M1L3d.txt

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Let's just use for a moment the result
we got from Heisenberg's equation of motion.
We know the classical result. We've
already derived for the classical spin,
the classical result for the expectation
value of the magnetic moment.
But now I want to sort of relate it
to something quantum mechanical because we
know that the classical solution equals
a quantum mechanical solution.
So, if you have a two-level system,
the z component of the magnetic moment
is the difference between spin down and spin up.
And because of conversation, probability, p up
and p down is unity, we can also write that as 2 times--
and let me now introduce e for excited state-- just,
I know it's hard to keep track of spin up and spin down--
I want to make sure that I mean now, the excited state.
So the excited state for the electron is spin up.
So we have this condition.
So therefore, the excited state friction of a two-level system
is related to the expectation value of the magnetic moment
in that way.
And now we want to use the classical result we derived.
We derived the classical result, when for t equals 0,
all the spins were in the ground state.
And by using the result we would derive previously,
we have the 1/2 from the previous line,
and then the magnetic moment mu z,
we found an expression which involved
the Rabi frequency and the off resonant Rabi frequency-- Sine
squared generalized Rabi frequency times time over 2.
So the two factors of one half and one half cancel,
and what we find now for the quantum mechanical system using
Heisenberg's equation of motion, is
that if you prepare a system initially
in the ground state, the excited state probability,
the fraction in the excited state oscillates
with the Rabi frequency.
And this is, I think, the second time in this course, and not
the last time that we obtain the Rabi transition probability.

OK.
So, but let's go further.
We have now discussed the classical spin.
We have sort of done classical-quantum correspondance
with Heisenberg's equation of motion.
We know that this, in generally, implies Rabi oscillations.