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\begin{center}
{Mathematical Physics --- PHZ 3113}\\\vspace {2mm} 
Homework 10 (April 10, 2013) \\ Pauli Matrices
\end{center}
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\bigskip
\noindent 1. Find three $2\times 2$ matrices $\sigma_i$,
$i=1,2,3$, which fulfill the relations
\begin{eqnarray} \label{epssigma}
  \sigma_i\,\sigma_j &=& i\,\epsilon_{ijk}\,\sigma_k
  ~~{\rm for}~~ i\ne j\,, \\ \label{sigma2}
  \sigma_i\,\sigma_j + \sigma_j\,\sigma_i &=& 
  2\,\delta_{ij}\,1_2\,,
\end{eqnarray} 
where $1_2$ is the $2\times 2$ unit matrix.
\bigskip
Solution: Simple matrices are found from the ansatz
\begin{eqnarray} \nonumber
  \sigma=\left(\begin{matrix}a&0\cr 0&b\end{matrix}\right)
  ~~{\rm or}~~
  \sigma=\left(\begin{matrix}0&c\cr d&0\end{matrix}\right)\,.
\end{eqnarray} 
From (\ref{sigma2}) $\left(\sigma_i\right)^2=1_2$ follows for $i=1,2,3$
and therefore
\begin{eqnarray} \nonumber
  a^2=b^2=1~~{\rm and}~~c\,d=1
\end{eqnarray} 
with solutions 
$$a=\pm 1,\ b=\pm 1,\ c=d=\pm 1,\ c=-d=\pm i\,.$$
Due to (\ref{epssigma}) the unit matrix is excluded as a solution
for $\sigma_i$. With the choice $c=1$, $d=-1$ for $\sigma_1$ 
and $c=-i$, $d=i$ for for $\sigma_2$ we have
\begin{eqnarray} 
  \sigma_1=\left(\begin{matrix}0&1\cr 1&0\end{matrix}\right)\,,~~~
  \sigma_2=\left(\begin{matrix}0&-i\cr i&0\end{matrix}\right)
\end{eqnarray} 
and find $\sigma_3$ from (\ref{epssigma}):
\begin{eqnarray} \nonumber
  \sigma_3 &=& -i\,\sigma_1\,\sigma_2\ =\ -i\,
  \left(\begin{matrix}0&1\cr 1&0\end{matrix}\right)\,
  \left(\begin{matrix}0&-i\cr i&0\end{matrix}\right)
  \\ \label{sigma3} &=&
  \left(\begin{matrix}1&0\cr 0&-1\end{matrix}\right)\,.
\end{eqnarray} 
Relation (\ref{epssigma}) follows by explicit multiplication
of the matrices and for $i\ne j$ relation (\ref{sigma2}) is 
a consequence of the antisymmetry of (\ref{epssigma}) in the
indices $i$ and $j$.
\bigskip

\vfil \newpage
\noindent 2. Find the transformation, which generates the general
solution $\sigma'_i$, $i=1,2,3$ from your previous special solution.
\bigskip

\noindent Solution: Let $A$ be a $2\times 2$ matrix with 
$$ |A|={\rm det}(A)\ne 0\,.$$
Then 
\begin{eqnarray} 
  \sigma'_i&=& A^{-1}\,\sigma_i\,A
\end{eqnarray} 
is the general solution to Eq.~(\ref{epssigma}) and (\ref{sigma2}).
\vfil

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