Chapter 10, Problem 10B.8P

Interpretation Introduction

Interpretation:

The multiplication table of the Pauli spin matrices has to be constructed.  Whether the four matrices form a group under multiplication or not has to be stated.

Concept introduction:

The Pauli spin matrices are the complex matrices generated in Pauli’s treatment of spin in quantum mechanics.  They are denoted by ${\text{σ}}_{\text{1}}{\text{,σ}}_{\text{2}}\text{and}{\text{σ}}_{\text{3}}$ as shown below.

  $\begin{array}{c}{\sigma }_1={\sigma }_{\text{x}}=\left[\begin{array}{cc}\begin{array}{c}0\\ 1\end{array}& \begin{array}{c}1\\ 0\end{array}\end{array}\right]\\ {\sigma }_2={\sigma }_{\text{y}}=\left[\begin{array}{cc}\begin{array}{c}0\\ i\end{array}& \begin{array}{c}-i\\ 0\end{array}\end{array}\right]\\ {\sigma }_3={\sigma }_{\text{z}}=\left[\begin{array}{cc}\begin{array}{c}1\\ 0\end{array}& \begin{array}{c}0\\ -1\end{array}\end{array}\right]\end{array}$