M1L3h.txt

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OK, so this is the famous Hamiltonian,
and of course, if it's the famous Hamiltonian,
we want to solve it.
As I said, the general solution is left to the homework,
but I want to sort of show you parts of the solution
to tell a story.
And the question is, well, how do we solve this Hamiltonian?
The answer is we do exactly as we
did in the classical problem.
We transform to the rotating frame.
In other words, this Hamiltonian is best
solved by doing-- you can actually solve it directly.
You can just put in a trial wave function and solve it.
But I want to sort of bring out the big idea here,
which is analogous to what we have
done in the last few classes.
Namely we have involved rotating frames.
So what solves this Hamiltonian is a unitary transformation,
and the unitary transformation is this one.

And so this unitary transformation,
it transforms the Hamiltonian to a time independent one.
We have now time independent off-diagonal matrix elements.
Our diagonal matrix elements have changed.
Delta is now the detuning of the drive
frequency from the energy splitting of the two level
system.
In particular, when we are on resonance,
the diagonal matrix elements have disappeared.
This is the result of the unitary transformation.

Let we just show you this transformation over here
can be actually written as an operator involving
the z component of the magnetic field.
And what I just wrote down for you is actually
the quantum mechanical operator, the rotation
operator for performing a rotation around the z-axis.
So by selecting the rotation angle to be omega t,
that's how I can generate the unitary transformation.
And this unitary transformation makes the Hamiltonian time
independent.
So in other words, everything is in the classical system.
We just go to a frame, which rotates with the Rabi
drive, and we find a time-independent problem.
So now this Hamiltonian can be easily solved.

And you will find as a special case
when you start with an amplitude,
initially you start in the ground state.
Then the excited state, amplitude squared,
the general is the Rabi oscillation, something
we discussed 40 minutes ago.

But before we got it from the classical quantum
mechanical correspondence using the Heisenberg equation
of motion, and here it comes out by explicitly
solving for the wave function for the dressed
atom Hamiltonian.