M3L13b.txt

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We can take this expression from the last lecture
and rewrite it using the Wigner-Eckart theorem
into a way which allows us to immediately formulate
selection rules.
Inert and prime are the quantum numbers of the state,
except for angular momentum, so matrixing
about principal quantum number of the hydrogen atom.
And we want to couple from a total angular momentum J-prime
to total angular momentum J. Actually,
we want to couple from J-prime, M-prime to a state, JM.
And what the Wigner-Eckart theorem
tells us, that we can factor out M-dependence.
The M-dependence just comes from orientation in space.
So M is just how you orient wave functions and vectors in space.
And you can start off, right, this matrix element
as a projection.
And this is called, this is nothing else
than the familiar Clebsch-Gordon coefficient.
And the Clebsch-Gordon coefficient--
or the Clebsch-Gordon coefficient,
to start with the initial shape J-prime, M-prime,
we have the L and the M of our operator.
And that should result in a total angular
momentum of J and M.
So we'll retrieve again, the formalism
of the addition of two angular momenta.
Sometimes you have two particles you couple with two angular
momenta and ask, you know, what is the total angular momentum
of the composite particles?
But what we do here for the selection rule.
We have the initial state.
We couple it with the angular momentum of the operator.
You can think the operator is a field which
can transfer angular momentum.
And then, of course, the final state
has to fulfill angular momentum conservation.
But one source of the momentum is now the operator.
It's the external field.
It's the photon or the microwave drive, whatever you apply.
This Wigner-Eckart theorem allows
us to write the matrix element as a reduced matrix element.
Which really decides whether the transition is non-vanishing
or not.
--times a factor, which is just the orientation of the wave
function and the operator in space.
So for the Clebsch-Gordon coefficient,
we have a simple selection rule.
And this is that, for the M quantum number,
the M of the final state has to be the M of the initial state
plus the little M of the operator.
And for both the Clebsch-Gordon and the reduced matrix element,
we have the triangle rule.
Well if you couple two angular momentum vectors
to two final angular momentums, the three vectors
have to form a triangle.
And the triangle rule says that-- let
me write it down and then you'll recognize it-- that the angular
momentum transferred by the field
has to fulfill the triangle rule that J-prime and J can
be connected.
Yes--
What is the meaning of the double bars?
It just doesn't look familiar.
It's just how, in many textbooks,
the reduced matrix element is written.
It's nothing else than a matrix element.
But what happens is, these are not states.
You know, J and J-prime are not states.
They have an M-dependence.
So we have taken out the M-dependence.
So this is sort of a matrix element between, you know,
a state which may have been stripped of its M-dependence.
So maybe, I don't know if that's 100% correct.
But if you have the YLM in certain states,
you have an E to the IM, an M part.
And this has probably been factored out.
So these are not really states, and the double line just
means it's reduced matrix element with the meaning I just
mentioned.
It's a standard way of factorizing matrix elements.
And, yeah, that means reduced matrix element.
So in other words, when we talked about selection rules,
we want to use the representation
in spherical tensors.
Because the rank of the spherical tensor just tells us
how much angular momentum is involved in the photon,
is involved in this transition.