L04E04.tex

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\begin{exercise} Verify the commutation relations of Eqs. $4.26$.
\end{exercise}
Let's first recall this set of equations:
\[ [\sigma_x, \sigma_y] = 2i\sigma_z; \]
\[ [\sigma_y, \sigma_z] = 2i\sigma_x; \]
\[ [\sigma_z, \sigma_x] = 2i\sigma_y; \]
For clarity, let's recall the commutator's definition, for two
observable $F$ and $G$:
\[
	[F, G] = FG - GF
\]
And finally, let's recall the Pauli matrices $\sigma_x$, $\sigma_y$
and $\sigma_z$ (from Eqs. $3.20$, at the end of section $3.4$)
\[
	\sigma_z = \begin{pmatrix}
		1 & 0 \\
		0 & -1 \\
	\end{pmatrix};\quad
	\sigma_x = \begin{pmatrix}
		0 & 1 \\
		1 & 0 \\
	\end{pmatrix};\quad
	\sigma_y = \begin{pmatrix}
		0 & -i \\
		i & 0 \\
	\end{pmatrix}
\]
Then this is just elementary matrix multiplication.
\[
	[\sigma_x, \sigma_y] = \begin{pmatrix}
		0 & 1 \\
		1 & 0 \\
	\end{pmatrix}\begin{pmatrix}
		0 & -i \\
		i & 0 \\
	\end{pmatrix} - \begin{pmatrix}
		0 & -i \\
		i & 0 \\
	\end{pmatrix}\begin{pmatrix}
		0 & 1 \\
		1 & 0 \\
	\end{pmatrix} = \begin{pmatrix}
		i & 0 \\
		0 & -i \\
	\end{pmatrix} - \begin{pmatrix}
		-i & 0 \\
		0 & i \\
	\end{pmatrix} = \begin{pmatrix}
		2i & 0 \\
		0 & -2i \\
	\end{pmatrix} = 2i\underbrace{\begin{pmatrix}
		1 & 0 \\
		0 & -1 \\
	\end{pmatrix}}_{\sigma_z} \qed
\]
\[
	[\sigma_y, \sigma_z] = \begin{pmatrix}
		0 & -i \\
		i & 0 \\
	\end{pmatrix}\begin{pmatrix}
		1 & 0 \\
		0 & -1 \\
	\end{pmatrix} - \begin{pmatrix}
		1 & 0 \\
		0 & -1 \\
	\end{pmatrix}\begin{pmatrix}
		0 & -i \\
		i & 0 \\
	\end{pmatrix} = \begin{pmatrix}
		0 & i \\
		i & 0 \\
	\end{pmatrix} - \begin{pmatrix}
		0 & -i \\
		-i & 0 \\
	\end{pmatrix} = \begin{pmatrix}
		0 & 2i \\
		2i & 0 \\
	\end{pmatrix} = 2i\underbrace{\begin{pmatrix}
		0 & 1 \\
		1 & 0 \\
	\end{pmatrix}}_{\sigma_x} \qed
\]
\[
	[\sigma_z, \sigma_x] = \begin{pmatrix}
		1 & 0 \\
		0 & -1 \\
	\end{pmatrix}\begin{pmatrix}
		0 & 1 \\
		1 & 0 \\
	\end{pmatrix} - \begin{pmatrix}
		0 & 1 \\
		1 & 0 \\
	\end{pmatrix}\begin{pmatrix}
		1 & 0 \\
		0 & -1 \\
	\end{pmatrix} = \begin{pmatrix}
		0 & 1 \\
		-1 & 0 \\
	\end{pmatrix} - \begin{pmatrix}
		0 & -1 \\
		1 & 0 \\
	\end{pmatrix} = \begin{pmatrix}
		0 & 2 \\
		-2 & 0 \\
	\end{pmatrix} = 2i\underbrace{\begin{pmatrix}
		0 & -i \\
		i & 0 \\
	\end{pmatrix}}_{\sigma_y} \qed
\]
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