5.1: Fórmulas útiles
- Page ID
- 110003
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)\[\begin{array}{c} {A = a_{0} 1+\vec{a} \cdot \vec{\sigma} \tilde{A} = a_{0} 1-\vec{a} \cdot \vec{\sigma} A^{\dagger} = a_{0}^{*} 1+\vec{a}^{*} \cdot \vec{\sigma} \bar{A} = \tilde{A^{\dagger}} = a_{0}^{*} 1-\vec{a}^{*} \cdot \vec{\sigma}} \nonumber \end{array}\]
\[\begin{array}{cc} {\frac{1}{2} Tr(A) = a_{0},}&{|A| = a_{0}^{2}-\vec{a}^{2} 1 \frac{1}{2} Tr(A \tilde{A})} \end{array}\]
\[\begin{array}{c} {\frac{1}{2} Tr(A \tilde{B}) = a_{0} b_{0}-\vec{a} \cdot \vec{b}} \end{array}\]
\[\begin{array}{ccc} {A^{-1} = |A|}&{for}&{|A| = 1:A^{-1} = \tilde{A}} \end{array}\]
\[\begin{array}{c} {(\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma} = \vec{a} \cdot \vec{b} 1+i (\vec{a} \times \vec{b}) \cdot \vec{\sigma}} \end{array}\]
\[\begin{array}{cccc} {For}&{\vec{a} \parallel \vec{b}}&{\frac{a_{1}}{b_{1}} = \frac{a_{2}}{b_{2}} = \frac{a_{3}}{b_{3}}}&{\vec{a} \times \vec{b} = 0} \end{array}\]
\[\begin{array}{c} {(\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma})-(\vec{b} \cdot \vec{\sigma})(\vec{a} \cdot \vec{\sigma}) = [(\vec{a} \cdot \vec{\sigma}), (\vec{b} \cdot \vec{\sigma})] = 0} \end{array}\]
\[\begin{array}{ccc} {For}&{A = a_{0}1+\vec{a} \cdot \vec{\sigma},}&{B = b_{0}1+\vec{b} \cdot \vec{\sigma}} \end{array}\]
\[\begin{array}{ccc} {[A, B] = 0}&{iff}&{\vec{a} \parallel \vec{b}} \end{array}\]
\[\begin{array}{c} {\text{For} \vec{a} \perp \vec{b},\vec{a} \cdot \vec{b}}\\ {\{\vec{a} \cdot \vec{\sigma}, \vec{b} \cdot \vec{\sigma}\} \equiv (\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma})+(\vec{b} \cdot \vec{\sigma})(\vec{a} \cdot \vec{\sigma}) = 0} \end{array}\]
\[\begin{array}{c} {A(\vec{b} \cdot \vec{\sigma}) = (\vec{b} \cdot \vec{\sigma}) \tilde{A}} \end{array}\]
\[\begin{array}{c} {U = U(\hat{u}, \frac{\phi}{2}) = \cos \frac{\phi}{2}1-\sin \frac{\phi}{2} \hat{n} \cdot \vec{\sigma} = \exp (-i \frac{\phi}{2} \hat{n} \cdot \vec{\sigma})} \end{array}\]
\[\begin{array}{c} {H = H(\hat{h}, \frac{\mu}{2}) = \cosh \frac{\mu}{2}1+\sinh \frac{\mu}{2} \hat{h} \cdot \vec{\sigma} = \exp (\frac{\mu}{2} \hat{h} \cdot \vec{\sigma})} \end{array}\]
U unimodular unitaria, H hermitiana y positiva.