Pauli Spin Matrices

#Physics

$$

\sigma_{x}=
\begin{bmatrix}
0, & 1 \\1 & 0
\end{bmatrix},,
\sigma_{y}=\begin{bmatrix}
0 & -i \\i & 0
\end{bmatrix},,
\sigma_{z}=\begin{bmatrix}
1 & 0 \\0 & -1
\end{bmatrix}
$$

$\displaystyle \hat{S}_{i}=\frac{\hbar \sigma_i}{2}$

• I know this is true for $\displaystyle i=3$ or $\displaystyle z$, need to check this later

$$

\lambda_{x}=\left{ -1,1 \right},\mathfrak{B}_{x}=\frac{1}{\sqrt{ 2 }}\left{ \begin{bmatrix}
1 \\-1
\end{bmatrix},\begin{bmatrix}
1 \\1
\end{bmatrix}\right}
$$

• Eigensystem for $\displaystyle x$ spinor

$$

\lambda_{y}=\left{ -1,1 \right},\mathfrak{B}_{y}=\frac{1}{\sqrt{ 2 }}\left{ \begin{bmatrix}
i \\1
\end{bmatrix},\begin{bmatrix}
-i \\1
\end{bmatrix}\right}
$$

• Eigensystem for $\displaystyle y$ spinor

$$

\lambda_{z}=\left{ -1,1 \right},\mathfrak{B}_{z}=\left{ \begin{bmatrix}
0 \\1
\end{bmatrix},\begin{bmatrix}
1 \\0
\end{bmatrix}\right}
$$

• Eigensystem for $\displaystyle z$ spinor