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}$$

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

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!