PROFESSOR: All right, so
our next thing will be
getting into the electron spin.
And I want to discuss of
two things, a little bit
of the Dirac equation
and the Pauli
equation for the electron.
We need to understand
electron a little better
and understand
the perturbations,
the relativistic
corrections so we'll
consider the Pauli equation.
So the Pauli equation is
what we have to do now.
So what is the Pauli equation?
It is the first
attempt there was
to figure out how the
Schrodinger equation should
be tailored for an electron.
So we can also think
of the Pauli equation
as a baby version of the
Dirac equation, which
is the complete equation
for the electron.
So it all begins
at the motivations
by considering what is the
magnetic moment of an electron?
So it's a question that
you've been addressing
in this course for a while,
from Stern-Gerlach experiment
and how magnetic fields
interact with electron.
So classically and
in Gaussian units,
if you have a current
loop and an area a,
the magnetic moment is
equal to i times the area
vector divided by c.
You can use that to imagine
an object that is rotating.
And as the object
rotates, if its charged,
it generates a magnetic
moment, and the magnetic moment
depends on the amount of
rotation of the orbit.
So it turns out that, with
a little classical argument,
you can derive that the
magnetic moment is related
to the angular momentum by
a relation of the form L,
so the charge of the object,
the mass of the object c,
and the angular momentum.
And that's perfectly
correct classically.
So people thought that
quantum mechanically
they expect that for an electron
or for an elementary particle,
you would get q times 2 mc
times the spin of the particle.
So this would be the intrinsic
magnetic moment of an electron
that it seems to have
times a fudge factor, g
that you would have to measure.
Because after all, the
world is not classical.
There's no reason why a
classical relation like that
would give you the right exact
value of the magnetic moment
of the electron.
As it turns out, this g
happens to be equal to 2.
And we get the following.
So for electrons g
happens to be equal to 2.
So mu is equal to 2 times minus
e-- the charge of the electron
is minus 2 mc, and the
spin of the electron
is h bar over 2 times
sigma, the Pauli matrices.
So this thing is minus
eh bar over 2 mc sigma
mu of the electron.
So if you have a magnetic
field, a B external,
the Hamiltonian that
includes the energetics
of the magnetic
field interacting
with the dipole moment,
it's always minus mu dot b.
So in this case it would be
eh bar over 2 mc sigma dot b.
That's the Hamiltonian
for an electron
in a magnetic field, something
you've used many, many times.
The most unpleasant part of
this thing is this g equal 2.
Why did it turn out
to be g equal 2?
People knew it was g equal 2.
But why?
And here we're going to see the
beginning of an explanation.
In fact, it comes very close
to an explanation of that
through the Pauli equation.
The Pauli equation is a
nice equation you can write,
and you can motivate.
And it predicts.
Once you admit the
Pauli equation,
it predicts that g
would be equal to 2.
And that, of course, happens
also for the Dirac equation.
And Dirac noticed that, and
he was very happy about it.
So it's quite remarkable
how these things showed up,
and the G equal to 2
was perfectly natural.
So let's see this
Pauli equation first.
What is it?
Well, when you do the
Schrodinger equation for a wave
function, you say
p squared over 2m--
say a free particle,
and you would
put the wave function is equal
to energies times the wave
function.
h on the wave function is
equal to energy terms the wave
function.
So for this
electron, you already
know it's a two-dimensional
Hilbert space, the state
space of the up and down.
So it's convenient to change
this and to say, you know what?
I'm going to put here
something I'll call a spinor.
Chi is a Pauli.
It has two things, a chi 1 and a
chi 2, and it's a Pauli spinor.
And this is reasonable so far.
The spin is defined by 2 degrees
of freedom, 2 basis is vectors.
So you've done wave
functions for spin.
And the wave functions for
spin have these things,
and each themself can be
a function of position.
So that's perfectly reasonable.
I don't think anyone of you is
very impressed by this so far.
But here comes the funny thing.
In a sense here, there is a 2 by
2 identity matrix sitting here.
So Pauli observes the
following identity.
If you have sigma dot
8 and sigma dot b,
it's equal to a dot b times 1
plus i sigma plot a cross b.
Look, these are two
vectors, a and b.
And if you multiply half
this product-- first,
this is a dot product.
It means sigma 1 times a1 sigma
2 times a2 sigma 3 times a3.
So this is a dot product.
This product here
is a matrix product
because this is
already a matrix,
this is already a matrix,
so this is a matrix product.
And this is the
result. This comes
from properties of the Pauli
matrices you already know.
In particular, their
commutator determines
this piece and
the anticommutator
determines this piece.
So here, if you
have, for example,
sigma dot p times sigma dot p--
see, these two vectors are
the same in this case--
you'll get just p
squared times 1 plus 0.
Because p times P, those are two
vector operators if you wish,
but still they commute.
So this is equal to 0.
So your p squared can be
replaced by sigma dot p sigma
dot p.
So h, the Hamiltonian-- that
is p squared over 2 m times
the identity matrix--
can be perhaps better
thought as sigma
dot p times sigma
dot p divided by 2m.
So this is the first step.
You haven't done
really much, but you've
rewritten the Hamiltonian
with a unit matrix
here perhaps in a
somewhat provocative way.
In quantum mechanics, when we
couple to electromagnetism,
there are simple
changes we have to do.
And we will study that in
detail in about three weeks,
but today I will
just bench on what
you're supposed to do in order
to couple to electromagnetism.
In order to couple
to electromagnetism,
you're supposed to
change p wherever you see
by an object you can call pi.
It's some sort of more
canonical momentum
in which it's got
to p minus q over c.
The vector potential
is a function of x.
Well, people write it
like this, qp minus q ac.
So you're supposed to
do this replacement.
When you're dealing
with a particle moving
in some electromagnetic field,
that's the change you must do.
We will study that in detail.
But I think an obvious
question at this moment
is the following.
You say, well, I'm
in quantum mechanics,
and I work with p and x.
Those are my opera-- what
is a? is it an operator?
Is it a vector?
Is it what? p is an
operator, but what is a?
You should think of
this vector potential.
In general, this
vector potential
will depend on the position.
So if you put a magnetic field,
you require a vector potential.
It has some position dependent.
So actually, this
a here should be
thought as a of x,
the same way as when
you have the potential
that depends on x.
In quantum mechanics, we just
think of that x as an operator.
So the a is an operator
because x is an operator,
and a has x dependents.
So the Pauli Hamiltonian, h
Pauli is nothing else but sigma
dot pi times sigma
dot pi over 2 m.
Because we said p must
be replaced by pi,
so this is the
Pauli Hamiltonian.
And it's equal to 1
over 2m pi squared
pi dot pi times 1 plus i
sigma dotted with pi cross pi.
OK, here is the
computation we need to do.
What is pi cross pi?
You know what pi is.
It's given by this thing.
So how much is pi cross pi?
Well, it takes a
little computation.
I'll tell you what
you have to do.
It's a very interesting
computation.
It's small.
It reminds you of
this computation
in angular momentum, l cross l.
Maybe you've written the
algebra of angular momentum
in this language.
It's equal to h bar l.
That's your computation in
relations of angular momentum
written like that.
So pi cross pi,
the kth component
is epsilon ijk pi i
pi j or 1/2 epsilon
ijk pi I commutator with pi j.
This last step, it
can be explained
by writing out the
commutator, which
is pi i pi j minus pi j pi i.
But with this epsilon, those
two terms are the same,
and it becomes this.
So how much is this thing?
I'll leave it for you to do it.
It's simple thing.
You have to commute
these things,
and you must think
of p as derivatives.
So this pi i pi j is i h bar
q over c di a j minus dj a i.
You can see it coming.
You see, you have a commutator
of two factors like that.
The a with a will commute.
The p with p will commute.
The cross products won't,
but that's just derivatives.
So therefore, you
get the derivatives
of a antisymmetrized.
And the derivatives
of a antisymmetrized
is the curl of a.
And the curl of a is
the magnetic field.
So that's why this happens.
So at the end of the day, when
you'll finish this computation,
get that pi cross pi is
just i q i h bar q over c
times the magnetic field.
Pretty nice equation.
So the Pauli Hamiltonian
includes h Pauli.
It includes this term,
which is i sigma over 2m
dotted with i hQ over
cb plus the other terms.
I'm just looking at this term
which has the magnetic field.
And therefore, look at this.
i with i is minus
1, but q is minus e.
So this is eh bar over 2 mc
sigma dot b, which is here,
which came from g equal 2.
Nowhere I had to say there
that g is equal to 2.
It came out of this calculation.
The Pauli Hamiltonian
knew of this.
And therefore, it's a
great progress, that Pauli
Hamiltonian, but
it suggests to us
that we can do things still
even better, as Dirac did.
