5
$\begingroup$

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?

$\endgroup$
3
  • $\begingroup$ 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. :) $\endgroup$
    – Lagerbaer
    Nov 29, 2013 at 16:10
  • $\begingroup$ 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 [15] arxiv.org/pdf/1106.4953v2.pdf $\endgroup$
    – user32462
    Nov 29, 2013 at 16:27
  • $\begingroup$ 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 :) $\endgroup$
    – Lagerbaer
    Nov 29, 2013 at 21:59

1 Answer 1

6
$\begingroup$

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 $\cos^{2}(\phi/2)$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.