TQM 11 Nov 2017 20:05
a) $\alpha \vert 0 \rangle + \beta \vert 1 \rangle = \begin{bmatrix} \alpha\\\beta \end{bmatrix} = \begin{bmatrix} \alpha & \beta \end{bmatrix}^{T} = \alpha \vert 0 \rangle + \beta \vert 1 \rangle$
b) $a \vert 0 \rangle + b \vert 1 \rangle \to \frac{b}{a}=\alpha$, $\alpha \to \frac{1}{\sqrt{1+\vert \alpha \vert ^2}} \vert 0 \rangle + \frac{\alpha}{\sqrt{1+ \vert \alpha \vert ^2}}$, and $\infty ↔ 1$
c) $(x, y, z) = (sin(\theta) cos(\phi), sin(\theta) sin(\phi), cos(\theta))$