How do spin measurements of particles relate to complex amplitudes?

[Grand]

Homework Statement

Say we want to fiddle with spin of a particle.

The state in which a measurement of $$s_x$$ is certain to yield +1/2 is:
$$|+,x\rangle=\sin{\frac{\pi}{4}}e^{i 0}|-\rangle+\cos{\frac{\pi}{4}}e^{-i 0}|+\rangle=\frac{1}{\sqrt{2}}(|-\rangle+\+|+\rangle)$$

Now, for this state, we want to find the amplitude for measuring $$s_z$$ to be either 1/2 or -1/2. We have to apply the bra
$$\langle+,s_z|=sin0e^{i\phi/2}\langle-|+cos0e^{i\phi/2}\langle+|$$

What I want to ask about is why in here $$\phi=0$$ as well - it is the z direction, which is determined only by $$\theta=0$$.

 

[tiny-tim]

Hi Grand!

If you multiply any eigenstate by an ordinary complex number, don't you get the same eigenstate?

 

[Grand]

Yes, we do. The probability is computed as mod squared of the amplitude. But here we have to show that the ratio of the complex amplitudes to measure the momentum $$s_z$$ to be 1/2 and -1/2 is 1 and for the same calculation, but for y it is i.