M1L2c.txt

#
# File:   content-mit-8-421-1x-subtitles/M1L2c.txt
#
# Captions for 8.421x module
#
# This file has 95 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------

So we want to go back to our classical magnetic moment,
and understand the motion of it.
But in addition to what we discussed so far,
a stationary magnetic field, we're
now going to add a rotating magnetic field.

So we have a magnetic moment.
We assume it's classical.
And we have a magnetic field, which we
assume points along the z-axis.
And then, of course, we know that--
from our previous discussion-- that the spin
undergoes precession.
It sort of precesses around the magnetic field
at the Larmor frequency.
And let's assume we have a rotating field.
We add a rotating field, B1.
And just to keep things simple, we
want to assume that the rotating field rotates
at the same frequency as a magnetic moment,
so we are on a resonance.
Because what happens now is we can simply do a transformation
to the rotating frame.

The rotating frame is now at the Larmor frequency.
And we've just learned that in the rotating frame-- of course,
in the rotating frame, the rotating field stands still.
So it becomes a static field which
points in the x direction.
And we have the static field in the z direction.
But now that's what I just reviewed.
We have a fictitious magnetic field,
which comes from the transformation to the rotating
frame.
And on resonance at the Larmor frequency,
this is exactly the negative of B0.

So in other words, we started out
to expose a magnetic moment to a time dependent field.

So we had a rotating field cosine omega L t sine omega L
t.

But in the rotating frame, this becomes now the x prime-axis,
and it's stationary in the rotating frame.

So this was the field in the lab frame.
In the rotating frame, we have an effective field,
which is the field in the lab frame
minus the fictitious field.
And the fictitious field was given by that.
It just cancels the B0 component.
And in the rotating frame, this vector and this vector
cancels, and we're just left with the field of strength
B1, which points in the x-axis.

Well, actually the x-axis in the rotating frame
is what I called x prime.
So therefore we have a very simple problem
that in the rotating frame we have
a static field of value B1.
And the good thing is we know already what a magnetic moment
does with a static field.
The magnetic moment is just precessing
around the static magnetic field.
So therefore our solution is now we have transformed away
the time-dependent field, and now we
know that mu precesses around this field,
and the precession frequency is the Rabi frequency, which
we discussed previously.

The Rabi frequency is the gyromagnetic ratio times B1.
So therefore if you would start out
with a magnetic moment aligned with the z-axis,
if you would wait half a Rabi cycle,
the magnetic moment would now be inverted.

So the situation is we have a magnetic moment which
points in the z-axis with a field in the z-axis.
But we expose it to a rotating field
which rotates in the xy-plane.
And the stationary frame-- so in the rotating frame,
we have a stationary field which points in x.
In this rotating frame, the spin is simply
precessing around what is now a steady field,
and after half a Rabi cycle, the spin points down.
So therefore we know, going back to the lab frame,
that this rotating field has caused--
in quantum mechanics I would say a spin flip-- a full reversal
of the magnetic moment.
And this is what we call a pi pulse.
It has rotated the spin by pi.
Or we call it a spin flip.
But it's a completely classical system.