None 2023
The Pauli spin matrices \(\sigma_x\), \(\sigma_y\), and \(\sigma_z\) are
defined by:
\[\sigma_x=
\begin{pmatrix}
0&1\\ 1&0\\
\end{pmatrix}
\hspace{2em}
\sigma_y=
\begin{pmatrix}
0&-i\\ i&0\\
\end{pmatrix}
\hspace{2em}
\sigma_z=
\begin{pmatrix}
1&0\\ 0&-1\\
\end{pmatrix}
\]
These matrices are related to angular momentum in
quantum mechanics. Prove, and become familiar with, the identities
listed below.
Show that \(\sigma_x \sigma_y = i\sigma_z\) and \(\sigma_y \sigma_x =
-i\sigma_z\). (Note: These identities also hold under a cyclic
permutation of \(\left\{x,y,z\right\}\), e.g. \(x\rightarrow y\),
\(y\rightarrow z\), and \(z\rightarrow x\)).
The commutator of two matrices \(A\) and \(B\) is defined by \(\left[A,
B\right]\buildrel \rm def \over = AB-BA\). Show that \(\left[\sigma_x,
\sigma_y\right] = 2i\sigma_z\). (Note: This identity also holds
under a cyclic permutation of \(\left\{x,y,z\right\}\), e.g.
\(x\rightarrow y\), \(y\rightarrow z\), and \(z\rightarrow x\)).
The anti-commutator of two matrices \(A\) and \(B\) is defined by
\(\left\{A, B\right\}\buildrel \rm def \over = AB+BA\). Show that
\(\left\{\sigma_x, \sigma_y\right\}
= 0\). (Note: This identity also
holds under a cyclic permutation of \(\left\{x,y,z\right\}\), e.g.
\(x\rightarrow y\), \(y\rightarrow z\), and \(z\rightarrow x\)).