M1L2a.txt

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We're spending quite some time in class just to figure out
that something is rotating.
So what we've figured out here is
that the effect of a rotating frame
is to simply add a fictitious magnetic field
to the real magnetic field.
And that's very handy.
For instance, we know that immediately if you have a real
magnetic field, and we pick the fictitious magnetic field
in such a way that the real magnetic field,
that the total field cancels-- that it cancels the total
magnetic field--
Well, then we have a system viewed from the rotating frame
where the effective field is zero.
Well, a spin of zero field does nothing,
and then we know when we go back into the lab frame,
all of the spin is doing, it rotates.
It's an exact solution.
I want to use it today again, but in a different way.
But I want to just point it out, today we're
not picking the fictitious field to make the magnetic field
zero.
Today we're talking about what happens
to a spin in a rotating field.
Complicated.
A time-dependent problem, time-dependent Hamiltonian.
But if you go in a rotating frame
and we rotate with the field, then in the rotating frame
it becomes a time independent problem,
which we can immediately solve.
So today we use the same transformation again,
but we pick our frequency not to cancel some static field,
we pick our frequency to co-rotate
with an external rotating field that in our rotating frame now,
everything is stationary.
So that's what you want to do today.
And finally, just to give you an outlook,
just try to of make you aware that often what we're doing
is the same thing in a different angle.
So this is one way to deal with rotation.
I give you an exact solution by going to rotating frame.
I will later show you today that quantum mechanically,
the solution to the Heisenberg equation of motion
for an angular momentum, or magnetic moment
in a magnetic field is exactly a rotation.
So I will again show you that a quantum mechanical solution,
the solution of a time-dependent problem is exactly rotation.
And later on, when he use Hamiltonian,
when we use the spin 1/2 Hamiltonian,
and we write down the wave function, we solve,
sometimes, the wave function by transforming the wave function.
And this will be, again, the rotating frame transformation.
It's not always called like this, but it's always the same.
You go to some form of rotating frame,
and we'll do that in three different ways.
This is the first way, just as a general classical physics
transformation to rotating frame.
We will do it again for the expectation
value as a solution of Heisenberg's equation
of motion, and then we do it again
when we transform the wave function
with a unitary transformation.

A lot of time for simple rotations,
but it's a good thing.
It really provides a lot of insight.