Calculate expectation value of entangled 2 state system?

[ianmgull]

Homework Statement

Homework Equations

I know that there are two eigenstates of the operator C:

|B> = (1 0) as a column vector with eigenvalue 1
|R> = (0 1) also a column vector with eigenvalue -1

The Attempt at a Solution

My work is shown here:

If anyone could point me in the right direction, I'd greatly appreciate it. I've been stuck for hours and just can't figure out what I'm doing wrong.

thanks

Ian

 

[DeathbyGreen]

Use a projection operator. A projector on the red basis would look like $P_R = |R\rangle\langle R|$. Then (dropping 1 and 2 subscripts, and assuming orthogonal states):

$Prob_R =\frac{1}{2} (\langle B|\langle R| - \langle R|\langle B|)|R\rangle\langle R|(|B\rangle|R\rangle -|R\rangle|B\rangle )=\frac{1}{2}$

The expectation value of C is the probability that the measurement produces either 1 or -1, so the average result will be 0.

 

[ianmgull]

I just worked it out and that makes much more sense.

I'm still a little unclear on (conceptually) what meaning I should attribute to taking the expectation value of an operator (like above) vs a projection operator.

Thanks so much!

 

[DeathbyGreen]

The eigenvalues are what can be measured in a lab; an expectation value will give you the average result that will be obtained over a large number of measurements, in this case 0. A projection operator projects the state onto a basis (red or blue) according to a probability, 1/2 here.

 

[ianmgull]

Awesome.

THANK YOU