6.3: Ejemplos típicos
- Page ID
- 124743
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Ejemplo 1
\ [\ begin {ecuación}
\ begin {alineado}
K^ {\ prime} =H K H,\ quad H=&\ exp\ izquierda (\ frac {\ mu} {2}\ hat {h}\ cdot\ vec {\ sigma}\ derecha)\\ vec {k} =
\ vec {k} =\ vec {k} _ _ {\ |} +\ vec {k} _ {\ perp}\ quad\ vec {k} _ {\ |} =(\ vec {k}\ cdot\ hat {h})\ hat {h}
\ end {alineado}
\ end { ecuación}\ label {1}\]
Al usar (6a) y (7b):
\ [\ begin {ecuación}
\ begin {array} {r}
\ vec {k} _ {\ |}\ cdot\ vec {\ sigma} H=H\ vec {k} _ {\ |}\ cdot\ vec {\ sigma},\ quad\ vec {k} _ _ {\ perp}\ cdot\ vec {\ sigma} ve=H^ {-1}\ c {k} _ {\ perp}\ cdot\ vec {\ sigma}\
\ vec {k} _ {\ |} ^ {\ prime} =\ vec {k} _ {\ |} =k\ sombrero {h}
\ end {array}
\ end {ecuación}\ etiqueta {2}\]
\ [\ begin {ecuación}
\ begin {alineada}
\ izquierda (k_ {0} ^ {\ prime} +\ vec {k} _ {\ |} ^ {\ prime}\ cdot\ vec {\ sigma}\ derecha) &=H^ {2}\ izquierda (k_ {0} +\ vec {k} _ _ {\ |}\ cdot\ vec {\ sigma} derecha)\\
& =(\ cosh\ mu+\ sinh\ mu\ hat {h}\ cdot\ vec {\ sigma})\ izquierda (k_ {0} +\ vec {k} _ {\ |}\ cdot\ vec {\ sigma}\ derecha)
\ end {alineado}
\ end {ecuación}\ label {3}\]
\ [\ begin {ecuación}
\ 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 {array}
\ end {ecuación}\ label {4}\]
Ejemplo 2
\ [\ begin {ecuación}
\ begin {alineado}
K^ {\ prime} =U K U^ {-1}, & U=\ exp\ izquierda (-i\ frac {\ phi} {2}\ hat {u}\ cdot\ vec {\ sigma}\ derecha)\\ vec {k} =
\ vec {k} =\ vec {k} _ _ {\ |} +\ vec {k} _ {perp}\ quad\ vec {k} _ {\ |} =(\ vec {k}\ cdot\ hat {u})\ hat {u}
\ end {alineado}
\ end { ecuación}\ etiqueta {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 {ecuación}
\ begin {alineado}
\ vec {k} _ {\ perp} ^ {\ prime}\ cdot\ vec {\ sigma} &=\ left (\ cos\ frac {\ phi} {2} 1-i\ sin\ frac {\ phi} {2}\ hat {u}\ cdot\ vec {\ sigma}\ derecha) ^ {2}\ vec c {k} _ {\ perp}\ cdot\ vec {\ sigma}\\
& =(\ cos\ phi 1-i\ sin\ phi\ sombrero {u}\ cdot\ vec {\ sigma}) \ vec {k} _ {\ perp}\ cdot\ vec {\ sigma}\
\ vec {k} _ {\ perp} ^ {\ prime} &=\ cos\ phi\ vec {k} _ {\ perp} +\ sin\ phi\ hat {u}\ veces\ vec {k} _ _ {\ perp}
\ end {alineado}
\ end {ecuación}\ final\ etiqueta {8}\]