Question

a) Check that the spin matrices (Equations 4.145 and 4.147) obey the fundamental commutation relations for angular momentum, Equation 4.134.

$$\begin{gathered} {\mathbf{S}}_{\mathbf{z}}\mathbf{=}\frac{\sout{\mathbf{h}}}{\mathbf{2}}\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\mathbf{}\mathbf{}\left(4.145\right)\mathbf{.} \\ {\mathbf{S}}_{\mathbf{x}}\mathbf{=}\frac{\sout{\mathbf{h}}}{\mathbf{2}}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathbf{s}}_{\mathbf{y}}\mathbf{=}\frac{\sout{\mathbf{h}}}{\mathbf{2}}\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)\mathbf{}\mathbf{}\mathbf{}\left(4.147\right)\mathbf{.} \\ \left[S_x,S_y\right]\mathbf{=}\mathbf{i}\sout{\mathbf{h}}{\mathbf{S}}_{\mathbf{z}}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\left[S_y,S_z\right]\mathbf{=}\mathbf{i}\sout{\mathbf{h}}{\mathbf{S}}_{\mathbf{x}}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\left[S_z,S_x\right]\mathbf{=}\mathbf{i}\sout{\mathbf{h}}{\mathbf{S}}_{\mathbf{y}}\mathbf{}\mathbf{}\mathbf{}\left(4.134\right)\mathbf{.} \\ \mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{}\mathbf{Show}\mathbf{}\mathbf{that}\mathbf{}\mathbf{the}\mathbf{}\mathbf{Pauli}\mathbf{}\mathbf{spin}\mathbf{}\mathbf{matrices}\mathbf{}\mathbf{(}\mathbf{Equation}\mathbf{}\mathbf{4}\mathbf{.}\mathbf{148}\mathbf{)}\mathbf{}\mathbf{satisfy}\mathbf{}\mathbf{the}\mathbf{}\mathbf{product}\mathbf{}\mathbf{rule} \\ {\mathbf{\sigma }}_{\mathbf{x}}\mathbf{\equiv }\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\mathbf{,}\mathbf{}{\mathbf{\sigma }}_{\mathbf{y}}\mathbf{\equiv }\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)\mathbf{,}\mathbf{}{\mathbf{\sigma }}_{\mathbf{z}}\mathbf{\equiv }\mathbf{\left(}\begin{array}{cc}\mathbf{1}& \mathbf{0}\\ \mathbf{0}& \mathbf{-}\mathbf{1}\end{array}\mathbf{\right)}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\left(4.148\right)\mathbf{.} \\ {\mathbf{\sigma }}_{\mathbf{j}}{\mathbf{\sigma }}_{\mathbf{k}}\mathbf{=}{\mathbf{\delta }}_{\mathbf{jk}}\mathbf{}\mathbf{+}\mathbf{i}{\underset{\mathbf{I}}{\mathbf{\sum }\stackrel{\mathbf{\text{'}}}{\mathbf{o}}}}_{\mathbf{jkl}}{\mathbf{\sigma }}_{\mathbf{I}}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\left(4.153\right)\mathbf{}\mathbf{}\mathbf{.}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{} \end{gathered}$$

$$\begin{gathered} \mathbf{Where}\mathbf{}\mathbf{the}\mathbf{}\mathbf{indices}\mathbf{}\mathbf{stand}\mathbf{}\mathbf{for}\mathbf{}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{,}\mathbf{}\mathbf{or}\mathbf{}\mathbf{z}\mathbf{,}\mathbf{}\mathbf{and}\mathbf{}{\stackrel{\mathbf{\text{'}}}{\mathbf{o}}}_{\mathbf{jkl}}\mathbf{}\mathbf{is}\mathbf{}\mathbf{the}\mathbf{}\mathbf{Levi}\mathbf{-}\mathbf{Civita}\mathbf{}\mathbf{symbol}\mathbf{:} \\ \mathbf{+}\mathbf{1}\mathbf{ifjkl}\mathbf{=}\mathbf{123}\mathbf{,}\mathbf{231}\mathbf{,}\mathbf{}\mathbf{or}\mathbf{}\mathbf{2}\mathbf{=}\mathbf{312}\mathbf{;}\mathbf{-}\mathbf{1}\mathbf{}\mathbf{if}\mathbf{}\mathbf{jkl}\mathbf{=}\mathbf{132}\mathbf{,}\mathbf{213}\mathbf{,}\mathbf{}\mathbf{or}\mathbf{}\mathbf{321}\mathbf{;}\mathbf{}\mathbf{otherwise}\mathbf{.} \end{gathered}$$

Short answer

$$\begin{gathered} \left(\mathrm{a}\right)\left[{\mathrm{S}}_{\mathrm{x}},{\mathrm{S}}_{\mathrm{y}}\right]=\mathrm{i}\sout{\mathrm{h}}{\mathrm{S}}_{\mathrm{z}}- \\ \left(\mathrm{b}\right){\mathrm{\sigma }}_{\mathrm{j}}{\mathrm{\sigma }}_{\mathrm{k}}={\mathrm{\delta }}_{\mathrm{jk}}+\mathrm{i}\underset{\mathrm{I}}{\sum \stackrel{\text{'}}{{\mathrm{o}}_{\mathrm{jkl}}}{\mathrm{\sigma }}_{\mathrm{I}}-} \end{gathered}$$

Step-by-step solution

(a) Checking the spin matrices obey the fundamental commutation relations for angular momentum.

The spin matrices are

$$\begin{gathered} S_z=\frac{\sout{h}}{2}\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right) \\ {S}_x=\frac{\sout{h}}{2}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),S_y=\frac{\sout{h}}{2}\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right) \\ \left[S_x,S_y\right]=S_x{S}_y-S_y{S}_x=\frac{{\sout{h}}^2}{2}\left[\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)-\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\right]. \\ =\frac{{\sout{\mathrm{h}}}^2}{4}\left[\left(\begin{array}{cc}\mathrm{i}& 0\\ 0& -\mathrm{i}\end{array}\right)-\left(\begin{array}{cc}-\mathrm{i}& 0\\ 0& \mathrm{i}\end{array}\right)\right] \\ =\frac{{\sout{\mathrm{h}}}^2}{4}\left(\begin{array}{cc}2\mathrm{i}& 0\\ 0& -2\mathrm{i}\end{array}\right)=\mathrm{i}\sout{\mathrm{h}\frac{{\sout{\mathrm{h}}}^2}{2}}\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right) \\ \left[{\mathrm{S}}_{\mathrm{x}},{\mathrm{S}}_{\mathrm{y}}\right]=\mathrm{i}\sout{\mathrm{h}}{\mathrm{S}}_{\mathrm{z}} \end{gathered}$$

(b) Showing the Pauli spin matrices satisfies the product rule.

$$\begin{gathered} {\sigma }_x{\sigma }_x=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)=1={\sigma }_y{\sigma }_y={\sigma }_z{\sigma }_z,So{\sigma }_j{\sigma }_j=1forj=x,y,orz. \\ {\sigma }_x{\sigma }_y=\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right)=i{\sigma }_z;{\sigma }_y{\sigma }_z=\left(\begin{array}{cc}0& i\\ i& 0\end{array}\right)=i{\sigma }_x;{\sigma }_z{\sigma }_x=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)=i{\sigma }_y. \\ {\sigma }_y{\sigma }_x=\left(\begin{array}{cc}-i& 0\\ 0& -i\end{array}\right)=i{\sigma }_z;{\sigma }_z{\sigma }_y=\left(\begin{array}{cc}0& -i\\ -i& 0\end{array}\right)=-i{\sigma }_x;{\sigma }_x{\sigma }_z=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)=-i{\sigma }_y. \end{gathered}$$

Equation 4.153 packages all this in a single formula.

${\mathrm{\sigma }}_{\mathrm{j}}{\mathrm{\sigma }}_{\mathrm{k}}={\mathrm{\delta }}_{\mathrm{jk}}+\mathrm{i}\sum _{\mathrm{I}}\stackrel{\text{'}}{{\mathrm{o}}_{\mathrm{jkl}}{\mathrm{\sigma }}_{\mathrm{I}}.}$