M1L2e.txt

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So with that, we have our Hamiltonian.
And a lot of what we discuss in this course,
understanding this Hamiltonian, understanding its solution,
understanding what is the physics described by that.
So this Hamiltonian has, this is the kinetic energy,
the canonical momentum minus the vector potential
is the mechanical velocity.
This is kinetic energy.
And we have separated the energy of the electromagnetic field
into the Coulomb energy, which is written against here,
again here, and the radiation field.
There is now one more term which we need later on, which
is this one here.
And at this point, I would say I'm just heuristically adding
it by hand.
This is now-- well, [? classically ?]
we don't have spin.
But our particles have spin.
And now we need [AUDIO OUT] how does this spin compare
to the rest of the [? world? ?]
Well, the spin, if we multiplied with the [INAUDIBLE]
on the [INAUDIBLE] a magnetic moment.
And what we simply add here is mu.B,
the interaction of a magnetic moment with the magnetic field.
So in that since we have been classical all the way.
Now at the end said, OK, we need the spin.
Let's put it on by simply taking an interaction which is mu.B.
If you don't like that because I try
to be very fundamental today and go from first principles,
for electrons you can get that by taking
the nonrelativistic limit of the [? Dirac ?] equation.
So if you have electrons, you can
start with the Dirac equation and do the [INAUDIBLE]
approximation, which is then non-relativistic limit
of the Dirac equation.
Any question?
So these are all the terms in the Hamiltonian.
I mentioned, but I don't want to dwell on
that the Coulomb energy has to be separated into a Coulomb
self energy and an interaction energy
between the charged particles.
But all of that becomes just one term in our Hamiltonian.
This is the atomic structure.
When we assume that atoms have energy levels,
all the Coulomb energy is included in that.
So we will not discuss any further the Coulomb energy.
We will simply assume we have an atom, which
has certain energy levels.
And that includes all the Coulomb terms.
But we will talk a lot about the Hamiltonian
for the radiation field.
The Hamiltonian for the radiation field can be
conveniently written in a and a [? dega. ?] But I have a few
equations up there I rigorously defined for you, operators,
E and B, in terms of a and a [? degas, ?] and those two
equations are identical.
So this looks very [? quantum, ?]
this looks very classical.
But if you interpret the electric and magnetic field
as operators, we have identical equations for the Hamiltonian
of the radiation field.

What you want to study are interactions
between light and atoms.
And of course, this comes from the vector potential.
And in particular when we square it out,
the canonical momentum of the atoms times the vector
potential has this [? cost ?] term,
[? P.A. ?] So let me now just take this Hamiltonian and write
it in the way how we will need it throughout this course.
So we want to take the Hamiltonian,
and I will often refer to that.
We want to take the Hamiltonian and split it
into three parts-- the atoms, the radiation field,
and the interaction between the two.
The Hamiltonian for the particles, H subscript p,
has the momentum squared, and it has
the part of the electromagnetic fields which are longitudinal,
the energy of which can be described by a Coulomb
[INAUDIBLE].

We have already discussed the radiation field.
The radiation field is nothing else than H [? by ?] omega
[? A ?] dega [? A. ?] But the new part, which we need now,
is the interaction term.
And I want to show you now or remind you
by just summarizing the term that the interaction part has
actually three different terms.

The first one is the [? cost ?] term between P and A.
The second one, when we hit P minus A and squared it is
[? CE ?] square term.
And the third one is the interaction
of the spin with the magnetic component of the radiation
field.
This is the mu.B interaction.

I don't think it's an exaggeration when I tell you
that with those three equations, you can understand
all of atomic physics.
This term here that's important is second order in A,
which of course also means it will be very important for very
strong [? laser ?] [? policies. ?] But there is