(Solved): Consider an isolated spin-1 system. The Hilbert space for the system is spanned by three basis vec ...

Consider an isolated spin-1 system. The Hilbert space for the system is spanned by three basis vectors \( |\uparrow\rangle \), \( |\downarrow\rangle \) and \( |0\rangle \) corresponding to the eigenvalues \( +\hbar,-\hbar \) and 0 ("zero \( \hbar \) ") of \( \hat{S}_{z} \) : \[ \begin{aligned} \hat{S}_{z}|\uparrow\rangle &=+1 \hbar|\uparrow\rangle \\ \hat{S}_{z}|\downarrow\rangle &=-1 \hbar|\downarrow\rangle \\ \hat{S}_{z}|0\rangle &=0 \hbar|0\rangle \end{aligned} \] (B-1) Explain why and how the operator \( \hat{S}_{z} \) can be represented in the above basis as \[ \hat{S}_{z}=\hbar\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{array}\right) \] (B-2) Now consider the observable \( \hat{A}=-D\left(\hat{S}_{z}\right)^{2} \). Write \( \hat{A} \) in matrix form. What can be said about the eigenspace of \( \hat{A} \) corresponding to its smallest eigenvalue? Write a spectral decomposition of \( \hat{A} \) in terms of two projectors (call them \( \hat{P}_{1} \) and \( \hat{P}_{0} \) ). Write the projectors in matrix form. (B-3) Consider the effective spin operator \( \hat{S}_{z}^{\text {eff }} \), constructed from the subset of basis vectors of \( \hat{S}_{z} \) that are in the eigenspace of \( \hat{P}_{1} \). What does \( \hat{S}_{z}^{\text {eff }} \) look like? What is the effective g-factor \( g_{\text {eff }} \) of the resulting magnetic moment.