Global and relative phases of kets in QM In one of the questions I'm trying to solve it is asked to, first, compute probabilities for the respective results of the Stern-Gerlach measurements performed on each state $\lvert\psi_1\rangle$, $\lvert\psi_2\rangle$, and $\lvert\psi_3\rangle$ in each of the three orthogonal directions $\hat{x}$, $\hat{y}$, and $\hat{z}$, which I know how to do. But the next part asks to observe something about the importance for computing probabilities of the global phase (in this case, the overall sign of the state vector) and the relative phase between the components of the state vector (in this case, the relative sign of terms in the superposition).
I don't understand what these global and relative phases mean and how they are related and to which signs exactly. I couldn't find it in the book either. The word "phase" is mentioned very vaguely there, perhaps assuming certain background. I'd appreciate some help with this. That is, how are probabilities related to these phases, and what are these phases and how to read them from bras and kets?
 A: Every complex number can be written in the form $re^{i\theta}$ for a real number $r$. We call $e^{i\theta}$ the phase. For example, if
$$|\psi \rangle = \frac{1}{\sqrt{2}} ( |0 \rangle + i |1 \rangle)$$
then the phases of the $|0 \rangle$ and $|1 \rangle$ components are $1$ and $i$, and their relative phase is $i$. Now consider 
$$|\psi' \rangle = \frac{1}{\sqrt{2}} ( i|0 \rangle - |1 \rangle).$$
This state is the same as $| \psi \rangle$ but has been multiplied by $i$. The components still have a relative phase of $i$, but the whole thing has also picked up a global phase of $i$.
A: The general description of the spin-$\tfrac{1}{2}$ particle is described by the  wavefunction $|\psi\rangle\in\mathbb{C}^2$,
\begin{equation}
|\psi\rangle\;=\;e^{i\frac{\gamma}{2}}
\begin{pmatrix}
e^{-i\tfrac{\alpha}{2}}\cos\left(\tfrac{\beta}{2} \right) \\
e^{i\tfrac{\alpha}{2}}\sin\left(\tfrac{\beta}{2} \right)
\end{pmatrix}
\end{equation}
The first 2 angles $\alpha$ and $\beta$ are the relative phases, and describe the orientation of the state vector on the Bloch sphere (the 2-sphere). Taking the outer product the density matrix of the state reads,
$$|\psi\rangle\langle\psi|\;=\;\frac{1}{2}\bigg(\sigma_1+\sin(\beta)\cos(\alpha)\sigma_x+\sin(\beta)\sin(\alpha)\sigma_y+\cos(\beta)\sigma_z\bigg)$$
where $\sigma_1$ is the identity and $\sigma_x,\sigma_y,\sigma_z$ are the Pauli matrices. These are the "observables" as the expectation values in each direction gives the orientation of the state vector on the 2-sphere;
$$\langle\psi|\sigma_x|\psi\rangle=\frac{1}{2}\sin(\beta)\cos(\alpha)\qquad
\langle\psi|\sigma_y|\psi\rangle=\frac{1}{2}\sin(\beta)\sin(\alpha)\qquad
\langle\psi|\sigma_z|\psi\rangle=\frac{1}{2}\cos(\beta)$$
The above is the 3-dimensional picture and in this case the global phase is a natural hidden variable, as it is not explicitly present in the 2-sphere dynamics. This is detailed in a J. Phys. A article (2015) which you can find on the arXiv at: https://arxiv.org/abs/1411.4999
The third angle $\gamma$ is the global phase and it is important as it describes the position of the state vector $|\psi\rangle$ "globally" on the 3-sphere. For example, if we consider some closed loop on the 2-sphere, and calculate that the global phase of one orbit is $\gamma=2\pi$, then the total value of the global phase coefficient is;
$$e^{i\tfrac{2\pi}{2}}=-1$$
The negative coefficient tells us that we have only traveled half of the total path on the 3-sphere. While the relative phases have returned to their initial values the global phase has not, and the state acquires a negative sign, i.e. $$|\psi\rangle\mapsto-|\psi\rangle$$
While it appears we have reached our starting point after one orbit on the 2-sphere, in fact we require a second orbit of the path to return to the initial point. This is the nature of the intrinsic spin of the fundamental particles; the magnetic moment is 4-dimensional, and the true "observable" in the Stern-Gerlach experiment is the global phase.
A: Let’s assume our quantum state is a superposition of $|0\rangle$ and $|1\rangle$:
$$
|\psi\rangle = \alpha |0\rangle + \beta |1\rangle,
$$
where $\alpha = r_1 e^{i \theta_1}$ and $\beta = r_2 e^{i \theta_2}$ are complex numbers. It is customary to say that the $\theta$ in the exponent is a phase, and the whole exponent $e^{i \theta}$ is a phase factor.
The relative phase is the difference between the phases of the coefficients of $|0\rangle$ and $|1\rangle$, so in our example the relative phase is $\theta_2 - \theta_1$ (or $\theta_1 - \theta_2$, whichever way you prefer to define it).
You can also extract a common factor from both coefficients:
$$
|\psi\rangle = r_1 e^{i \theta_1} |0\rangle + r_2 e^{i \theta_2} |1\rangle = r_3 e^{i \theta_3}\left(\frac{r_1}{r_3} e^{i (\theta_1 - \theta_3)}|0\rangle + \frac{r_2}{r_3} e^{i (\theta_2 - \theta_3)}|1\rangle\right),
$$
and now $\theta_3$ is the global phase. (Note that before we extracted this factor, the global phase was $0$.)


how are probabilities related to these phases

Well, the probability of getting a 0 as the result of your measurement of $|\psi\rangle$ is:
$$
|\alpha|^2 = r_1^2,
$$
so you see where this is going.
