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# 6.3: Typical Examples

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Example 1

\begin{aligned} K^{\prime}=H K H, \quad H=& \exp \left(\frac{\mu}{2} \hat{h} \cdot \vec{\sigma}\right) \\ \vec{k}=\vec{k}_{\|}+\vec{k}_{\perp} \quad \vec{k}_{\|}=(\vec{k} \cdot \hat{h}) \hat{h} \end{aligned} \label{1}

By using (6a) and (7b):

$\begin{array}{r} \vec{k}_{\|} \cdot \vec{\sigma} H=H \vec{k}_{\|} \cdot \vec{\sigma}, \quad \vec{k}_{\perp} \cdot \vec{\sigma} H=H^{-1} \vec{k}_{\perp} \cdot \vec{\sigma} \\ \vec{k}_{\|}^{\prime}=\vec{k}_{\|}=k \hat{h} \end{array} \label{2}$

\begin{aligned} \left(k_{0}^{\prime}+\vec{k}_{\|}^{\prime} \cdot \vec{\sigma}\right) &=H^{2}\left(k_{0}+\vec{k}_{\|} \cdot \vec{\sigma}\right) \\ &=(\cosh \mu+\sinh \mu \hat{h} \cdot \vec{\sigma})\left(k_{0}+\vec{k}_{\|} \cdot \vec{\sigma}\right) \end{aligned} \label{3}

$\begin{array}{l} k_{0}^{\prime}=k_{0} \cosh \mu+k \sinh \mu \\ k^{\prime}=k_{0} \sinh \mu+k \cosh \mu \end{array} \label{4}$

Example 2

\begin{aligned} K^{\prime}=U K U^{-1}, & U=\exp \left(-i \frac{\phi}{2} \hat{u} \cdot \vec{\sigma}\right) \\ \vec{k}=\vec{k}_{\|}+\vec{k}_{\perp} \quad \vec{k}_{\|}=(\vec{k} \cdot \hat{u}) \hat{u} \end{aligned} \label{5}

$\vec{k}_{\|} \cdot \vec{\sigma} U^{-1}=U^{-1} \vec{k}_{\|} \cdot \vec{\sigma}, \quad \vec{k}_{\perp} \cdot \vec{\sigma} U^{-1}=U \vec{k}_{\perp} \cdot \vec{\sigma}\label{6}$

$\vec{k}_{\|}^{\prime}=\vec{k}_{\|}\label{7}$

\begin{aligned} \vec{k}_{\perp}^{\prime} \cdot \vec{\sigma} &=\left(\cos \frac{\phi}{2} 1-i \sin \frac{\phi}{2} \hat{u} \cdot \vec{\sigma}\right)^{2} \vec{k}_{\perp} \cdot \vec{\sigma} \\ &=(\cos \phi 1-i \sin \phi \hat{u} \cdot \vec{\sigma}) \vec{k}_{\perp} \cdot \vec{\sigma} \\ \vec{k}_{\perp}^{\prime} &=\cos \phi \vec{k}_{\perp}+\sin \phi \hat{u} \times \vec{k}_{\perp} \end{aligned} \label{8}

6.3: Typical Examples is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by László Tisza (MIT OpenCourseWare) .