WEBVTT

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PROFESSOR: All right, so
our next thing will be

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getting into the electron spin.

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And I want to discuss of
two things, a little bit

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of the Dirac equation
and the Pauli

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equation for the electron.

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We need to understand
electron a little better

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and understand
the perturbations,

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the relativistic
corrections so we'll

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consider the Pauli equation.

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So the Pauli equation is
what we have to do now.

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So what is the Pauli equation?

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It is the first
attempt there was

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to figure out how the
Schrodinger equation should

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be tailored for an electron.

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So we can also think
of the Pauli equation

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as a baby version of the
Dirac equation, which

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is the complete equation
for the electron.

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So it all begins
at the motivations

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by considering what is the
magnetic moment of an electron?

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So it's a question that
you've been addressing

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in this course for a while,
from Stern-Gerlach experiment

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and how magnetic fields
interact with electron.

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So classically and
in Gaussian units,

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if you have a current
loop and an area a,

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the magnetic moment is
equal to i times the area

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vector divided by c.

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You can use that to imagine
an object that is rotating.

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And as the object
rotates, if its charged,

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it generates a magnetic
moment, and the magnetic moment

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depends on the amount of
rotation of the orbit.

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So it turns out that, with
a little classical argument,

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you can derive that the
magnetic moment is related

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to the angular momentum by
a relation of the form L,

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so the charge of the object,
the mass of the object c,

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and the angular momentum.

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And that's perfectly
correct classically.

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So people thought that
quantum mechanically

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they expect that for an electron
or for an elementary particle,

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you would get q times 2 mc
times the spin of the particle.

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So this would be the intrinsic
magnetic moment of an electron

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that it seems to have
times a fudge factor, g

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that you would have to measure.

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Because after all, the
world is not classical.

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There's no reason why a
classical relation like that

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would give you the right exact
value of the magnetic moment

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of the electron.

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As it turns out, this g
happens to be equal to 2.

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And we get the following.

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So for electrons g
happens to be equal to 2.

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So mu is equal to 2 times minus
e-- the charge of the electron

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is minus 2 mc, and the
spin of the electron

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is h bar over 2 times
sigma, the Pauli matrices.

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So this thing is minus
eh bar over 2 mc sigma

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mu of the electron.

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So if you have a magnetic
field, a B external,

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the Hamiltonian that
includes the energetics

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of the magnetic
field interacting

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with the dipole moment,
it's always minus mu dot b.

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So in this case it would be
eh bar over 2 mc sigma dot b.

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That's the Hamiltonian
for an electron

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in a magnetic field, something
you've used many, many times.

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The most unpleasant part of
this thing is this g equal 2.

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Why did it turn out
to be g equal 2?

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People knew it was g equal 2.

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But why?

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And here we're going to see the
beginning of an explanation.

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In fact, it comes very close
to an explanation of that

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through the Pauli equation.

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The Pauli equation is a
nice equation you can write,

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and you can motivate.

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And it predicts.

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Once you admit the
Pauli equation,

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it predicts that g
would be equal to 2.

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And that, of course, happens
also for the Dirac equation.

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And Dirac noticed that, and
he was very happy about it.

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So it's quite remarkable
how these things showed up,

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and the G equal to 2
was perfectly natural.

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So let's see this
Pauli equation first.

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What is it?

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Well, when you do the
Schrodinger equation for a wave

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function, you say
p squared over 2m--

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say a free particle,
and you would

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put the wave function is equal
to energies times the wave

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function.

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h on the wave function is
equal to energy terms the wave

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function.

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So for this
electron, you already

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know it's a two-dimensional
Hilbert space, the state

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space of the up and down.

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So it's convenient to change
this and to say, you know what?

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I'm going to put here
something I'll call a spinor.

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Chi is a Pauli.

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It has two things, a chi 1 and a
chi 2, and it's a Pauli spinor.

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And this is reasonable so far.

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The spin is defined by 2 degrees
of freedom, 2 basis is vectors.

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So you've done wave
functions for spin.

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And the wave functions for
spin have these things,

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and each themself can be
a function of position.

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So that's perfectly reasonable.

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I don't think anyone of you is
very impressed by this so far.

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But here comes the funny thing.

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In a sense here, there is a 2 by
2 identity matrix sitting here.

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So Pauli observes the
following identity.

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If you have sigma dot
8 and sigma dot b,

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it's equal to a dot b times 1
plus i sigma plot a cross b.

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Look, these are two
vectors, a and b.

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And if you multiply half
this product-- first,

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this is a dot product.

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It means sigma 1 times a1 sigma
2 times a2 sigma 3 times a3.

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So this is a dot product.

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This product here
is a matrix product

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because this is
already a matrix,

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this is already a matrix,
so this is a matrix product.

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And this is the
result. This comes

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from properties of the Pauli
matrices you already know.

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In particular, their
commutator determines

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this piece and
the anticommutator

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determines this piece.

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So here, if you
have, for example,

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sigma dot p times sigma dot p--

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see, these two vectors are
the same in this case--

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you'll get just p
squared times 1 plus 0.

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Because p times P, those are two
vector operators if you wish,

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but still they commute.

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So this is equal to 0.

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So your p squared can be
replaced by sigma dot p sigma

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dot p.

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So h, the Hamiltonian-- that
is p squared over 2 m times

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the identity matrix--

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can be perhaps better
thought as sigma

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dot p times sigma
dot p divided by 2m.

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So this is the first step.

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You haven't done
really much, but you've

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rewritten the Hamiltonian
with a unit matrix

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here perhaps in a
somewhat provocative way.

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In quantum mechanics, when we
couple to electromagnetism,

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there are simple
changes we have to do.

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And we will study that in
detail in about three weeks,

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but today I will
just bench on what

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you're supposed to do in order
to couple to electromagnetism.

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In order to couple
to electromagnetism,

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you're supposed to
change p wherever you see

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by an object you can call pi.

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It's some sort of more
canonical momentum

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in which it's got
to p minus q over c.

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The vector potential
is a function of x.

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Well, people write it
like this, qp minus q ac.

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So you're supposed to
do this replacement.

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When you're dealing
with a particle moving

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in some electromagnetic field,
that's the change you must do.

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We will study that in detail.

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But I think an obvious
question at this moment

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is the following.

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You say, well, I'm
in quantum mechanics,

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and I work with p and x.

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Those are my opera-- what
is a? is it an operator?

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Is it a vector?

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Is it what? p is an
operator, but what is a?

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You should think of
this vector potential.

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In general, this
vector potential

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will depend on the position.

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So if you put a magnetic field,
you require a vector potential.

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It has some position dependent.

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So actually, this
a here should be

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thought as a of x,
the same way as when

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you have the potential
that depends on x.

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In quantum mechanics, we just
think of that x as an operator.

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So the a is an operator
because x is an operator,

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and a has x dependents.

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So the Pauli Hamiltonian, h
Pauli is nothing else but sigma

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dot pi times sigma
dot pi over 2 m.

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Because we said p must
be replaced by pi,

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so this is the
Pauli Hamiltonian.

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And it's equal to 1
over 2m pi squared

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pi dot pi times 1 plus i
sigma dotted with pi cross pi.

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OK, here is the
computation we need to do.

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What is pi cross pi?

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You know what pi is.

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It's given by this thing.

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So how much is pi cross pi?

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Well, it takes a
little computation.

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I'll tell you what
you have to do.

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It's a very interesting
computation.

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It's small.

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It reminds you of
this computation

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in angular momentum, l cross l.

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Maybe you've written the
algebra of angular momentum

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in this language.

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It's equal to h bar l.

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That's your computation in
relations of angular momentum

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written like that.

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So pi cross pi,
the kth component

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is epsilon ijk pi i
pi j or 1/2 epsilon

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ijk pi I commutator with pi j.

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This last step, it
can be explained

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by writing out the
commutator, which

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is pi i pi j minus pi j pi i.

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But with this epsilon, those
two terms are the same,

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and it becomes this.

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So how much is this thing?

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I'll leave it for you to do it.

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It's simple thing.

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You have to commute
these things,

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and you must think
of p as derivatives.

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So this pi i pi j is i h bar
q over c di a j minus dj a i.

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You can see it coming.

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You see, you have a commutator
of two factors like that.

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The a with a will commute.

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The p with p will commute.

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The cross products won't,
but that's just derivatives.

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So therefore, you
get the derivatives

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of a antisymmetrized.

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And the derivatives
of a antisymmetrized

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is the curl of a.

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And the curl of a is
the magnetic field.

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So that's why this happens.

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So at the end of the day, when
you'll finish this computation,

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get that pi cross pi is
just i q i h bar q over c

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times the magnetic field.

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Pretty nice equation.

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So the Pauli Hamiltonian
includes h Pauli.

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It includes this term,
which is i sigma over 2m

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dotted with i hQ over
cb plus the other terms.

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I'm just looking at this term
which has the magnetic field.

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And therefore, look at this.

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i with i is minus
1, but q is minus e.

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So this is eh bar over 2 mc
sigma dot b, which is here,

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which came from g equal 2.

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Nowhere I had to say there
that g is equal to 2.

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It came out of this calculation.

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The Pauli Hamiltonian
knew of this.

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And therefore, it's a
great progress, that Pauli

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Hamiltonian, but
it suggests to us

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that we can do things still
even better, as Dirac did.