Number of Independent Parameters for a Spin 1/2 State Ket I was reading over Sakurai's Modern Quantum Mechanics textbook and had the following question. He introduces the spinor representation where a spin state can be represented as
$$\chi = c_{+}\chi_{+} + c_{-}\chi_{-},$$
where $\chi_{+}=(1,0)^{T}$ and $\chi_{-} = (0,1)^{T}$, and $c_+$ and $c_-$ are general complex numbers. If this is the case, it seems like that leaves three independent variables needed to specify for a spin state (4 for the two complex numbers and minus 1 for the normalization condition). However, later he shows a spin state which is an eigenstate of $\vec{S}\cdot \hat{n}$ (where $\hat{n}$ is characterized by polar and azimuthal angles $\alpha$ and $\beta$, respectively) is given by $\left[\cos(\beta/2)e^{-i\alpha/2}, \sin(\beta/2)e^{i\alpha/2}\right]^{T}$. In this case, you only need two numbers to characterize a spin state.
So do you need two numbers or three to specify a spin state?
 A: There are only two numbers needed to specify a spin-$\frac{1}{2}$ state.  Each of the coefficients in the general spinor
$$\chi=\left[\begin{array}{c}
c_{+} \\
c_{-}
\end{array}\right]$$
is a complex number, giving four real parameters.  However, one real parameter can be eliminated by requiring normalization, $|c_{+}|^{2}+|c_{-}|^{2}=1$. Moreover, one more real parameter is superfluous, since the phase of $\chi$ does not affect any physical observables. A phase change $\chi\rightarrow e^{i\alpha}\chi$ does not change a spin observable such as $\chi^{\dagger}O\chi$.
This means that only two real parameters are needed to specify a spin state. Equivalently, every spin state is an eigenvalue of some Pauli spin operator $\sigma_{n}=\vec{\sigma}\cdot\hat{n}$, where $\hat{n}$ is a real unit vector.  A real unit vector is parameterized by two angles, which give the spatial orientation of the quantization axis corresponding to $\sigma_{n}$.
You cannot choose two any two parameters from the four real numbers that described $c_{+}$ and $c_{-}$.  However, the single complex number $c_{+}/c_{-}$ is enough to uniquely specify the state.
A: Quantum state or state-vector? The spin vector requires  three numbers: two polar angles $\theta,\phi$ parametrizing  the direction of the spin  and an overall-phase. The quantum state does not depend on the overall phase, as states are specified by rays in the Hilbert space.
