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1. The Hilbert space

The quantum system is represented in a two-dimensional complex Hilbert space, which we realize as $\Cset^2$ - the set of all column vectors
a=\pmatrix{ a_1 \cr
\end{displaymath} (1)

where $a_1$ and $a_2$ are complex numbers, and with scalar product $(a,b)$ defined by
(a,b)={\bar a}_1 b_1+{\bar a}_2 b_2
\end{displaymath} (2)

where the bar ${\bar c}$ stands for the complex conjugation of the complex number $c.$