PROOFS PROGRAMME

qm/que/06011

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Given that :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 $$  \sigma_x =\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
\sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix}, \qquad
\sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$ Answer the following question
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Question :      What vector would represent the state of the system if it is known that
the system has definite value $+1$ for the dynamical variable $X$? What vector
would represent the state if the system has definite value $-1$ for the variable
$Y$.

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