5.6.1 (a) Using the two spin functions \( \varphi_{1}=\alpha \) and \( \varphi_{2}=\beta \) as an orthonormal basis (so \( \langle\alpha \mid \alpha\rangle=\langle\beta \mid \beta\rangle=1,\langle\alpha \mid \beta\rangle=0) \), and the relations
\[
S_{x} \alpha=\frac{1}{2} \beta, \quad S_{x} \beta=\frac{1}{2} \alpha, \quad S_{y} \alpha=\frac{1}{2} i \beta, \quad S_{y} \beta=-\frac{1}{2} i \alpha, \quad S_{z} \alpha=\frac{1}{2} \alpha, \quad S_{z} \beta=-\frac{1}{2} \beta,
\]
construct the \( 2 \times 2 \) matrices of \( S_{x}, S_{y} \), and \( S_{z} \).
(b) Taking now the basis \( \varphi_{1}^{\prime}=C(\alpha+\beta), \varphi_{2}^{\prime}=C(\alpha-\beta) \) :
(i) Verify that \( \varphi_{1}^{\prime} \) and \( \varphi_{2}^{\prime} \) are orthogonal,
(ii) Assign \( C \) a value that makes \( \varphi_{1}^{\prime} \) and \( \varphi_{2}^{\prime} \) normalized,
(iii) Find the unitary matrix for the transformation \( \left\{\varphi_{i}\right\} \rightarrow\left\{\varphi_{i}^{\prime}\right\} \).
(c) Find the matrices of \( S_{x}, S_{y} \), and \( S_{z} \) in the \( \left\{\varphi_{i}^{\prime}\right\} \) basis.
