PROOFS PROGRAMME

qm/que/06017

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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 $$  \sigma_x =\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad \sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix}, \qquad \sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$ State which of the following operators can represent an observable quantity and which ones cannot represent an observable.

  1.  $X_P = X+iY$
  2.  $X_M=X-iY$ 
  3.  $R=3X + 12Y + 4Z$
  4.  $T= X^2+Y^2+Z^2$

 

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