M1L1f.txt

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So with that, let me make the transition
to another simple system.
And we want to spend some time on it.
And these are rotating systems.

So a system which rotates, well what do you think,
will it behave using the discussion we
just, more a classical system or more
like a kind of mechanical system?

Of course, I gave you a very special definition
what brings out quantum mechanics in a system.

The harmonic oscillator is always linear.
You can drive it as hard as you want ,
you drive hundred times stronger,
and the reaction is hundred times more,
everything is linear.
The quantumness of a two level system
comes from saturation.

What about a rotating system?

Something which can go in a circle.
Sometimes hard to ask a simple question, without giving
the answer away, but what I had in mind
was the gyroscope, a gyroscope which is precessing.
And what I wanted to sort of lead you
with the question is, if you have something which rotates,
the amplitude is limited.
A rotating object, let's assume a magnetic classical magnetic
moment, it can have a precession angle,
which is 180 degrees, but that's a maximum.
In other words, the rotating system when you excite it,
has a maximum amplitude.
Exactly as a two-level system.
So that's what a rotating system and a two-level system--
but could actually then-- I'm now
specializing on more rotating gyroscope.
If you have a free rotator, this of course
can rotate with every type of angular momentum
and the excitation spectrum would not be bound.
So I think I have to rephrase the question the next time I
teach the class, I wanted to ask you here
about a special rotating system, which
was a processing gyroscope.
So rotating system, if you think about, processing gyroscope,
it has a bound
on the amplitude it can be excited.
So what I want to show you today,
and in the next lecture is, that the motion
of classical magnetic moments.

When you think about the motion of classical magnetic moments,
think about a compass needle, a magnetized needle,
which is angular momentum.

And the system is acted upon with a magnetic field,
so this is our system.
And if angular momentum, magnetic moments, it
all come into play, we have the physics of classical rotation.
But the excitation spectrum here is limited,
because you can flip a compass needle
and this is a [? maximum ?] excitation.
When the North Pole points in the opposite direction, that's
the maximum excitation you can give it.
So therefore, it has a limited amplitude of its excitation,
unlike a harmonic oscillator.
And at this point, you may say, but maybe
somewhat similar or analogous to a two-level system.

But the surprisingly result is, at least
it was surprising when I first learned about it, that it's not
just somewhat analogous to a two-level system,
it actually kept just exactly a lot
of the properties of the dynamics of a two-level system.
So you can write that down.
The motion of classical magnetic moments provides a model,
it's actually an exact model, which
captures essentially all features
of the quantum mechanical two-level system.
So I want to show you today and the next lecture
that concepts of Rabi] frequency,
of generalized off resonant Rabi frequency,
that all of that, you find in the classical motion
of a magnetic moment.
Or for instance, the physics of Rapid Adiabatic Following
Landau-Zener crossing.
A lot of physics, we would usually
associate with a quantum system,
you find it here in a purely classical system.
What aspects of the two-level system will we not find?
Any ideas?
Spontaneous emission definitely, yes.
But actually in a two-level system,
in a quantum mechanical two-level system,
which we drive with a single frequency,
spontaneous emission is also missing.
Spontaneous emission as we will discuss later, only
comes into play when see the excited state of the system
enter x with many, many modes, and not just the one mode
we apply.
And typically, if you go to high frequency,
we have an optical oscillator we cannot avoid spontaneous
emission.
Whereas for a quantum mechanical spin one half
into acting with microwaves, we can completely
eliminate spontaneous emission.
So spontaneous emission I would say
comes into play at high frequency.
So that's correct.
But there is one aspect, even at low frequency,
one aspect of quantum mechanics which we cannot capture.

[Student]: different g-factors in a magnetic field?
Differing g-factors, yes.
That's we see a more quantitative aspect,
but there's one very important feature about quantum mechanics
you will never get in a classical system.
[Student]: spin?
Spin.
[Stident]: Superposition

OK, I think you're skirting around.
It's a quantum measurement process
and projection.
If you perform a measurement on a compass needle,
it can be at any angle.
But if you do a measurement on a quantum system,
you do a projection.
After the measurement projects a system,
you either spin up or spin down.
So the probabilistic nature, the projection
occurring in a quantum measurement
is of course absent in a classic system.

But when you say, spin one half in quantum levels,
this is sort of implied in it, that there are only two levels.
There's only up and down and not an infinite number of angles.
So what we will actually see is that we
find an exact analogy between the classical system
and the kind of quantum mechanical system.
Then we compare expectation values.
But the individual measurement, the individual quantum
measurement, because it is projective is different.