In the basis of spin-up and spin-down states, the operator \(\hat{S}_x\) has matrix representation \[\hat{S}_x = \frac{\hbar}{2}\, \begin{pmatrix}0&1\\1&0\end{pmatrix}.\] The eigenvalues and eigenvect...In the basis of spin-up and spin-down states, the operator \(\hat{S}_x\) has matrix representation \[\hat{S}_x = \frac{\hbar}{2}\, \begin{pmatrix}0&1\\1&0\end{pmatrix}.\] The eigenvalues and eigenvectors are \[\begin{align} \begin{aligned}s_x = \;\;\frac{\hbar}{2},\; &\;\;\; |\!+\!x\rangle = \frac{1}{\sqrt{2}}\Big(|\!+\!z\rangle + |\!-\!z\rangle\Big) \\ s_x = -\frac{\hbar}{2}, &\;\;\; |\!-\!x\rangle = \frac{1}{\sqrt{2}}\Big(|\!+\!z\rangle - |\!-\!z\rangle\Big).\end{aligned}\end{align}\] Convers…
