Chapter 6, Problem 6.32P

(a)

To determine

To show that $\left(\overrightarrow{a}\cdot \overrightarrow{\sigma }\right)\left(\overrightarrow{b}\cdot \overrightarrow{\sigma }\right)=\overrightarrow{a}\cdot \overrightarrow{b}+i\left(\overrightarrow{a}\times \overrightarrow{b}\right)\cdot \overrightarrow{\sigma }$ .

(b)

To determine

To show that $\text{exp}\left[-i\frac{\phi }{\hslash }\widehat{n}\cdot S\right]=\cos \left(\frac{\phi }{2}\right)-i\sin \left(\frac{\phi }{2}\right)\widehat{n}\cdot \overrightarrow{\sigma }$ .

(c)

To determine

To show that the result of part (b) becomes the matrix $R_n=\cos \left(\frac{\phi }{2}\right)\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)-i\sin \left(\frac{\phi }{2}\right)\left(\begin{array}{cc}\cos \theta & \sin \theta e^{-i\varphi }\\ \sin \theta e^{i\varphi }& -\cos \theta \end{array}\right)$ in the standard basis of spin up and spin down along the z axis.

(d)

To determine

To verify that $R_n$ is unitary.

(e)

To determine

To compute the matrix ${S^{\prime }}_x=R^{†}{S}_{x}R$ .explicitly and to verify that it returns the expected result.

(c)

To determine

To construct the matrix for a $\pi$ rotation about the x axis and to verify that it turns an up spin into a down spin.

(g)

To determine

To construct the matrix for a $2\pi$ rotation about the z axis and the reason why the result is surprising.