问题及解答

Pauli spin 矩阵指的是

\[
\sigma^1=
\begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix},\quad
\sigma^2=
\begin{pmatrix}
0 & i\\
-i & 0
\end{pmatrix},\quad
\sigma^3=
\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix},
\]

它们之间的关系是

\[(\sigma^i)^2=I,\quad i=1,2,3.\]

\[\sigma^i\sigma^j=-\sigma^j\sigma^i\]

\[
\begin{aligned}
\sigma^1\sigma^2=-i\sigma^3,\\
\sigma^2\sigma^3=-i\sigma^1,\\
\sigma^3\sigma^1=-i\sigma^2.
\end{aligned}
\]