6.1: Useful formulas
- Page ID
- 31978
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)\(A=a_{0} 1+\vec{a} \cdot \vec{\sigma} \tilde{A}=a_{0} 1-\vec{a} \cdot \vec{\sigma} A^{\dagger}=a_{0}^{*} 1+\vec{a}^{*} \cdot \vec{\sigma} \bar{A}=\tilde{A}^{\dagger}=a_{0}^{*} 1-\vec{a}^{*} \cdot \vec{\sigma}\)
\[\frac{1}{2} \operatorname{Tr}(A)=a_{0}, \quad|A|=a_{0}^{2}-\vec{a}^{2} 1 \frac{1}{2} \operatorname{Tr}(A \tilde{A})\label{1}\]
\[\frac{1}{2} \operatorname{Tr}(A \tilde{B})=a_{0} b_{0}-\vec{a} \cdot \vec{b}\label{2}\]
\[A^{-1}=\frac{\tilde{A}}{|A|} \quad \text { for } \quad|A|=1: A^{-1}=\tilde{A}\label{3}\]
\[(\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma})=\vec{a} \cdot \vec{b} 1+i(\vec{a} \times \vec{b}) \cdot \vec{\sigma}\label{4}\]
\[\text { For } \vec{a} \| \vec{b} \quad \frac{a_{1}}{b_{1}}=\frac{a_{2}}{b_{2}}=\frac{a_{3}}{b_{3}} \quad \vec{a} \times \vec{b}=0\label{5}\]
\[(\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma})-(\vec{b} \cdot \vec{\sigma})(\vec{a} \cdot \vec{\sigma})=[(\vec{a} \cdot \vec{\sigma}),(\vec{b} \cdot \vec{\sigma})]=0\label{6}\]
\[\text { For } \quad A=a_{0} 1+\vec{a} \cdot \vec{\sigma}, \quad B=b_{0} 1+\vec{b} \cdot \vec{\sigma}\label{7}\]
\[[A, B]=0 \quad \text { iff } \quad \vec{a} \| \vec{b}\label{8}\]
\[\begin{equation}
\begin{array}{ll}
\text { For } \quad & \vec{a} \perp \vec{b}, \quad \vec{a} \cdot \vec{b} \\
& \{\vec{a} \cdot \vec{\sigma}, \vec{b} \cdot \vec{\sigma}\} \quad \equiv \quad(\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma})+(\vec{b} \cdot \vec{\sigma})(\vec{a} \cdot \vec{\sigma})=0
\end{array}
\end{equation}\label{9}\]
\[A(\vec{b} \cdot \vec{\sigma})=(\vec{b} \cdot \vec{\sigma}) \tilde{A}\label{10}\]
\[U=U\left(\hat{u}, \frac{\phi}{2}\right)=\cos \frac{\phi}{2} 1-i \sin \frac{\phi}{2} \hat{n} \cdot \vec{\sigma}=\exp \left(-i \frac{\phi}{2} \hat{n} \cdot \vec{\sigma}\right)\label{11}\]
\[H=H\left(\hat{h}, \frac{\mu}{2}\right)=\cosh \frac{\mu}{2} 1+\sinh \frac{\mu}{2} \hat{h} \cdot \vec{\sigma}=\exp \left(\frac{\mu}{2} \hat{h} \cdot \vec{\sigma}\right)\label{12}\]
\(U\) unitary unimodular, H Hermitian and positive.