Saltar al contenido principal
LibreTexts Español

5.3: Ejemplos típicos

  • Page ID
    110004
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    Ejemplo 1

    \[\begin{array}{cc} {K' = HKH,}&{H = \exp (\frac{\mu}{2} \hat{h} \cdot \vec{\sigma})}\\ {\vec{k} = \vec{k}_{\parallel}+\vec{k}_{\perp}}&{\vec{k}_{\parallel} = (\vec{k} \cdot \hat{h}) \hat{h}} \end{array}\]

    Mediante el uso de (6a) y (7b):

    \[\begin{array}{cc} {\vec{k}_{\parallel} \cdot \vec{\sigma} H = H \vec{k}_{\parallel} \cdot \vec{\sigma},}&{\vec{k}_{\perp} \cdot \vec{\sigma} H = H^{-1} \vec{k}_{\perp} \cdot \vec{\sigma}}\\ {}&{\vec{k}'_{\parallel} = \vec{k}_{\parallel} = k \hat{h}} \end{array}\]

    \[\begin{array}{c} {(k'_{0}+\vec{k}'_{\parallel} \cdot \vec{\sigma}) = H^{2} (k_{0}+\vec{k}_{\parallel} \cdot \vec{\sigma})}\\ {(\cosh \mu+\sinh \mu \hat{h} \cdot \vec{\sigma})(k_{0}+\vec{k}_{\parallel} \cdot \vec{\sigma})} \end{array}\]

    \[\begin{array}{c} {k'_{0} = k_{0} \cosh \mu+ k \sinh \mu}\\ {k' = k_{0} \sinh \mu+ k \cosh \mu} \end{array}\]

    Ejemplo 2

    \[\begin{array}{cc} {K' = UKU^{-1},}&{U = \exp (-i \frac{\phi}{2} \hat{u} \cdot \vec{\sigma})}\\ {\vec{k} = \vec{k}_{\parallel}+\vec{k}_{\perp}}&{\vec{k}_{\parallel} = (\vec{k} \cdot \hat{u}) \hat{u}} \end{array}\]

    \[\begin{array}{cc} {\vec{k}_{\parallel} \cdot \vec{\sigma} U^{-1} = U^{-1} \vec{k}_{\parallel} \cdot \vec{\sigma},}&{\vec{k}_{\perp} \cdot \vec{\sigma} U^{-1} = U \vec{k}_{\perp} \cdot \vec{\sigma}}\\ {}&{\vec{k'}_{\parallel} = \vec{k}_{\parallel}} \end{array}\]

    \[\begin{array}{c} {\vec{k}'_{\perp} \cdot \vec{\sigma} = (\cos \frac{\phi}{2} 1-i \sin \frac{\phi}{2} \hat{u} \cdot \vec{\sigma})^{2} \vec{k}_{\perp} \cdot \vec{\sigma}}\\ {= (\cos \phi 1-i \sin \phi \hat{u} \cdot \vec{\sigma}) \vec{k}_{\perp} \cdot \vec{\sigma}} \end{array}\]

    \[\begin{array}{c} {\vec{k}'_{\perp} \cdot \vec{\sigma} = \cos \phi \vec{k}_{\perp}+\sin \phi \hat{u} \times \vec{k}_{\perp}} \end{array}\]


    This page titled 5.3: Ejemplos típicos is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by László Tisza (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.