Question

Consider a system whose Hamiltonian $H$ and an operator $A$ are given by the matrices $$H=\mathcal{E}_{0}\left(\begin{array}{ccc} 0 & -i & 0 \\ i & 0 & 2 i \\ 0 & -2 i & 0 \end{array}\right), \quad A=a_{0}\left(\begin{array}{ccc} 0 & -i & 0 \\ i & 1 & 1 \\ 0 & 1 & 0 \end{array}\right)$$ (a) If we measure energy, what values will we obtain? (b) Suppose that when we measure energy, we obtain a value of $\sqrt{5} \mathcal{E}_{0}$. Immediately afterwards, we measure $A$. What values will we obtain for $A$ and what are the probabilities corresponding to each value? (c) Calculate the expectation value $\langle A\rangle$.

          Consider a system whose Hamiltonian $H$ and an operator $A$ are given by the matrices
$$H=\mathcal{E}_{0}\left(\begin{array}{ccc}
0 & -i & 0 \\
i & 0 & 2 i \\
0 & -2 i & 0
\end{array}\right), \quad A=a_{0}\left(\begin{array}{ccc}
0 & -i & 0 \\
i & 1 & 1 \\
0 & 1 & 0
\end{array}\right)$$
(a) If we measure energy, what values will we obtain?
(b) Suppose that when we measure energy, we obtain a value of $\sqrt{5} \mathcal{E}_{0}$. Immediately afterwards, we measure $A$. What values will we obtain for $A$ and what are the probabilities corresponding to each value?
(c) Calculate the expectation value $\langle A\rangle$.

Added by Cesar O.

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