Understanding Bell's Inequality and QM with \vec{n}

[Thrice]

I'm trying to understand a part of the text where they prove QM doesn't satisfy Bell's inequality. I get how he derives the inequality. Apparently it's same as Sakuri (1985) and Townsend (2000). Problem is I lose him as soon as he starts the main part. Quoting almost directly,

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Consider a qubit oriented in an arbitrary direction. Consider a unit vector \vec{n} = \sin \theta \cos\phi \hat{x} + \sin \theta \sin\phi \hat{y} + \cos \theta \hat{z}. The eigenvectors of \sigma \cdot \vec{n} are..

|+_n \rangle = \cos \frac{\theta }{2} |0 \rangle + e^{i\phi }\cos \frac{\theta }{2} |1 \rangle

|-_n \rangle = \cos \frac{\theta }{2} |0 \rangle - e^{i\phi }\cos \frac{\theta }{2} |1 \rangle

He then goes on to interpret |\langle 0|+_n\rangle |^2 as a probability & i can follow from there. I just don't get the bit I posted. The hint is to consider the x & y axes, eg to get the eigenvectors of \sigma \cdot \hat{x} set \theta =\pi /2 & \phi =0

Thanks.

 

[badphysicist]

The key is to understand what \sigma \cdot \vec{n} means. This is a dot product of matrices (Pauli matrices) times scalars (components of \vec{n}. Let's just look at the first component. You will take the x Pauli matrix times by the x component of \vec{n} to get \sigma_{x} n_{x}=\left(\begin{array}{cc}0 & 1 \\ 1 & 0\end{array} \right) \sin \theta \cos\phi=\left(\begin{array}{cc}0 & \sin\theta\cos\phi \\ \sin\theta\cos\phi & 0\end{array} \right). You do the same for the y and z components and add them all up to get your matrix, \sigma \cdot \vec{n}. Find the eigenvectors of it and you should get what you gave from the book.