M1L3b.txt

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We want to now talk about quantized spin
in a magnetic field.

And one of the first things I will be telling you
is that everything we have learned
about the classical magnetic moment,
you don't have to unlearn or relearn.
It exactly applies to the quantized-- to the quantum
spin.
Before we look at standard Hamiltonian
for two-level system, let us first
look at something more general, which is Heisenberg equations
of motion.
So we want to write down the differential
equation, the equation of motion for expectation values.

So for an atom in a magnetic field,
our Hamiltonian is simply the Zeeman Hamiltonian,
which involves the gyromagnetic ratio, the angular momentum
operator, and the magnetic field.
And you know from quantum mechanics
that Heisenberg's equation of motion for any operator
is simply that the time derivative of the operator
equals the commutator with the Hamiltonian.
In some cases, if you have an explicit time
derivative of the operator, you have to add it.
But we are talking here about the angular momentum
operator, which has no explicit time dependence.
So we are interested in the operator which
describes the magnetic moment.
But the magnetic moment is nothing else
than the gyromagnetic ratio times the angular momentum.
So therefore, the relevant commutator
is the commutator of the Hamiltonian with the angular
momentum operator.
And just to remind you, the Hamiltonian
was proportional to Lz.
So what we are talking about is commutators, not surprisingly,
between the angular momentum operators.
And those commutators are just this cyclic permutation
involving the epsilon tensor.
So if you just put that in component by component,
you realize immediately that the operator equation for the time
derivative of the magnetic moment
is nothing else than the cross product, the vector product,
of the operator of the magnetic moment
times the magnetic field.
Two comments.
This is exact, but also it looks exactly
like the classical result. Where you
would say it's an operator equation, usually
operator equation some of them are pretty useless,
because you can't calculate the operators.
But in this case, we can immediately
take the expectation value, so we
can get some immediately meaningful equation,
namely that the expectation value follows
the same equation.
And this tells us that whenever we have a quantum system,
it has a magnetic moment.
In an applied external magnetic field,
the result is simply rotation, precession.
And this is exact.
It's not a classical approximation,
it's an exact result for quantum mechanics.
The way how we derived it makes it obvious
that it's an exact result which is validate
not only for spin 1/2, but it is valid for any spin.
If you have a magnetic moment corresponding
to a spin of 10 h bar, this spin follows the same equation
of motion as spin 1/2.
Of course, a special case is valid for spin 1/2.
But spin 1/2 is isomorphous to a two-level system.
Any two-level system can be regarded as spin 1/2 system.
Therefore, this geometric interpretation
that the dynamics of the quantum system
is just a precession rigorously, exactly
applies to any two-level system.

It's also valid-- and this will be relevant for atoms--
if you have composite angular momentum.

For instance, we will encounter the total angular momentum, F,
of an atom which has components from the orbital motion
of the electron, the spin of the electron,
and the spin of the nucleus.
But if you have such an angular momentum F,
the equation of motion is it will precess around
a magnetic field.
Well the small print here is unless the B field is so strong
that it decouples the components,
or that it breaks up the coupling of the different parts
of the angular momentum.
In other words, we have simply assumed here
that angular momentum -- a magnetic moment is
gamma times angular momentum.
And that requires that the angular momenta
are coupled in a certain way.
And if you don't fully understand
what coupling of angular momenta is,
we'll really talk about that when
we talk about atomic structure.
So as long as the spins stay coupled to one total spin,
this total spin will just precess.
This picture of precession will also
be valid for a system of N two-level systems
coupled to an external field.
And this will be the example of Dicke superradiance, which
we will discuss towards the end of the course.

OK, so very simple result, but very powerful.
And this is your permission, whenever
you encounter any of the systems,
to see a vector precessing in your head.
This is exact.

So we've talked about Heisenberg equation of motion
for general spin.
But you have a question, Nancy?
What did you mean by N two-level systems here?
Are we talking about correlated systems
or non-correlated systems?

We talk about N two-level systems.
And to be more specific, the coupling
comes because they all talk to the same electromagnetic field.
So we have N two-level systems connected to the modes
of the electromagnetic field.
We start with this symmetric state.
The coupling is symmetric.
And that preserves the symmetry of the atomic state.
In other words, we will have a situation where the angular
momentum is the maximum angular momentum we
can get of N two-level system.
And the dynamics of this two-level system,
the description of Dicke superradiance,
has this geometric visualization of this precessing motion.

I know I'm not explaining it exactly.
It's more I want to sort of wet your appetite for what
comes later, and also sort of prep you
that some of the simple pictures will really carry through the course.