Probabilities of measuring ##\pm \hbar/2## along ##\hat{n}##?

[happyparticle]

Homework Statement:

Probabilities of measuring $\pm \hbar/2$ along $\hat{n}$?

Relevant Equations:

$|+\rangle_n = cos(\theta/2)e^{-i\phi/2}|+\rangle +sin(\theta/2)e^{i\phi/2}|-\rangle$
$|-\rangle_n = sin(\theta/2)e^{-i\phi/2}|+\rangle -cos(\theta/2)e^{i\phi/2}|-\rangle$

Hi,

Given a spin in the state $|z + \rangle$, i.e., pointing up along the z-axis what are the probabilities of measuring $\pm \hbar/2$ along $\hat{n}$?

My problem is that I'm not sure to understand the statement. It seems like I have to find the probabilities of measuring an eigenvalue along $\hat{n}$. What does that mean exactly? Is it the probability to measure $\pm \hbar/2$ in the $n$ basis?

I tried to find $|+\rangle$ in the $n$ basis, which I think this is $|z+\rangle$ in the $n$ basis. I thought maybe it could help me.
I got $|+\rangle = 1/2 (cos (\theta/2) e^{i\theta /2} |+ \rangle_n + sin (\theta/2)e^{i\theta /2} |- \rangle_n)$

I'm really confuse with this statement. I'm not sure to understand the difference between $|z + \rangle$ and $|+\rangle$

 

[PeroK]

My problem is that I'm not sure to understand the statement. It seems like I have to find the probabilities of measuring an eigenvalue along $\hat{n}$. What does that mean exactly?

The question is this. Imagine you have a system that produces electrons in the $\ket{z+}$ state. You can confirm this by setting up a measurement aparatus to measure the spin component about the z-axis and finding that every electron produced by your system behaves in the same way, corresponding to a measurement of $+\frac \hbar 2$. Note that the z direction is simply some direction you have chosen in your experimental set-up.

Now, you leave your electron production system in place and re-orient your measurement aparatus along an axis $\hat n$, represented by azimuthal and polar angles $\theta, \phi$.

Each electron will behave in one of two ways, corresponding to the measurements of $\pm \frac \hbar 2$. The question is: what is the probability that the electron is measured to have $+\frac \hbar 2$; and what is the probability that the electron is measured to have $-\frac \hbar 2$?