5.1: Fórmulas útiles
- Page ID
- 110003
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\[\begin{array}{c} {A = a_{0} 1+\vec{a} \cdot \vec{\sigma} \tilde{A} = a_{0} 1-\vec{a} \cdot \vec{\sigma} A^{\dagger} = a_{0}^{*} 1+\vec{a}^{*} \cdot \vec{\sigma} \bar{A} = \tilde{A^{\dagger}} = a_{0}^{*} 1-\vec{a}^{*} \cdot \vec{\sigma}} \nonumber \end{array}\]
\[\begin{array}{cc} {\frac{1}{2} Tr(A) = a_{0},}&{|A| = a_{0}^{2}-\vec{a}^{2} 1 \frac{1}{2} Tr(A \tilde{A})} \end{array}\]
\[\begin{array}{c} {\frac{1}{2} Tr(A \tilde{B}) = a_{0} b_{0}-\vec{a} \cdot \vec{b}} \end{array}\]
\[\begin{array}{ccc} {A^{-1} = |A|}&{for}&{|A| = 1:A^{-1} = \tilde{A}} \end{array}\]
\[\begin{array}{c} {(\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma} = \vec{a} \cdot \vec{b} 1+i (\vec{a} \times \vec{b}) \cdot \vec{\sigma}} \end{array}\]
\[\begin{array}{cccc} {For}&{\vec{a} \parallel \vec{b}}&{\frac{a_{1}}{b_{1}} = \frac{a_{2}}{b_{2}} = \frac{a_{3}}{b_{3}}}&{\vec{a} \times \vec{b} = 0} \end{array}\]
\[\begin{array}{c} {(\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma})-(\vec{b} \cdot \vec{\sigma})(\vec{a} \cdot \vec{\sigma}) = [(\vec{a} \cdot \vec{\sigma}), (\vec{b} \cdot \vec{\sigma})] = 0} \end{array}\]
\[\begin{array}{ccc} {For}&{A = a_{0}1+\vec{a} \cdot \vec{\sigma},}&{B = b_{0}1+\vec{b} \cdot \vec{\sigma}} \end{array}\]
\[\begin{array}{ccc} {[A, B] = 0}&{iff}&{\vec{a} \parallel \vec{b}} \end{array}\]
\[\begin{array}{c} {\text{For} \vec{a} \perp \vec{b},\vec{a} \cdot \vec{b}}\\ {\{\vec{a} \cdot \vec{\sigma}, \vec{b} \cdot \vec{\sigma}\} \equiv (\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma})+(\vec{b} \cdot \vec{\sigma})(\vec{a} \cdot \vec{\sigma}) = 0} \end{array}\]
\[\begin{array}{c} {A(\vec{b} \cdot \vec{\sigma}) = (\vec{b} \cdot \vec{\sigma}) \tilde{A}} \end{array}\]
\[\begin{array}{c} {U = U(\hat{u}, \frac{\phi}{2}) = \cos \frac{\phi}{2}1-\sin \frac{\phi}{2} \hat{n} \cdot \vec{\sigma} = \exp (-i \frac{\phi}{2} \hat{n} \cdot \vec{\sigma})} \end{array}\]
\[\begin{array}{c} {H = H(\hat{h}, \frac{\mu}{2}) = \cosh \frac{\mu}{2}1+\sinh \frac{\mu}{2} \hat{h} \cdot \vec{\sigma} = \exp (\frac{\mu}{2} \hat{h} \cdot \vec{\sigma})} \end{array}\]
U unimodular unitaria, H hermitiana y positiva.