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##?