Quelltext der Seite Benutzer:MovGP0/Physik/Pauli-Matrizen

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== [[Pauli-Matrizen]] ==

=== Definition === 
:<math>\sigma_0 = \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}</math>
:<math>\sigma_1 = \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}</math>
:<math>\sigma_2 = \begin{pmatrix}
0 & -i \\
i & 0
\end{pmatrix}</math>
:<math>\sigma_3 = \begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}</math>
wobei:
:<math>\sigma_0^2 = \sigma_1^2 = \sigma_2^2 = \sigma_3^2 = \mathbb{1}</math> (Einheitsmatrix)

=== Multiplikation ===
:<math>
\begin{pmatrix}
\sigma_0 \, \sigma_0 & \sigma_0 \, \sigma_2 & \sigma_0 \, \sigma_3 & \sigma_0 \, \sigma_3 \\
\sigma_1 \, \sigma_0 & \sigma_1 \, \sigma_1 & \sigma_1 \, \sigma_2 & \sigma_1 \, \sigma_3 \\ 
\sigma_2 \, \sigma_0 & \sigma_2 \, \sigma_1 & \sigma_2 \, \sigma_2 & \sigma_2 \, \sigma_3 \\ 
\sigma_3 \, \sigma_0 & \sigma_3 \, \sigma_1 & \sigma_3 \, \sigma_2 & \sigma_3 \, \sigma_3 \\ 
\end{pmatrix}
=
\begin{pmatrix}
\sigma_0 & \sigma_1 & \sigma_2 & \sigma_3 \\
\sigma_1 & \sigma_0 & \mathrm{i}\,\sigma_3 & -\mathrm{i}\,\sigma_2 \\ 
\sigma_2 & -\mathrm{i}\,\sigma_3 & \sigma_0 & \mathrm{i}\,\sigma_1 \\ 
\sigma_3 & \mathrm{i}\,\sigma_2 & -\mathrm{i}\,\sigma_1 & \sigma_0 \\ 
\end{pmatrix}
</math>

=== Dekomposition von Matrizen ===
<math>\mathbf{A} 
= 
\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} 
= 
\begin{pmatrix}
z_{0}+z_{3} & z_{1}-\mathrm{i}\,z_{2} \\
z_{1}+\mathrm{i}\,z_{2} & z_{0}-z_{3} \\
\end{pmatrix} 
=
z_0\,\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + 
z_1\,\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} + 
z_2\,\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} + 
z_3\,\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
=
z_0\,\sigma_0 + z_1\,\sigma_1 + z_2\,\sigma_2 + z_3\,\sigma_3
</math>

mit: 
:<math>a_{00} = z_0 + z_3,\quad a_{01} = z_1 - \mathrm{i}\,z_2,\quad a_{10} = z_1 + \mathrm{i}\,z_2,\quad a_{11} = z_0-z_3</math>

und: 
:<math>
z_0 = \frac{a_{00} + a_{11}}{2},\quad 
z_1 = \frac{a_{10} + a_{01}}{2},\quad 
z_2 = -\mathrm{i}\frac{a_{10} - a_{01}}{2},\quad 
z_3 = \frac{a_{00} - a_{11}}{2}
</math>

=== Inverse einer Matrix ===
:<math>\mathbf{A}^{-1} = \frac{z_0\,\sigma_0 - z_1\,\sigma_1 - z_2\,\sigma_2 - z_3\,\sigma_3}{z_0^2 - z_1^2 - z_2^2 - z_3^2}</math>