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Physics LibreTexts

6.3: Typical Examples

  • Page ID
    31980
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    Example 1

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

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

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

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

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

    Example 2

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

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

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

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


    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) .

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