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Hi, I can't get my head around this question.

The Bell state:

[tex]|\psi\rangle = \frac{1}{\sqrt{2}}\left(|\uparrow_1\rangle|\downarrow_2\rangle +|\downarrow_1\rangle |\uparrow_2\rangle \right)[/tex]

Find the correlation coefficient of the measurement of the spins in the directions [tex]z,\phi[/tex].

[tex] C(\phi) = \langle S_{z1} S_{\phi 2} \rangle [/tex]

S takes the values [tex]\pm 1[/tex]

I know that I need to calculate the probability of the states being parallel and subtract the probability that the states are anti-parallel. But how do you calculate those?

[tex]Pr(\uparrow_{z1}\uparrow_{\phi 2}) = \left|\langle\uparrow_{z1}\uparrow_{\phi 2} \left |\frac{1}{\sqrt{2}}\left(|\uparrow_1\rangle|\downarrow_2\rangle +|\downarrow_1\rangle |\uparrow_2\rangle \right)\right| \uparrow_{z1}\uparrow_{\phi 2} \rangle \right|^2[/tex]

Where do I go from here?

Thanks

## Homework Statement

The Bell state:

[tex]|\psi\rangle = \frac{1}{\sqrt{2}}\left(|\uparrow_1\rangle|\downarrow_2\rangle +|\downarrow_1\rangle |\uparrow_2\rangle \right)[/tex]

Find the correlation coefficient of the measurement of the spins in the directions [tex]z,\phi[/tex].

## Homework Equations

[tex] C(\phi) = \langle S_{z1} S_{\phi 2} \rangle [/tex]

S takes the values [tex]\pm 1[/tex]

## The Attempt at a Solution

I know that I need to calculate the probability of the states being parallel and subtract the probability that the states are anti-parallel. But how do you calculate those?

[tex]Pr(\uparrow_{z1}\uparrow_{\phi 2}) = \left|\langle\uparrow_{z1}\uparrow_{\phi 2} \left |\frac{1}{\sqrt{2}}\left(|\uparrow_1\rangle|\downarrow_2\rangle +|\downarrow_1\rangle |\uparrow_2\rangle \right)\right| \uparrow_{z1}\uparrow_{\phi 2} \rangle \right|^2[/tex]

Where do I go from here?

Thanks

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