U1L5d.txt

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So there's a standard way of describing measurements
in quantum mechanics.
A measurement operator is a Hermitian matrix.

M equals summation lambda i v sub i v sub i.

The measurement operator gives us lambda sub i
and projects the state onto the eigenspace corresponding
to lambda sub i.

So I'm going to do this.
I'm going to explain this for sigma-x, sigma-y, sigma-z.

Sigma-x equals 1 1 0 0.
The eigenvectors are plus with eigenvalue 1
and minus with eigenvalue minus 1.

And 1/2 sigma-x is--

I'm going to say this is a measurement
operator corresponding to spin along the x-axis.

So let's draw on the blocks sphere.

Plus is spin 1/2 along the x-axis,
and minus is spin minus 1/2 along the x-axis.

So apply-- OK, so what I want to do is I want to say,
apply the measurement corresponding to--

corresponding to sigma-x, get plus 1/2 or minus 1/2.
And you get plus 1/2, you know, this is the eigenstate.

And eigenstate plus.
And if you get minus 1/2, it's going to be eigenstate minus.

And this is just because sigma-x equals summation--
or 1/2 sigma-x equals summation.
OK, I don't want to.
1/2 plus plus minus 1/2 minus minus.

So this corresponds to the measurement
in eigenvectors plus and minus.

So if M is a measurement operator,
the size of quantum state--

actually, what quantum people call this in quantum mechanics,
these are called observables.
So let's call them observable.
If psi is a quantum state, then psi
M psi is the expectation value of the observable.
Expected value of observable.

Proof.
M is equal to summation lambda sub i v sub i v sub i.

Psi M psi is equal to summation lambda i psi v sub i v sub i
psi is equal to sum lambda i v sub i psi squared equals
summation of lambda i times the probability of observing
lambda sub i.

So one thing I'm going to say is that so far, we've
only talked about something called von Neumann measurements
or projective measurements.
And there is a more general class of measurements called
positive-operator valued measurements or POVMs,
and we are not going to deal with them until later
in the course.
Because, well, basically because they're much too--
they're much more confusing than this kind of measurement
and we don't need them until we get