Benutzer:MovGP0/Physik/Pauli-Matrizen

Definition

\sigma _{0}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}

\sigma _{1}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}

\sigma _{2}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}

\sigma _{3}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}

wobei:

\sigma _{0}^{2}=\sigma _{1}^{2}=\sigma _{2}^{2}=\sigma _{3}^{2}=\mathbb {1}

(Einheitsmatrix)

Multiplikation

{\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}}

Dekomposition von Matrizen

$\mathbf {A} ={\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}={\begin{pmatrix}z_{0}+z_{3}&z_{1}-\mathrm {i} \,z_{2}\\z_{1}+\mathrm {i} \,z_{2}&z_{0}-z_{3}\\\end{pmatrix}}=z_{0}\,{\begin{pmatrix}1&0\\0&1\end{pmatrix}}+z_{1}\,{\begin{pmatrix}0&1\\1&0\end{pmatrix}}+z_{2}\,{\begin{pmatrix}0&-i\\i&0\end{pmatrix}}+z_{3}\,{\begin{pmatrix}1&0\\0&-1\end{pmatrix}}=z_{0}\,\sigma _{0}+z_{1}\,\sigma _{1}+z_{2}\,\sigma _{2}+z_{3}\,\sigma _{3}$

mit:

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}

und:

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}}

Inverse einer Matrix

\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}}}

Land Brandenburg

Benutzer:MovGP0/Physik/Pauli-Matrizen

Inhaltsverzeichnis

Pauli-Matrizen

Definition

Multiplikation

Dekomposition von Matrizen

Inverse einer Matrix

What are your Feelings

Wissen

Automated Chatbot

Data Security

Virtual Reality

Communication

Support

Services

FAQ's

Privacy Policy

Terms & Condition

Team

Contact Us

Company

About Us

Services

Features

Our Pricing

Latest News

Benutzer:MovGP0/Physik/Pauli-Matrizen

Pauli-Matrizen

Definition

Multiplikation

Dekomposition von Matrizen

Inverse einer Matrix

What are your Feelings

Share This Article :

Wissen

Automated Chatbot

Data Security

Virtual Reality

Communication

Support

Services

FAQ's

Privacy Policy

Terms & Condition

Team

Contact Us

Company

About Us

Services

Features

Our Pricing

Latest News