### Problem: Spin-polarization principle

Any state of a single spin is an eigenvector of some component of the spin. In his book Quantum mechanics: The theoretical minimum, Leonard Susskind refers to this statement as the spin-polarization principle. Let us scrutinize the statement by considering the following problem.

### Problem

Consider a spin-$$\tfrac{1}{2}$$ particle in a generic spin-state given by $\Psi = \begin{pmatrix} \psi_1 \\[0.1cm] \psi_2 \end{pmatrix} = \begin{pmatrix} e^{i \alpha} \cos \delta \\[0.1cm] e^{i \beta} \sin \delta \end{pmatrix} \,,$ where $$\alpha$$, $$\beta$$, $$\delta$$ are some real numbers. The state is specified in the $$z$$-representation; that is, the states $\begin{pmatrix} 1 \\ 0 \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ have well-defined $$z$$-components of the spin equal to $$+\hbar/2$$ and $$-\hbar/2$$, respectively.

Find a unit vector $$\boldsymbol{n}$$ (in three-dimensional Euclidean space) such that $$\Psi$$ has a well-defined spin component along $$\boldsymbol{n}$$ equal to $$+\hbar/2$$.

### Solution

For a solution, please see Issue #9 - Spin-polarization principle of Quantum Newsletter.