Chapter 2.2, Problem 17P

The Pauli spin matrices ${\sigma }_1,{\sigma }_2\text{ and }{\sigma }_3$ are defined by

${\sigma }_1=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],{\sigma }_2=\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right]$,

and ${\sigma }_3=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$.

Verify that they satisfy

${\sigma }_1{\sigma }_2=i{\sigma }_3,{\sigma }_2{\sigma }_3=i{\sigma }_1,{\sigma }_3{\sigma }_1=i{\sigma }_2$.

If A and B are $n\times n$ matrices, we define their commutator, denoted $\left[A,B\right]$, by

$\left[A,B\right]=AB-BA$

Thus, $\left[A,B\right]=0$ if and only if A and B commute. That is, $AB=BA$. Problems 19-22 required the commutator.