normalize things to a single atom.
And by just using the square root,
if you have a similar particle, which is in the S equals M
equals 1/2 state, then you see that this square root
is just 1.
So, therefore, for when I discuss
now the relative strengths off the transitions
between those eigenstates, I've always
normalized to single particle.