M3L13a.txt

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What they're talking about is actually matrix elements.
If you want to do anything interesting in atomic physics,
you have to couple or induce transitions from one state
to another state.
For many phenomena which we'll cover
in the rest of the semester-- spontaneous emission,
coherent systems, free level systems, and super radiance--
all we need is a matrix element.
And this matrix element will just
run through all the equations and be
responsible for a lot of interesting phenomena.
And for most of the description of those phenomena,
we don't have to know where this matrix element comes from.
The only thing we have to know-- there
is a non zero matrix element which drives the process.
And as you know, the matrix element with an external field
is called the Rabi frequency.
And a lot of physics just depends on the Rabi frequency.
But what is behind?
The engine behind about the Rabi frequency is a matrix element.
So in the unit I started to teach,
the week before spring break, we talked about matrix elements.
And, well, for each, for the Hamiltonian,
we used a coupling of an atom to the electromagnetic field.
And we calculated what is the matrix element introduced
by the electric field.
We made the dipole approximation and that's
your plain, vanilla, genetic dipole operator which
can connect two states.
But we also consider what happens
when we go beyond the dipole approximation.
And we found extra ways of coupling two states.
For instance, we can couple two states
which have the same parity with a quadrupole transition.
Or we can couple them with a magnetic dipole transition.
So these are other ways to get into matrix elements.
For most of the course, you don't
have to understand what is behind the matrix.
I mean, you just know there is a number which
drives the process.
So what I want to finish today is to discuss--
and these are called selection rules, which tells us
when are those numbers-- when is this matrix element which
couples two states, when is it zero
or when is it non vanishing?

And what is helpful here is, well,
as always in physics, use symmetry.
And if you have an operator-- and I will give you examples
immediately.
But just think for a moment about the electric dipole.
The electric dipole is the position operator R.
And you want to know-- can the position operator
R induce a transition between two states?
The way to analyze it is now in terms of symmetry.
And the symmetry which is always fulfilled for isolated atoms
is angular momentum.
Angular momentum is a conserved quantum number.
We have rotation symmetry.
So therefore, we want to now understand matrix elements
in the language of rotation symmetry.
And therefore, we don't want to use the position operator
xyz because xyz do not have the rotation symmetry.
We're going to lose linear superpositions of x, y,
and z-- I'll give you a free example in a moment-- in such
a way that the operator becomes an element
of a spherical tensor.
And spherical tensor I gave you the definition
in the last lecture.
The element of a space for spherical tensor lm
is defined by-- well, I connect it with something you know,
that it transforms under rotation
like the spherical harmonics ylm.
So it is pretty much for an operator
what the ylm, what the spherical harmonics,
are for the functions.
I think I can do it more formal.
And professor knows much more about it.
I think these are sort of elements
of the rotational symmetry group.
But I don't want to go there.
So if you take the position vector R,
you can expand it into a basis which is x and y.
But if you use the spherical pieces x plus minus iy,
then what appears are the spherical harmonics.

So in that case, it's rather simple.
The position vector has actually in this representation
components which you can even see
are the spherical harmonics.
And therefore, they transform like the spherical harmonics.
Or just to give you another example, if you
have the operator which is responsible for the quadrupole
transition, well, you get the gist.
It's a product of two coordinates.
So therefore, it's a spherical tensor of rank two.
And it so happens, but I'm not deriving
that it is a superposition of two components with lm
quantum number 2 plus 1, 2 minus 1.
So that's how you should think about it.
So we want to ask-- we want to extend
the operator into operators which have rotational symmetry.
And these are those or these are those three.
So instead of using the vector and Cartesian coordinate,
we use its spherical components.