6.3: Typical Examples
- Last updated
- Mar 22, 2021
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( \newcommand{\kernel}{\mathrm{null}\,}\)
Example 1
K′=HKH,H=exp(μ2ˆh⋅→σ)→k=→k‖+→k⊥→k‖=(→k⋅ˆh)ˆh
By using (6a) and (7b):
→k‖⋅→σH=H→k‖⋅→σ,→k⊥⋅→σH=H−1→k⊥⋅→σ→k′‖=→k‖=kˆh
(k′0+→k′‖⋅→σ)=H2(k0+→k‖⋅→σ)=(coshμ+sinhμˆh⋅→σ)(k0+→k‖⋅→σ)
k′0=k0coshμ+ksinhμk′=k0sinhμ+kcoshμ
Example 2
K′=UKU−1,U=exp(−iϕ2ˆu⋅→σ)→k=→k‖+→k⊥→k‖=(→k⋅ˆu)ˆu
→k‖⋅→σU−1=U−1→k‖⋅→σ,→k⊥⋅→σU−1=U→k⊥⋅→σ
→k′‖=→k‖
→k′⊥⋅→σ=(cosϕ21−isinϕ2ˆu⋅→σ)2→k⊥⋅→σ=(cosϕ1−isinϕˆu⋅→σ)→k⊥⋅→σ→k′⊥=cosϕ→k⊥+sinϕˆu×→k⊥
