U1L5f.txt

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So when we major S sub z, you get--

well, we get one of three values.

We get-- so you can see what the eigenvalues are here.
They're either 1 or 0 or minus 1.

And 1, the eigenspace is 0 0.
Minus 1, the eigenspace is 1 1.
And 0, the eigenspace is the span of 0 1 and 1 0.
So if we get 0, the spin gets projected into this eigenspace.

OK.
So plus 0 1 x z y, which was 0 plus i1 over root 2.

So let's think of a point here, which is x, y, z.

So this is a block sphere.
And I want to claim the spin.
OK.

Alpha-x, alpha-y, alpha-z.

Along-- I'm going to say the observable
for the spin along the alpha-x, alpha-y, alpha-z axis.

Alpha-x, alpha-y, alpha-z axis is alpha-x sigma-x
plus alpha-y sigma-y plus alpha-z sigma-z,
and let's divide by 1/2.

OK, I don't want to call this a theorem,
but I'm going to call it something.
So 1/2 alpha-x sigma-x plus alpha-y sigma-y
plus alpha-z sigma-z is Hermitian with eigenvalues
plus or minus 1.

So let's call this sigma in the direction alpha.

Proof-- 2 sigma alpha squared is equal to the identity.

OK, let's-- I'm sorry.
If we take the 1/2--
yeah.
If we had 1/2 here, which would be the observable for spin
on the alpha direction, then the eigenvalues
would be plus or minus 1/2, but let's call sigma
alpha just this.
And it's has eigenvalues plus or minus 1.
So the proof is that sigma alpha squared
is equal to the identity because it's equal to, well, we
have alpha squared plus alpha-x squared plus alpha-y squared
plus alpha-z squared.

So sigma-x squared is sigma-y squared
is sigma-z squared is the identity.
So we take this term squared, this term squared,
and this term squared, we get this times the identity.
And because this is a unit vector,
because it's at a point of the surface of a sphere,
this is 1 plus alpha-x alpha-y sigma-x sigma-y
minus sigma-y sigma-x plus dot-dot-dot.
And these terms, sigma-x and sigma-y, anticommute.
Remember, sigma-x sigma-y was i sigma-z,
and sigma-y sigma-x was minus i sigma-z.
So these terms all go away.
So we have a matrix whose square is 1.
That means its eigenvalues must be--
and it's also Hermitian because we're adding 3 Hermitian
matrices, so we get--
it's Hermitian because its adjoint is itself.

So we get a matrix with eigenvalues plus or minus 1.
And trace of sigma alpha is equal to alpha trace.

Sum over x, y, x.

Alpha key trace sigma key.
And the trace of all these Pauli matrices is 0, so equals 0.
So that means one of its eigenvalues must be plus 1
and the other must be minus 1.
So this is observable for spin in some direction.
And you know, with some calculation,
you can show that the direction is alpha-x, alpha-y, alpha-z.
But, you know, the other thing is that with this formula,
it essentially has to be because, you know,
it's true at x, at y, and at z, and--

I don't want to do this calculation because it's
rather messy, but I think it's in the textbook.

So we have the observable for spin in an arbitrary direction.
And if you want to calculate what
state at this point on the block sphere corresponds to,
one way to do it is, you know, compute this matrix
and then find its eigenvalues.
And another way to do it is just with trigonometry.