PROOFS PROGRAMME

qm/que/06012

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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 $$  \sigma_x =\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad \sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix}, \qquad \sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$ Show that the allowed values of $$ X_{\hat{n}} = n_1 X + n_2 Y + n_3 Z $$ are given by $\pm \sqrt{n_1^2+n_2^2+n_3^2}.$

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