$$ \vert \psi \rangle = \alpha \vert 0 \rangle + \beta \vert 1 \rangle $$

$$ \left| \alpha \right|^{2}+\left| \beta \right|^{2} = 1 $$

$$ \vert \varphi \rangle = \eta \vert\uparrow \rangle + \theta \vert \downarrow \rangle $$

$$ H\vert 0 \rangle =\frac{1}{\sqrt{2}}\vert 0 \rangle + \frac{1}{\sqrt{2}}\vert 1 \rangle $$