M3L12e.txt

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This leads us to a discussion of selection rules.
Selection rules is nothing else than a classification
of possible transitions according to symmetry.
And it's a way using, well, CG coefficients,
angular momentum coupling, or using you would say symmetry
to figure out of matrix elements are non vanishing or vanishing.
And I gave you already one example
and I want to formalize it now.
If you go between two S states which have zero angular
momentum, you cannot have an operator which is a quadropole
operator because, and this is what I want to tell you now,
the quadropole operator is a spherical tensor with two units
of angular momentum.
And this would forbid the triangle rule.
This would forbid conservation of angular momentum.
So that's what I want to discuss now in the next chapter
or at least get started for the next five minutes
by discussing selection rules.
So the introduction to selection rules
is that we have forbidden transitions.
Forbidden transitions are suppressed
by-- because we are forced to go to higher order.
And this is usually higher order in alpha.
So forbidden transitions are weaker by some power of alpha.
And that means they require higher approximations.

And, of course, the comparison is always dipole transition.
This is the dominant transition.
This is the industrial strength transition.
And from there on, it can get weaker.
So it can get weaker by multipole expansion.
We've just discussed that.
It can get weaker because you have
to emit-- you have to have that cascade of dipole transitions.
This would be multiphoton processes
as we discuss later in the course.
They can get weaker because they are exactly
zero in a non relativistic approximation
and require relativistic effects.
The example we have encountered in this course
is the singlet to triplet transition in helium.
Or they're transitions which would not be allowed just
for the electron, but if we invoke hyperfine interactions
with the nucleus, then they become allowed.
So it's a rich subject and I'm not an expert and I cannot do
full justice to it.
But I went to at least give you some general rules how
we discuss matrix elements.
So what is always a good quantum number,
what is always a label for our atomic states,
is angular momentum.
Because atoms live in free space and there is rotation
invariance.
So we always categorize our atoms
with angular momentum quantum numbers JM.
And we are asking are there transitions between a state
JM to a state J prime M prime.
And all other quantum numbers we can now summarize with a label
N. And now we have an operator.
I gave you examples for the operator--
the magnetic moment, the electric dipole,
the quadropole operator.
But in general, every operator can be written,
can be expanded, in a sum of spherical tensors.
So what is discussed in the classification of matrix
elements are matrix elements involving components--
the operator of components of a spherical tensor.
So T is a spherical tensor of rank L.
And if you want the simple definition what
is a spherical tensor, you try to write
an operator like the position operator R.
You try to write it as a sum of terms.
And each term transforms like a spherical harmonic YLM.
So in other words, we can write every operator
as the sum of spherical tensors.
And the spherical tensor is characterized
that it transforms under rotations exactly
as the spherical harmonics YLM.

So every operator is now a sum of spherical tensors.
I don't want to get too much interest into symmetry
classification, but the story is that you
know the YLM functions are complete functions.
Every function can be expanded in spherical harmonics.
And similarly, if you have an operator,
you can decompose it into objects which transform
under rotation as the YLMs.
So you have a part which transforms
according to an object.
YLM is a classification of wave functions
with angular momentum.
And so therefore, each operator may not have a specific angular
momentum, but can be written as the sum of operators,
each of which has the same symmetry,
the same transformation properties,
as the state of angular momentum.

And I think I should stop here, but let me just
give you the final message.
So by doing that, by separating the operator
into a sum of spherical tensors, we are actually now back
to angular momentum.
If we have a matrix element and just
think how you calculate in the Schroedinger representation,
you have a wave function operator
and a wave function to indicate over it.
So what you have is you have the product of the objects.
And now we have to-- we can classify them
by angular momentum.
And in the end, and this is what I will show you
in the next class is that ultimately,
the question whether this matrix element is zero or not will
boil down to the question whether J prime and N
prime, angular momentum, can be added to L and M.
And there is overlap with angular momentum of JM.
So we are back to the rules for adding
angular momenta we get the triangle rule
and CG coefficients.