PROFESSOR: I want to just
elucidate a little more what
are the eigenstates here.
So with angular momentum,
we measure L squared
and we measure Lz.
So with spin, we'll measure
spin squared and Sz.
And Sz is interesting.
It would be spin or angular
momentum in the z direction.
So let's look at
that, Sz, this is
the operator, the measurable.
It's this time nothing
else than a simple matrix.
It's not the momentum operator.
It's not angular momentum
operator with derivative.
It's an angular
momentum operator,
but it seems to have
come out of thin air.
But it hasn't.
So here it is.
Oh, and it's diagonal already.
So the eigenstates
are easily found.
I have one state--
I don't know how I
want to call it--
I'll call it 1, 0.
It's one state.
And that's an eigenstate of it.
We'll call it, for
simplicity, up.
We'll see why.
Sz, acting on up, is equal to
h bar over 2, 1, 0; 0, minus 1
acting on 1, 0.
That's h bar over 2.
And the matrix is at 1, 0.
So it is an eigenstate
because it's h bar over 2 up.
The 1, 0 state again.
So this thing, we call it up,
because it has up component
of the z angular momentum.
So it's a spin up state.
What is the spin down state?
It Would be 0, 1.
It's a spin down.
And Sz on the spin
down, it's also
an eigenstate, this time with
minus h bar over 2, spin down.
And we call it spin 1/2
because of this 1/2.
And you'd say, no, you
just put that constant
because you want it there.
Not true, if I would have
put a different constant here
in defining this, I would
not have gotten this
without any constant, that it's
how angular momentum works.
So if I use two-by-two matrices,
I'm forced to get spin 1/2.
You cannot get anything else.
The 1/2 of the spin
is already there.
The component of angular
momentum is h bar over 2.
If you have a photon,
it has spin 1.
The components of angular
momentum is plus h or minus h,
if you have the two
circularly polarized waves.
So this is actually interesting.
But it begs for another
question because we
have a good intuition.
And this is spin up
along the z direction
because it has a Sz component,
eigenvalue h over 2.
So the last question
I want to ask
is, how do I get a spin state
to point in the x direction
or in the y direction.
You see, the interpretation
of this spin state
is that it's a spin state
that has the spin up
in the z direction,
because that's
what you can
measure, or spin down
in the down direction of Sz.
Can I get spin states that
point along the x direction or y
direction?
And here's where the problem
seems to hit you and you say,
I'm in trouble.
I have this state spin
up and spin down along z.
And it's a two-dimensional
vector space,
because two-by-two matrices, and
Sx, Sy, Sz is three dimensions.
How am I going to get
three dimensions out
of two dimensions?
You just have spin states
along z, up and down.
Now the spin up and
spin dow, moreover,
are orthogonal states.
These two are orthogonal states.
You see, you do the inner
product, transpose this,
you get this, and times that.
So they are
orthogonal, unless you
imagine this vector
plus this vector
is a full basis for
the vector space,
because the vector
space is a, b.
And now you see that this is
a times up plus b times down.
So anything is a
superposition of up and down.
So how do I ever
get something that
points along x, or something
that points along y?
Well, let's try to see that.
Well, consider Sx, you
have an Sx operator, which
is h bar over 2, 0, 1, 1, 0.
And then you can
try to analyze this,
but it's more entertaining to
imagine other things, to say,
look, if I've gotten this
vector 1, 0, which is up,
and 0, 1, which is down,
I can try maybe a vector
that has the up and the down.
Maybe the up and the down is
a vector that points nowhere.
Who knows, whatever.
If I want to normalize
it, I have to put a 1
over square root of 2.
And now I know, it's 1
over square root of 2,
up, plus down.
That's what this vector is.
But let's see what
Sx does on it.
Sx on 1 over square root
of 2, 1, 1 is h bar over 2,
1 over square root of
2, 0, 1, 1, 0, on 1, 1.
So h bar over 2, 1
over square root of 2.
And let's see, that gives
1, that gives me another 1.
Oops, I got the same
vector I started with.
It an eigenstate.
So this thing, this plus
and down, superimposed,
is an eigenstate of Sx.
So this is actually a
spin that points up,
but in the x direction.
Whenever we don't put anything,
we're talking about z.
But this is the spin
up in the x direction.
And these appeared
as the sum of a spin
up and spin down
in the z direction.
It may not be too
surprising for you
to imagine that if you put
1 over square root of 2,
1 minus 1, that vector is
orthogonal to this one.
Yes, you do the transpose.
And this one is orthogonal.
So this is 1 over square
root of 2, up, minus, down.
That is the down spin along x.
So the up and down spins
along x come out like that.
We form the linear combinations.
So finally, you would say,
well, I'm going to push my luck
and try to get spins
along the y direction.
But I now form those
linear combinations.
What else could I do?
These linear
combinations are there.
And I've got already two things.
And you say, well, that's fair,
you're a two-dimensional vector
space, so you're getting two
things, spin states along x
and spin states along z.
But actually, we didn't
run out of things to try.
We could try a state of the
form 1 over square root of 2,
something like this.
We could try the state up.
And then, we've put
a plus, but now we
could put a plus i, state down.
So this would be a
state of the form 1, i.
And what does it do?
Well, let's see what
it does with Sy.
1, i.
And the Sy matrix is h bar
over 2 minus i, i, 0, 0, 1, i.
And there's 1 over
square root of 2.
So it's 1 over square root
of 2, or h bar over 2,
1 over square root of 2.
And let's see what we get.
Minus i times i is one 1.
And the second one is i.
We get the same state.
Yes, it is an eigenstate.
So with a plus i here,
this is this spin up
along the y direction.
And the spin down
along the y direction
would be up, minus i, down.
This is orthogonal
to that vector.
It's 1 minus i.
And it's the spin down
in the y direction.
You can calculate
the eigenvalue,
it's minus h bar over 2,
and it's pointing down.
So your complex numbers
play the crucial role.
If you didn't have
complex numbers,
there was no way you could ever
get a state that this pointing
in all possible directions.
And you also see,
finally, that this thing
has nothing to do with your
usual wave functions, functions
of x, theta, phi.
No, spin is an additional
world with two degrees
of freedom, an extra thing.
It doesn't have a
simple wave function.
The spin wave functions are
these two column vectors.
But there is angular momentum in
there, as you discovered here.
There is a commutation
relations of angular momentum,
the units of angular
momentum, the eigenvalues
of angular momentum.
And this great thing is
such a nice simple piece
of mathematics.
It has an enormous utility.
It describes the
spins of particles.
So it's an introduction,
in some sense,
to what 805 is all about.
Spin systems are
extremely important,
practical applications.
These things, because they
have basically two states,
are essentially qubits
for a quantum computer.
Within these systems,
we understand,
in the simplest
way, entanglement,
Bell inequalities,
superposition,
all kinds of very, very
interesting phenomena.
So it's a good place to stop.
