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PROFESSOR: Good afternoon.
Yes.
Let's have an on time departure.
The topic of last class was the ac Stark effect.
And I said what I've been saying to you,
was additional [INAUDIBLE] of the ac Stark difference
from perturbation theory.
But then I have three points for discussion.
The first one was [INAUDIBLE] for the [INAUDIBLE] problem,
[INAUDIBLE].
The second one was the oscillator strength,
which we almost finished.
And today, I want to say a few words about [INAUDIBLE]
in [INAUDIBLE].
And eventually how the ac Stark effect
leads to the absorption coefficient, which many of you
use [INAUDIBLE].
So I mentioned that the oscillator strength is actually
one way to parametrize the matrix element.
We'll actually talk most of the lecture
today about matrix element, but I
come to that in a few minutes.
From here, the matrix element is nothing,
but you get on with [INAUDIBLE].
Sure, it's a dimension of the Bohr radius
but we don't really think in terms of lengths.
There are two ways how you want to use a matrix element.
One is the matrix elements squared,
having some other effect with which
to [INAUDIBLE] this likeness, an actual likeness
against a number you should know how to [INAUDIBLE].
Or another way to parametrize matrix elements
is by this [INAUDIBLE] number, which
is what we will see in this class.
And that's, of course, also related
to semi-classical physics.
And I mentioned that in the last class,
that the oscillator strength allows
us to connect the quantum mechanical treatment of an atom
to the response of a classical harmonic oscillator
through the charge of an external [INAUDIBLE].
So I used this expression, it was the end of class last.
I used this expression for the matrix element
in terms of the oscillator strengths
to give you a nice relationship for strong transition
as in the alkaline E lines, where the oscillator strength
is close to 1.
You see that the matrix element is
1 over square of 2 times the geometric mean
of the complete wave lengths of the electron,
and the wave lengths of the transition.
So that's actually important for the dipole approximation,
because the concurrent wave lengths of the electron
is very, very small.
So therefore if the matrix element
is the geometric mean of the two,
that means that the matrix element
is smaller than the optical wave lengths.
And therefore, the matrix element
is the length scale, the relevant length
scale of the atom for transition.
It is smaller than the optical wave lengths.
And that means-- and we'll talk about it today--
that in the expression E to the IKR, KR is really small.
So we can do the dipole approximation.
Now I have a question for you.
I did it, frankly, because we don't
have many questions to discuss with.
But I mentioned that there is a sum rule for the oscillator
strengths.
And the sum rule limits it to 1.
And if there's a really strong transition,
it kind of exhausts the sum rule,
and this is what I was talking about.
But because the thought crossed my mind after last lecture,
if I don't add something to it, you
may actually use it as a proof, which would be wrong.
That the matrix element cannot be larger than what you see
on the right hand side.
On the other hand, we discussed in the context of Rydberg atoms
that Rydberg atoms can have huge matrix elements.
Matrix elements which [INAUDIBLE]
but they can really become huge.
So how do you reconcile the fact that Rydberg atoms
in large instates, in states with large strengths
of [INAUDIBLE] numbers can have huge matrix elements.
How do you reconcile it with our discussion
of oscillator strengths and sum rules?
The answer is simple but subtle.
So is it a proof that the matrix elements are impossible?
Or if big matrix elements, huge matrix elements,
are possible, what property of the oscillator strengths
did I not emphasize?
I mean, keep watching.
So first, are there atoms which have huge matrix elements?
Yes.
Rydberg atoms.
So the left hand side can be really huge.
On the right hand side, we have the wave lengths
of the transition which can get large.
You have to count the wave lengths.
But then you have the oscillator strengths.
And the oscillator strengths, the sum
of all oscillator strengths is 1.
But you have to be careful that not all oscillator
strengths have to be possible.
The definition of the oscillator strengths
involves frequency from one state to the next.
And depending whether you go up or down,
the frequency has positive or negative side.
So therefore we have a sum rule.
For the ground state, all frequencies
are positive because you can only go up.
So therefore the sum rule is only
very useful for the ground state,
because all oscillator strengths are positive.
And if one oscillator strength is 1,
you know no other transition has any strength at all.
However when you have an excited state
and you find an oscillator strength which is 1,
you could still have a lot of other strong transitions,
but their oscillator strengths are positive and negative
and they compensate for each other in the sum rule.
So just be careful about that.
For the ground state-- and this is what I've proven to you
here, I think fairly rigorously--
if you are in the ground state, the transition matrix element
cannot be larger than the geometric mean
of the [INAUDIBLE] wave lengths and the [INAUDIBLE] wave
lengths.
But for excited states, you have to be careful,
because the oscillator strengths can have cosines.
OK.
Any questions about that?
Then let's talk quickly about the third point
I wanted to discuss.
And this is the relationship to the index of refraction.
The polarizability here is responsible for the index
of refraction.
Well, the polarizability determines [INAUDIBLE]
evaluate the constant and use whatever relation you want,
but for a common physics purpose,
this is sort of the useful relation.
This is the index of refraction.
It's related to the polarizability.
And assuming that the polarizability is small,
you get this, the parallel extension.
So roughly saying n minus 1.
The difference of the index of refraction from the vacuum
is proportionate to the atomic density
times the polarizability.
OK.
And we know this is what we derive from the perturbation
theory, that the polarizability depends
on a matrix element squared and 1 over the cube.
Now I want to sort of show you that by just using that concept
and putting in phenomenologically dissipation,
we can get a full expression what happens to a laser
beam crossing an atomic medium, like a [INAUDIBLE] condensate.
How much of the laser beam is absorbed?
And what is the feature of the laser beam.
In order to get interesting or simple formula,
I want to now parametrize the matrix element
by this correlated gamma.
This will turn out with an actual [INAUDIBLE],
but right now, since we haven't talked
about spontaneous emission, just say
I replace the matrix element by gamma.
And then I can write the index of refraction
in the following way.
But the matrix element squared is now parametrized by gamma.
I have also introduced the coordinate sigma
0, which will later be the lesson in absorption
for section for atoms.
But again here, just use it as a parametrization.
I mean, I've not introduced any new concepts.
I've just rewritten this expression in this way,
involved sigma 0, which is a cross-section, and gamma, which
is an actual light.
OK.
The polarizability depended-- that was perturbation
theory-- 1 over the energy denominator,
or 1 over the [INAUDIBLE].
Now we have used, with the oscillator strengths,
the analogy to a classical harmonic oscillator.
And at some level, you know that every harmonic oscillator
must have little bit of damping.
And so the same applies to atomic oscillators.
And if you can account for the damping
by putting into the frequency denominator of the [INAUDIBLE],
1 over delta.
Give it an imaginary part.
So I'm just telling you here, yes.
Every oscillator has some damping,
and I can phenomenologically accounting for the damping
by making the resonant frequency, or the tuning,
slightly complex by adding imaginary part.
Later on I want to express all the tunings normalized
to this line which is parallel to a normalized [INAUDIBLE]
imaginary part.
OK.
So we have the situation that this expression, gamma
over delta-- this was the expression
for the polarizability-- now acquires
a small imaginary part, and that means it's a complex number.
And I can now take this expression separate it
into an imaginary part and a real part.
So why do I do that?
Well, the index of refraction appears
in the propagation of the plane wave.
If you're a plane wave, the key vector
is no longer the key vector in vacuum.
It's multiplied with the index of refraction,
and that's what I do here.
And now you see that immediately,
an imaginary part of the index of refraction
leads to absorption.
And the real part leads to [INAUDIBLE].
OK.
So what we have now is we have this expression for the plane
wave after it is propagated through the medium.
And we see there is exponential absorption.
Exponential absorption.
On resonance, we have an optimal density
which is given by this expression.
And the second term here give us a phase shift.
And it's clear since we have one medium which
has a certain thickness that the optical density and the phase
shift are related.
Yes.
If you're on resonance, you have the maximum optical density
and the maximum absorption.
And the optical density, of course,
is 1 over the [INAUDIBLE] squared.
[INAUDIBLE]
Whereas for large detuning, the phase shift goes as 1
over delta.
It's the dispersive scaling with the tuning,
and this is also what we had when we started out.
When we would have started out with the polarizability and not
a decaying dissipation, because far away from resonance,
you simply have a regional of dispersive effect,
and dissipation doesn't occur.
So anyway, I thought I would just
show that to you because this is a full understanding of what
happens to optical views when they pass the atomic
[INAUDIBLE].
It's nothing else than the polarizability
or the harmonic oscillator response
of an atom to electromagnetic radiation.
The only thing which is non-derivative here is--
and that's what we will discuss later in this course--
is in order to get the final result, which
you may want to use in analyzing your data,
you have to set the dissipative, the damping
of the harmonic oscillator, equal with this gamma
parameter.
And that's something I cannot tell you without talking about
spontaneous emission.
So just to remind you, I introduce big gamma just
as a parametrization of the matrix element.
Then I introduce phenomenologically little gamma
as just the damping of the harmonic oscillator
of those two quantities for two levels [INAUDIBLE]
are identical.
And that's something I cannot show you here,
because this requires discussion of spontaneous emission
and the other modes of the electromagnetic field.
This is not covered by a discussion of the ac
polarizability.
OK.
Any questions?
But still-- I know I'm repeating myself--
I always find it interesting that the whole physics
of absorption, the beams are absorbed,
you can pretty much pull it out of the response
of a harmonic oscillator.
It's only the damping rate of the harmonic oscillator
that this is simply spontaneous emission.
That's the only point where you have
to go beyond the ac Stark shift and bring in the [INAUDIBLE]
nature of the electromagnetic field
and all the empty modes of the vacuum.
Any questions about waves, or in general interactions
of atoms with electric and magnetic fields?
Well, then let's move on.
We have talked about electric and magnetic fields
intimately for very good reasons,
because low frequency fields are either electric or magnetic
in nature.
But ultimately, we want to understand
what happens when atoms interact with light.
And this is our discussion for today.
The outline of this chapter is as follows.
Today, all I want to discuss is the coupling matrix in an atom.
In other words, when we have a Hamiltonian between state A
and B, we have an off-diagonal matrix element.
And this off-diagonal matrix element causes transitions.
Or when we calculate it, the effect of electric field--
the ec Stark effect or the ac Stark effect,
we assume there was a quantity, state 1, state 2,
and we put some operator in between.
So this was a matrix element which connects the two states.
And so far, we have always assumed
using the classical multiple expansion
of electromagnetic field energy that this matrix element has
the form electric field times dipole matrix element.
And therefore, the relevant matrix element
was the position operator of the electron connecting the two
states.
So in a way, we have used it all the time.
But in any more advanced discussion of atomic physics,
you have two states.
And as long as they are coupled, you
have something which you call H1 2
or you call it matrix element N1 2,
and you're not even asking where it comes from.
When two levels are coupled, they undergo Rabi oscillations.
You have transitions.
If you're in the excited state and you have a matrix element,
you get spontaneous emission.
So a lot of the things we want to discuss
about atomic physics, all they require
a number which is a matrix element between state
1 and state 2.
And today, I want to talk to you about different mechanisms
which can lead to matrix element.
We talked about the dipole operator,
but you also talk about higher order possibilities,
magnetic dipole coupling, or electric [INAUDIBLE] coupling.
And a little bit later-- this is an outlook--
we will talk about twofold uncoupling.
But in the end, even if it's a twofold uncoupling,
for a lot or phenomena, all you need is a coupling.
A coupling means a Rabi frequency.
A Rabi frequency means Rabi oscillation, and so on.
So today the focus is, what is the structure, what
are the principles behind this number, which
is a relevant matrix element.
OK.
So that's today.
Next week is spring break, but when we resume,
we then want to talk about what is the matrix element doing?
And there are two cases which we have to consider.
One is we can see it's narrow band and broad band.
This matrix element can couple just
to one move of the electromagnetic field.
Everything is coherent.
We have coherent Rabi oscillation.
Or you can couple to many modes, and then you
have a broad band situation, which may be described
[INAUDIBLE] and this [INAUDIBLE] unit.
Very, very different behavior.
So that's what we then want to discuss.
When atoms interact with light, we
have two very different limiting cases, narrow band
and broad band.
And eventually, we want to go beyond the semi-classical
formulation of the electromagnetic field
and we'll talk about the quantized electromagnetic
field.
OK.
So let's talk about the coupling between atoms
and electromagnetic fields.
I want to give you very brief derivation
of the canonical coupling between electromagnetic fields
and atoms.
I know this is covered in many textbooks,
and we have a very deep discussion
about it using the full QED formulas in A422,
but I feel this course A421 would not
be complete without a discussion like that.
And secondly-- this is so important
if you hear it twice from two different angles, that's
probably useful-- I just want to remind you
that if you use classical electromagnetism
and the Lagrangian formalism, you find the very
elegant result that the coupling to the electromagnetic field
can be introduced by modifying the momentum.
That the canonical momentum which appears in the Lagrangian
equation is no longer the mechanical momentum
which gives rise to kinetic energy, But?
It is modified by the vector potential.
So therefore the Hamiltonian-- at this point
it's the classical Hamiltonian, but then we
use the same expression in quantum mechanics.
The Hamiltonian has kinetic energy and potential energy.
But the kinetic energy is no longer p square.
Because by p, I mean now the canonical momentum.
We have to correct for the vector potential
to go from the canonical momentum
to the mechanical momentum.
And if we squared it, this gives the kinetic energy.
If you worry about this sign here,
I will assume now that the electron has a charge.
The charge q f the electron was minus e,
so that's why we have a plus sign here and not a minus sign.
OK.
So we will use this fact that we have to substitute the momentum
operator by the canonical momentum, which
is no longer the mechanical momentum.
We used it also in our quantum mechanical Hamiltonian
in the Schrodinger equation.
If you ask me, can you prove it?
No.
Nobody can prove it.
We can never prove what a quantum mechanical equation is.
We can just use physical understanding,
physical analogies.
And that's how quantum mechanics was developed in [INAUDIBLE].
Based on the classic quantum mechanical correspondence,
this seems to be a very reasonable assumption.
And ultimately, this very reasonable assumption
has now withstood the test of time for many, many decades,
over almost 100 years.
So that's why we assume that doing the same substitution
in the Schrodinger equation for the momentum operator
gives the right result, but there is no way
how you can rigorously derive fundamental equations
in physics.
You can observe nature, postulate them,
and verify them.
So therefore, this line, which was the classical Hamiltonian,
by interpreting p as an operator,
is now our quantum mechanical Hamiltonian.
And the one thing we have to consider now
is that once we have quantum mechanical operator,
we have to be careful about compute order.
That certain quantities have to be ordered.
It matters which one we put first.
What is convenient now is to use the Coulomb gauge.
In the Coulomb gauge, diversions of A is zero.
But the del operator, the deliberative operator,
is the operator of the canonical momentum.
And therefore, if you use the Coulomb gauge,
then we have commutativity between the canonical momentum,
p, and the vector potential.
And that's sort of nice, because if you take this square
and square it out, we get p dot A plus A dot p,
but they are the same in the Coulomb gauge.
So therefore our Hamiltonian has now the following terms.
We call it H naught.
And we assume that the Coulomb potential of the nucleus,
which eventually gets the electronic structure
of the atom, is included here.
So this is the Hamiltonian part which
describes the structure of atoms.
And then we have two terms which couple
to the electromagnetic field.
So this is p dot A or A dot p.
It doesn't matter in the Coulomb gauge.
We call this the interaction Hamiltonian.
And now we get a second order term,
which is the A square term.
Well, since it's second order, we designate it.
We use the symbol H2.
In the following, we neglect the second order term.
The argument is for weak fields.
It doesn't matter.
It's not so important because it's higher order.
However, in A422, we look at it more carefully
and we can actually do a canonical transformation which
eliminates the A square term.
So to drop it is not necessary.
You can actually show with a canonical transformation
that I'm doing is actually more exact than just
saying small and delineated.
But anyway, I don't want to spend too much time on it.
All I wanted to remind you, what are the steps to obtain-- just
give me one second-- to obtain the interaction Hamiltonian,
and that's what I want to use for the remainder
of this class.
That we have the coupling of momentum
to the vector potential.
We assume that the vector potential-- we are not
quantizing the electromagnetic field.
It's a semi-classical field.
It's therefore a classical vector.
And we will investigate what is now
the coupling between an atom for this Hamiltonian
to a plane wave of the electromagnetic field.
So therefore we are introducing for the vector potential
the expression that the vector potential has an Hamiltonian
naught.
The polarization p hat, and then the plane wave factor
e to the ikr minus i omega t.
So this can be divided in a much more rigorous way,
but this is now the starting point of our discussion.
[INAUDIBLE]
OK.
Any questions about that?
Yes.
Matt.
AUDIENCE: Is it a little bit weird
that you're going to take the divergence of its particular A
that's not 0?
Since it's divergence of e to the ik dot r, just k dot A?
PROFESSOR: Sorry.
You said the divergence of A-- so you're
asking what is the divergence of A,
if A is a plane wave e to the ikr?
AUDIENCE: [INAUDIBLE] the propagation of [INAUDIBLE]?
PROFESSOR: Yeah.
What happens is, the polarization is
A, and if you take the divergence of the plane wave,
you get a K vector, so you get this [INAUDIBLE] product of e
dot k.
And electromagnetic waves propagate.
The polarizations [INAUDIBLE] propagation.
So ultimately what you find is with Coulomb gauge
and the radiation field is transverse.
It is only polarization [INAUDIBLE] propagation.
OK.
So we want to talk about matrix element.
So the matrix element between two states.
Du to the interaction Hamiltonian for one plane wave
has the e to the minus i omega t dependence, which is trivial.
I mean, in terms of we will assume
that there is a monochromatic wave.
And what you want to discuss now is
this time independent factor, H ba.
Let me get the factors.
Let me get the polarization.
So your relevant operator which connects the two states A and B
is the momentum operator.
And now for the plane wave, we want
to do an expansion in orders of kr.
And the leading term will be the dipole approximation
and the next order term will give rise
to magnetic dipole and electric quadrupole transitions.
So first, the fact that I can do this expansion
requires that k dot A is much smaller than 1.
So it's a long wave length approximation,
which you can say is valid as long as the atom is smaller
than the wave length we are talking about.
And I will show you later when I discuss the next higher order
term that for atoms in the ground state,
this is actually also an expansion alpha.
So again we retrieve the fine-structure constant.
So when we do this expansion of this plane wave exponential,
every term is smaller than the previous term
by the fine-structure constant alpha, which is 1 over 137.
OK.
So the leading term is the 1, and this gives rise
to the dipole approximation.
So what I want to show you now in one or two
minutes is that this is indeed the dipole approximation.
It leads to a matrix element which
is electric field times r, the dipole moment.
What we have right now is A dot p, the vector potential,
times the momentum.
So in other words, I want to show you
that the A dot p matrix element is equivalent to an e dot r
matrix element.
This can be done with a canonical transformation, which
we'll do in A422, but here let me sort of just show you
the elementary discussion.
We want to replace the vector potential
by the electric field.
So therefore, we use the fact that the electric field
is the derivative of the vector potential.
And the derivative of the vector potential
means since we have an E the minus i omega t dependence,
that we simply multiply with a factor of omega
of the frequency.
And what we obtain here is then the amplitude
of the electric field, E naught.
So with that, we have a matrix element
which now involves the electric field.
But still the matrix element of the momentum operator.
But we can easily go from the momentum operator
to the position operator by taking the fact
that the momentum operator is nothing
else than the commutator of r with the Hamiltonian H naught.
The kinetic energy in H naught is p square,
and the commutator of r with p square is simply p.
So this is what I'm using here.
And these are the p factors.
So therefore if you take the momentum operator between two
states, we have this relation between the momentum operator
and the position operator.
But we have not the momentum operator.
We have the commutator.
And r H naught gives us r.
But when H naught acts on the a, it gives the energy in a.
For the other part of the commutator, which
is H naught r, we can have H naught act on the left hand
side on b, and this gives us the energy ab.
For dividing by H bar, what we get
is the energy difference, eb minus ea.
Or in frequency units, omega ba.
So this is now how we have implemented the commutator.
So that by inserting this into this equation,
we now finally obtain our result for the matrix element
in the dipole approximation.
We have e, the electric field amplitude, the polarization,
the matrix element b r a, omega ba,
and here we have an 1 over omega in the denominator, which
came from the derivative of the vector potential da dt.
This here is the matrix element of the dipole operator.
So in other words, I have derived for you
that the interaction matrix element
in the dipole approximation is the dipole operator
times the electric field as long as I make the approximation
that this frequency factor's on the order of [INAUDIBLE].
Now this is clearly the case in your resonance,
so you should be safe.
However, if you do the more rigorous derivation
of the dipole Hamiltonian using canonical transformation,
this factor of omega ba over omega does not appear.
So again, the dipole approximations
is better than I presented today.
The two approximations topping the A squared term
and approximating this ratio of frequencies with 1
is not necessary if you use the other methods using
canonical transformation.
Any questions about that?
Let me make one comment.
I've shown you in this derivation
that the A dot p term, the A dot p interaction
within the dipole approximation is identical to the e dot r
interaction.
So these are the same operators, and if you would exactly
calculate them in atomic structure calculation,
you would get the same coupling, the same matrix element.
However, in practice, there are important differences.
Because the E dot r operator, the r operator,
has a lot of weight where the electron is
far away from the nucleus because it's multiplied with r.
Whereas the p operator emphasizes the derivative
of the wave function, and this is usually
strong at close distances.
So in the end, the results have to be the same.
But if you make an approximation to your wave function,
one formulation may be numerically much more accurate
than the other one.
But fundamentally, the two terms are the same
and they are related the way how I derived it for you.
Yes?
AUDIENCE: So when we say it's the dipole approximation,
does it mean that it's a long wave length,
or be the small field or everything together.
So what's the absolute fundamental assumption of this?
PROFESSOR: The absolute fundamental assumption
of the dipole approximation is that we
assume-- let me just scroll up.
In the equation above, we had an expression
for the vector potential.
And the dipole approximation means
that we can neglect the position dependence of the vector
potential A of r, so we approximate A
of r as being A evaluated at the origin of the atom.
And over the extent, over the size of the atom,
we do not have to consider a spacial dependence
of the vector potential.
So this is sufficient.
The other things that e square is small
and that we're near resonance-- I
have to make those assumptions if I
want to use this elementary derivation.
But with a canonical transformation,
as is discussed in atom-photon interactions,
you do not need those additional assumptions.
The only assumption behind the dipole approximation
is that one, and this means that the extent of the atom
is much smaller than the optical wave lengths.
OK.
But there are situations where the leading approximation
vanishes.
For instance, if two levels have the same parity,
then the dipole operator between the two of them is 0.
And then if you want to have a transition between,
or if you want to consider a transition between those two
levels, it will come from next order terms.
So let's now discuss higher order radiation processes.
So the motivation why I want to discuss higher order radiation
processes is because in some basic courses,
you only need the dipole approximation.
And you think the dipole coupling
is the only coupling which exists in the world.
And by going to higher order, I want
to show you that this is not the case.
Also I want to sort of give you an idea what
it means if you have leading order
transitions are forbidden.
I want to sort of show you how other terms come
in which can couple two levels.
And also, actually, when you drive transitions
within the hyper-fine structure using radio frequency fields,
you are not driving them with the electric dipole
approximation.
You're driving them with a magnetic dipole,
and this is actually the next approximation.
So there's a number of reasons why
I want to show you what the next order terms are,
and how they actually lead to beautiful result,
magnetic dipole, and electric quadrupole transitions.
So our coupling term-- let me just rewrite the equation.
Let me now simplify the notation by assuming
that the polarization is in the z direction
and the propagation is in the x direction.
So then the coupling had the dipole term.
And the next order term is ikx, and this
is what we want to investigate now.
So the term kx or kr is smaller.
And now I want to show you explicitly
that it's smaller by alpha.
The k vector of the photon is related to the frequency H bar
omega divided by H bar z.
If I approximate for r, the relevant r
when we indicate over the wave function
will be the Bohr radius.
The relevant frequency, H bar omega, well, it's a Rydberg.
And the Rydberg is e square over the Bohr radius.
So now if I insert it, the Bohr radius cancels out.
And what we find is, well, kr is dimensionless.
We have to find something which is dimensionless,
but expressed by the fundamental constants of atomic physics.
And the only quantity which is available for that-- it's not
a surprise-- is alpha.
The fine-structure constant.
So therefore the dipole approximation
is actually the result of an expansion of the plane wave
e the ikr factor in units of expansion
in the fine-structure constant.
OK.
We want to look now at the second term.
And well, sometimes if you deal with a term,
we first make it more complicated
and then we simplify it.
I want to sort of symmetrize and anti-symmetrize it
in the following way.
This is p z x.
So let me subtract z p x.
But then, of course, I have to add it.
So now we have two terms.
1 has a minus sign, one has a plus sign.
And as will see in just a minute,
is the first one is magnetic dipole transition.
The second one is electric quadrupole transition.
And we see that the first one can be regarded as-- p
z x is like p dot r.
Its the y component of the vector product and
is therefore the y component of the orbital angular momentum.
OK.
So let's focus for now on this part.
The second term, which is the electric quadrupole term,
we'll do in a few moments.
The relevant matrix element is now the matrix element
of the orbital angular momentum Ly.
The p factor-- let me just collect
all of the constant imaginary unit ehr AK over 2 mc.
That looks complicated, but it immediately
simplifies when we realize that this here is the Bohr magneton.
And well, we still have the vector potential.
But the magnetic field is the curl of the vector potential,
and that means we will assume that we have a vector
potential propagating in x, polarized in z.
That means that our magnetic field is that.
So therefore ka, which appears in our expression
for the coupling, is just the magnetic field.
So therefore we find the result that the next order
term in the coupling of the atoms
to the electromagnetic field have the simple form
to the electromagnetic field.
That it is the magnetic field part
of the electromagnetic wave times the Bohr magneton
times the matrix element, due to the orbital angular momentum
operator.
And actually, if you take the orbital angular momentum
and multiply it with the Bohr magneton,
this is actually the operator of the magnetic moment.
Well, with a minus sign because the electron is negative
charged.
Remember we had introduced the operator
for the magnetic moment.
And the magnetic moment was the g factor
times the Bohr magneton times the orbital angular momentum.
And the g factor for the orbital motion is 1.
So therefore the interaction we are talking about
is the Bohr magneton times the magnetic field.
And, of course, what we realize is the operator-- maybe
I should back up for a second and say what we actually
realize is that operator which couples
to the electromagnetic field has actually the form u dot b.
This is exactly what we use for the Zeeman effect in a DC
magnetic field.
But now the same form u dot b appears
for time dependent magnetic field.
And time depending magnetic fields
are not only creating level shifts.
They can also use transitions through the matrix element.
So in other words, this whole exercise
show you that the form u dot b, which appeared naturally
in the formulation for DC magnetic field,
also applies to AC magnetic fields.
But with that, I can say, wait a moment.
There are now two sources for the magnetic movement
of the atom.
One is due to orbital angular momentum,
and the other one is due to spin angular momentum.
But the spin angular momentum has
a g factor, which is different.
The approximation of the [INAUDIBLE] creation is 2.
So therefore I mean, we will never
get spin out of a semi-classical discussion.
Remember, we started with a classical canonical treatment
of the electromagnetic field, canonical momentum.
And now we are running with it and we
find that there is a coupling between the magnetic field
and the magnetic moment of the atom.
But, of course, we only get the magnetic moment to the extent
that it comes from orbital motion.
But in a semi-classical way, I'm waving my hands now and say,
well, what is valid for the coupling
to orbit to the magnetic moment of the orbital angular momentum
also applies to the speed.
And I'm simply adding the spin here.
And with that, I've derived for you
the expression for the interaction matrix element
called M1.
M1 is the magnetic dipole transitions.
Let me just write it down here.
So this is nothing else than u dot b.
Any questions?
Let me just summarize.
You may find it sort of interesting,
when we discussed static electric and static magnetic
fields.
For the static electric field, we
had an electric field times the dipole.
And we find this now as a time independent term
which can drive transitions for interactions with time
dependent electromagnetic field, but we find it
in the dipole approximation.
The magnetic part, u dot b, we find
when we go to the next order.
We find it as a magnetic dipole term.
But there are more terms, and I just
want you to illustrate that and then
I'll stop with that multiple expansion.
We had the second term, the kr term, was this one.
But then we sort of anti-symmetrized it
and symmetrized it.
The first one here, we could relate to orbital angular
momentum and to the magnetic moment.
And now I would want to discuss the second term.
So that term uses a mixture of position and momentum
operators.
But we know already how we can get rid of momentum operators,
namely by expressing momentum operators as commutators
with the Hamiltonian.
So this gives us the commutator of z with H naught times x
plus z times the commutator of x with H naught.
OK.
So we have two commutators.
Each of them has two terms.
That means a total of four terms.
And if you write them down, you see
that two are opposite but equal and cancel.
So therefore we are left with two terms, which
are minus H naught zx plus zx times H naught.
So this term has now the following contribution
to the coupling between the levels A and B.
So we have this coupling matrix element.
We have b factor mc Am.
OK.
So using the same approach we used for the dipole matrix
element, the H naught can act on the state A on the right side,
and can act on the state B on the left hand side.
So this gives us simply the energy difference.
And what is left is now a matrix element
which only involves position operators,
but it uses a product of 2.
And now, again, we express the vector potential
by the electric field, as we had done before.
And therefore we have now expressed our coupling yes,
by an electric field.
And we assume that in the near resonant approximation,
as we had used for the dipole coupling,
that this is the frequency omega.
OK.
So what we are realizing is that we
have one part of the interaction Hamiltonian
which couples levels A and B, has--
we call it E2, this electric quadrupole
because it involves elements of the quadrupole tensor,
or products of coordinates x, y, y, z, x, z.
And for this geometry, which we assume with our plane wave,
it is z, x.
It couples to the electric field.
And this is the p factor.
For a more general geometry of plane waves
with different polarization going in different directions,
we would have obtained different products of coordinates.
So let me indicate that, that what we have picked out here
is one specific component of a tensor, which
is sort of the tensor form by using the position vector
r twice.
So in this derivation, we found when
we go beyond the dipole approximation,
when we take the kr term in leading order, that we
have two contributions.
One is M1, magnetic dipole.
The other one was electric quadrupole.
We realized-- I tried to keep of all the p factors--
that the electric quadrupole matrix element is imaginary,
whereas the magnetic dipole was u dot b.
There was no imaginary unit.
It's real.
That means that you will never have any interference
effect between magnetic dipole and electric quadrupole.
Or in other words, when we have processes
like spontaneous emission where we
take the square of the matrix element,
the square of the matrix element will
be the sum of the squares of the matrix element
for magnetic dipole and electric quadrupole transitions.
So let me summarize.
We have discussed three different ways
how we can have coupling matrix elements between two
states, electric dipole, magnetic dipole,
electric quadrupole, E1, M1, E2.
The operator was the electric dipole operator.
Here it was the operator of the magnetic movement, which
is orbital angular momentum and spin angular momentum.
And for the quadrupole, it was the quadratic expression
in the spatial coordinates.
You can also ask what is the parity?
The electric dipole operator connects
states with opposite parity, whereas both magnetic dipole
and electric quadrupole connect states
even with the same parity.
Magnetic dipole and electric quadrupole transition
are often called forbidden transitions.
Well, you would say it's a misnomer because they
are transitions, so they are allowed, but they are weak.
But this is the language we use.
Weak transitions are forbidden, which
simply means they don't appear.
They are forbidden in leading order,
but when you go to higher order, they are allowed.
You can, of course, say if they were completely forbidden,
there would be no need to discuss them.
But since they are only forbidden
at a certain level, then, of course,
it's interesting to discuss them.
And a lot of narrow transitions which
are relevant for atomic clocks are highly forbidden
transitions.
The strength of them, which usually scales with transitions
with the square of the matrix element,
is on the order of 5 times 10 to the minus 5.
So those transitions are four or five orders
of magnitude weaker Yeah than an allowed E1 transition.
So that's what they are.
Questions?
AUDIENCE: When people talk about highly forbidden transitions,
does this mean that's like a optical transition, or how
do we distinguish it from just regularly forbidden?
PROFESSOR: Actually I would say forbidden transitions
are weaker by alpha to the power n.
Here we have situations where the matrix
element is just smaller by alpha.
But, yes.
Sometimes, yes.
Some transitions are highly forbidden.
For instance-- I try to remember--
if you have hydrogen 1s and 2s.
Because of s states-- actually, you
are asking question about the next chapter, namely about
selection rules.
Let me give it in words.
The s, if you connect two s states,
they have both 0 angular momentum.
AUDIENCE: Yeah.
PROFESSOR: So you cannot have a quadrupole operator connecting
the two.
It would violate the triangle rule.
You cannot have angular momentum of 0 and angular momentum of 2
and get angular momentum of 0.
So therefore you have a transition
between two states of the same parity.
There is no dipole operator.
There is no quadrupole operator.
So you soon run into a situation where it's highly forbidden.
Sometimes you have the situation that something is forbidden
in non-relativistic physics, but they're
a relativistic term, which makes it allowed.
Well, there's also that relativistic terms
are fine-structure terms is also an expansion alpha squared.
So the symmetry may allow it, but only
in connection with relativistic terms.
I'm not an expert on forbidden transitions,
but usually you have transitions which are multiply forbidden.
They are forbidden by spacial symmetry.
For instance, in the helium atom,
singular triplet are forbidden by spin symmetry.
So if you have multiple layers of being forbidden,
then you get extremely weak transitions.
And one example, actually, are the singular triplet transition
in helium, or the 1s to 2s transition in hydrogen.
They are not allowed by simply going
to the next order in a multiple expansion.
Yes?
AUDIENCE: In ordinary situations when
you do have to possibly do at least even more [INAUDIBLE]
for the other transitions?
PROFESSOR: I'm not an expert on that.
I'm not sure if there is an atom which
has a relevant transition which is a [INAUDIBLE] transition.
I've not heard about that.
At least relevant examples which are the fundamental atoms,
helium and hydrogen.
For hydrogen, at the 2s to 1 transition,
the leading order is the emission,
the simultaneous emission of two photons.
So you have not just one photon, you
have two photons which, of course, requires
in the perturbation expansion the immediate step.
We discuss two photon transitions
at the end of this course.
So in that case, it's not in higher order
in the single photon multiple expansion,
it becomes a multi-photon transition.
So this is one relevant case.
And for helium, triplet to singlet,
this involves relativistic physics.
Actually, I mentioned it in the other class on helium.
I tried to look it up and I want to show here,
you have to go to this order to get a transition.
But when I tried to look into the literature,
I couldn't find a clear answer.
Ultimately, it was a relativistic term
in the fully relativistic formulation
of the coupling of electromagnetic fields
to the atom.
I'm not sure if you can put a label on it and it would say,
this is this and this order term.
It may actually involve-- and we know
that this happens in the [INAUDIBLE] creation--
that spin and spacial degrees become treated together
in the [INAUDIBLE] creation, and maybe it's one of those terms.
Yes?
AUDIENCE: So what's the actual meaning
of interacting with different Hamiltonians?
Like, you wrote square root of Hamiltonians and then--
PROFESSOR: No.
I meant actually the square of the matrix element.
AUDIENCE: Oh.
Sorry.
PROFESSOR: If you have, for instance-- actually
I will talk about it in the next lecture when
we discuss Fermi's golden rule.
The transition strength in Fermi's golden rule
is proportionate to the square of the matrix element.
But now you would ask the question, well,
could we have some interference between the two
different processes?
And I wanted to point out at this level
that we don't, because one matrix element is imaginary,
the other one is real.
And if you take the complex matrix element between two
states and calculate the square, the square
of the complex matrix element is the square of the real part
plus the square of the imaginary part,
and there's no interference trend between the two.
So in other words, if you have an atom which has a weak decay
through M1 and a weak decay through quadrupole,
the two parts cannot destructively interfere,
because one is real, one is imaginary.
They add up in quadrature, That's what I wanted to say.
OK.
So these were examples of higher order transitions.
And as the questions have shown, 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 of using well, [INAUDIBLE] coefficients,
angular momentum coupling, or using, you would say,
symmetry to figure out if 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 0 angular momentum,
you cannot have an operator which is a quadrupole operator
because-- and this is what I want to tell you now--
the quadrupole 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
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
the dipole transition.
This is the dominant transition.
This is the industrial strengths transition.
And from there on, it can get weaker.
So if can get weaker by multiple expansion.
We've just discussed that.
It can get weaker because you have
to have a cascade of dipole transitions.
This would be multi-photon processes,
as we discuss later in the course.
It can get weaker because they are exactly
0 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 there are transitions which would not be allowed just
for the electron, but if we invoke hyperfine interactions
with a 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 want to at least give you some general rules how
we discussed matrix elements.
So what is always a good quantum number, what is always
a label for our atomic states is angular momentum.
Because atoms [INAUDIBLE] through space,
and there is rotation in variance.
So we always categorize our atoms with angular momentum
with the 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 quadrupole 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.
Our matrix elements involving components,
the operator are components of a spherical tensor.
So T is a spherical tensor of rank l.
And if you want a simple definition
of 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 a sum of spherical tensors.
And a 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 into symmetry classification,
but the story is that you know the Ylm functions are
compensator functions.
Every function can be expanded and swell to harmonics.
And similarly, if you have an operator,
you can decompose it into objects
which transform under rotation, and it's the Ylms.
So you have a part which transforms
according to an object.
And Ylm is the 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 a 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
in angular momentum.
If you have a matrix element-- and just
think how you calculate it in the Schrodinger representation.
You have a wave function, operator,
and a wave function we indicate go by.
So what you have is, you have the product of the objects.
And now we can classify them by angular momentum.
And this is what I will show you in the next class,
is that ultimately the question whether this matrix element is
0 or not will boil down to the question
whether J prime and M 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 momentum.
We get the triangle rule.
We get the [INAUDIBLE] quote unquote [INAUDIBLE].
OK.
Enjoy the spring break, and we meet on Monday
after the spring break.