(a) The possible energies are given by the eigenvalues of H. A diagonalization of H yields three nondegenerate eigenenergies E_{1}=0,E_{2}=-\varepsilon _{0},and E_{3}=2\varepsilon . The respective eigenvectors are
|\phi _{1}〉=\frac{1}{\sqrt{2}}\left(\begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \right), |\phi _{2}〉=\left(\begin {matrix} 0 \\ 0 \\ 1 \end{matrix} \right), |\phi _{3}〉=\frac{1}{\sqrt{2}}\left(\begin{matrix} -1 \\ 1 \\ 0 \end{matrix} \right); (3.183)
these eigenvectors are orthonormal.
(b) If a measurement of the energy yields -\varepsilon _{0}, his means that the system is left in the state |\phi _{2}〉. When we measure the next observable, A, the system is in the state |\phi _{2}〉. The result we obtain for A is given by any of the eigenvalues of A. A diagonalization of A yields three nondegenerate values: a_{1}=-\sqrt{17}a, a_{2}=0, and a_{3} = \sqrt{17}a;their respective eigenvectors are given by
|a_{1}〉=\frac{1}{\sqrt{34}}\left(\begin{matrix} 4 \\ -\sqrt{17} \\ 1 \end{matrix} \right), |a_{2}〉=\frac{1}{\sqrt{17}} \left(\begin{matrix} 1 \\ 0 \\ -4 \end{matrix} \right), |a_{3}〉=\frac{1}{\sqrt{2}}\left(\begin{matrix} 4 \\ \sqrt{17} \\ 1 \end{matrix} \right). (3.184)
Thus, when measuring A on a system which is in the state |\phi _{2}〉, the probability of finding -\sqrt{17}a is given by
P_{1}(a_{1})=\left|〈a_{1}|\phi _{2}〉\right| ^{2}=\left|\frac{1}{\sqrt{34}}\left(\begin{matrix} 4 & -\sqrt{17} & 1\end{matrix} \right) \left(\begin{matrix} 0 \\ 0 \\ 1 \end{matrix} \right) \right| ^{2}=\frac {1}{34}. (3.185)
Similarly, the probabilities of measuring 0 and \sqrt{17}a are
P_{2}(a_{2})=\left|〈a_{2}|\phi _{2}〉\right| ^{2}=\left|\frac{1}{\sqrt{17}}\left(\begin{matrix} 1 & 0 & -4\end{matrix} \right)\left (\begin{matrix} 0 \\ 0 \\ 1 \end{matrix} \right) \right| ^{2}=\frac{16}{17}, (3.186)
P_{3}(a_{3})=\left|〈a_{3}|\phi _{2}〉\right| ^{2}=\left|\frac{1}{\sqrt{34}}\left(\begin{matrix} 4 & \sqrt{17} & 1\end{matrix} \right)\left(\begin{matrix} 0 \\ 0 \\ 1 \end{matrix} \right) \right| ^{2}=\frac{1}{34}. (3.187)
(c) Since the system, when measuring A is in the state |\phi _{2}〉, the uncertainty Δ A is given by \Delta A =\sqrt{\left|〈\phi _{2}|A^{2}|\phi _{2}〉\right| ^{2}-\left|〈\phi _{2}|A|\phi _{2}〉\right| ^{2}} , where
〈\phi _{2}|A|\phi _{2}〉=a\left(\begin{matrix} 0 & 0 & 1\end{matrix} \right)\left(\begin{matrix} 0 & 4 & 0 \\ 4 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} \right)\left(\begin{matrix} 0 \\ 0 \\ 1 \end{matrix} \right)=0, (3.188)
〈\phi _{2}|A^{2}|\phi _{2}〉=a^{2}\left(\begin{matrix} 0 & 0 & 1\end{matrix} \right)\left(\begin{matrix} 0 & 4 & 0 \\ 4 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} \right)\left(\begin{matrix} 0 & 4 & 0 \\ 4 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} \right)\left(\begin{matrix} 0 \\ 0 \\ 1 \end{matrix} \right)=a^{2}. (3.189)
Thus we have Δ A = a.