M1L1j.txt

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We have only a few minutes left today,
and I want to spend those few minutes to talk
about another factor of two.
Now, let me ask you the following.
If you have an electron in the magnetic field-- well,
you know that the electron goes in circles.

It's the cyclotron motion of the electron.
Now, for any classical charge distribution,
the Larmor frequency is the charge
of the particle-- in case of the electron, it's e-- divided
by 2m times b.
So the Larmor frequency is e over 2m times b.
So therefore we know that when we
have an ensemble of positive and negative charges,
and there is an effective magnetic moment,
that this magnetic moment would precess
at the Larmor frequency, which is given by this expression.
So who knows what the frequency of the cyclotron motion is?

So when we have a free electron, what
is the frequency at which it goes in circles?
It's two times the Larmor frequency.

OK, I just wanted to mention it--
there's an important factor of two,
which you should know about.
In previous classes I spent 10 or 20 minutes
to teach you about a theorem, which
is called Larmor's theorem, but I summarized the argument
on the atomic physics wiki, and I can't say more here in class
than I've written on the wiki.
So please read on our atomic physics wiki
about Larmor's theorem.
Larmor's theorem shows you that under certain assumptions,
you can transform away the effect of a magnetic field
by going to the Larmor frequency.

That looks exactly like what we have discussed here,
but what we discussed here was exact.
There no approximation, whereas the derivation
of Larmor's theorem, which talks about charge distributions,
not about magnetic moments-- charge distributions--
has to make certain approximations.
So just wanted to point your attention
to that there are two derivations about Larmor frequency.
One is exact, which I gave to you.
There IS another one, which is Larmor's theorem, which
applies to isolated charges, which is not exact.
But they both conclude that you can transform away
the effect of a magnetic field by going to rotating
frame at the Larmor frequency.
And the fact that Larmor's theorem is not exact
is actually illustrated by this example of a free electron,
where you have a factor of two.
And this comes because the term which
you neglect when you derive Larmor's theorem
is negligible if the situation is
that you have electrons and charges forming
magnetic moments.
But if you have a free electron, the neglected term
is exactly 1/2 of the dominant term,
and that is why the cyclotron frequency is twice
the Larmor frequency.
So never confuse the cyclotron and the Larmor frequency.
And the factor of two is not related to a G
factor of the electron or such.
It's really the difference between the physics
of a free charge and the physics of a magnetic moment.