[QUE/QM-06010]

 The vector space needed to describe a particular physical system is two
dimensional complex vector space. The  states are therefore represented by 2
component complex column vectors. The observables for this system are $2\times2$
matrices. A set of three dynamical variables of the system, $X,Y,Z,$ are be
represented by $2\times 2$ matrices denoted by $\sigma_x, \sigma_y,\sigma_z$,
where  $$  \sigma_x
=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
\sigma_y=\begin{pmatrix} 0&-i\\i&0\end{pmatrix}, \qquad
\sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$     What values are
experimentally allowed if one measures the dynamical variable

1.        $X$ 
2.       $Z$ 
3.       $T= X^2+Y^2+Z^2$