(a) Find the eigenvalues and eigenspinors of S_y .
(b) If you measured S_y on a particle in the general state χ (Equation 4.139), what values might you get, and what is the probability of each? Check that the probabilities add up to 1. Note: a and b need not be real!
\chi=\left(\begin{array}{l} a \\ b \end{array}\right)=a \chi_{+}+b \chi (4.139).
(c) If you measured S_{y}^{2} what values might you get, and with what probabilities?
(a) S_{y}=\frac{\hbar}{2}\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right) ; \quad\left|\begin{array}{cc} -\lambda & -i \hbar / 2 \\ i \hbar / 2 & -\lambda \end{array}\right|=\lambda^{2}-\frac{\hbar^{2}}{4} \Rightarrow \lambda=\pm \frac{\hbar}{2} (of course).
\frac{\hbar}{2}\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right)\left(\begin{array}{l} \alpha \\ \beta \end{array}\right)=\pm \frac{\hbar}{2}\left(\begin{array}{l} \alpha \\ \beta \end{array}\right) \Rightarrow-i \beta=\pm \alpha ; \quad|\alpha|^{2}+|\beta|^{2}=1 \Rightarrow|\alpha|^{2}+|\alpha|^{2}=1 \Rightarrow \alpha=\frac{1}{\sqrt{2}} .
\chi_{+}^{(y)}=\frac{1}{\sqrt{2}}\left(\begin{array}{l} 1 \\ i \end{array}\right) ; \quad \chi_{-}^{(y)}=\frac{1}{\sqrt{2}}\left(\begin{array}{c} 1 \\ -i \end{array}\right) .
(b)
c_{+}=\left(\chi_{+}^{(y)}\right)^{\dagger} \chi=\frac{1}{\sqrt{2}}(1-i)\left(\begin{array}{l} a \\ b \end{array}\right)=\frac{1}{\sqrt{2}}(a-i b) ; \quad+\frac{\hbar}{2}, \text { with probability } \frac{1}{2}|a-i b|^{2} .
c_{-}=\left(\chi_{-}^{(y)}\right)^{\dagger} \chi=\frac{1}{\sqrt{2}}(1 i)\left(\begin{array}{l} a \\ b \end{array}\right)=\frac{1}{\sqrt{2}}(a+i b) ; \quad-\frac{\hbar}{2}, \text { with probability } \frac{1}{2}|a+i b|^{2} .
P_{+}+P_{-}=\frac{1}{2}\left[\left(a^{*}+i b^{*}\right)(a-i b)+\left(a^{*}-i b^{*}\right)(a+i b)\right] .
=\frac{1}{2}\left[|a|^{2}-i a^{*} b+i a b^{*}+|b|^{2}+|a|^{2}+i a^{*} b-i a b^{*}+|b|^{2}\right]=|a|^{2}+|b|^{2}=1 .
(c) \frac{\hbar^{2}}{4}, \text { with probability } 1 .