# How do I measure the spin along an arbitrary direction?

I need to measure the atoms spin along a direction perpendicular to the z-axis but at an angle $$\phi$$ with the x-axis.

They then get the output of the spin measured is + with probability $$\cos^2(\phi/2)$$ Is there a general formula I can use to compute the + spin state and then form that determine this probability?

• Can you provide more detail? The answer depends on what the atom's spin is initially. Also, this sounds like a homework problem, so I'll tag it accordingly. :) – Lagerbaer Nov 29 '13 at 16:10
• Its from a paper that im studying. Here is the link to it. All the information needed is on the first page or in reference  arxiv.org/pdf/1106.4953v2.pdf – user32462 Nov 29 '13 at 16:27
• Okay. I'd still keep the homework tag because it's a somewhat basic / instructional question. Now just look at vnb's answer and use his hint :) – Lagerbaer Nov 29 '13 at 21:59

Take a spin $1/2$ particle with its spin pointing along $\hat{n}$ defined by
$$\hat{n}=(\sin{\phi}\cos{\theta},\sin{\phi}\sin{\theta},\cos{\phi})$$
We are measuring the spin along $\hat{n}$ and the operator corresponding to this observable is $\vec{S}\cdot\hat{n}$.
$$\vec{S}\cdot\hat{n}=\frac{\hbar}{2}\begin{pmatrix} \cos{\phi} & \sin{\phi}e^{-i\theta} \\ \sin{\phi}e^{i\theta} & -\cos{\phi} \end{pmatrix}$$
Since in the mean time this was label as homework, I will not provide a full answer. From this point, all you have to do is find the eigenvalues and eigenvectors of $\vec{S}\cdot\hat{n}$. After you find the eigenvectors, you should be able to arrive quickly at the desired probability of $\cos^{2}(\phi/2)$.