# 问题：

Verify the commutation relations $[X, Y]=2 i Z ; \quad[Y, Z]=2 i X ; \quad[Z, X]=2 i Y$.

There is an elegant way of writing this using $\epsilon_{j k l}$, the antisymmetric tensor on three indices, for which $\epsilon_{j k l} =0$ except for $\epsilon_{123}=\epsilon_{231}=\epsilon_{312}=1$ and $\epsilon_{321}=\epsilon_{213}=\epsilon_{132}=-1$:$\left[\sigma_{j}, \sigma_{k}\right]=2 i \sum_{l=1}^{3} \epsilon_{j k l} \sigma_{l}$.

## 解答

$X Y=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\left[\begin{array}{cc}0 & -i \\ i & 0\end{array}\right]=\left[\begin{array}{cc}i & 0 \\ 0 & -i\end{array}\right]=i Z$;

$Y X=\left[\begin{array}{rr}0 & -i \\ i & 0\end{array}\right]\left[\begin{array}{rr}0 & 1 \\ 1 & 0\end{array}\right]=\left[\begin{array}{rr}-i & 0 \\ 0 & i\end{array}\right]=-i Z$;

$Y Z=\left[\begin{array}{rr}0 & -i \\ i & 0\end{array}\right]\left[\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right]=\left[\begin{array}{cc}0 & i \\ i & 0\end{array}\right]=i X$;

$Z Y=\left[\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right]\left[\begin{array}{rr}0 & -i \\ i & 0\end{array}\right]=\left[\begin{array}{rr}0 & -i \\ -i & 0\end{array}\right]=-i X$;

$Z X=\left[\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right]\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]=\left[\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right]=i Y$;

$X Z=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\left[\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right]=\left[\begin{array}{rr}0 & -1 \\ 1 & 0\end{array}\right]=-i Y$;

#### 参考

[1]www.qtumist.com