推 sunev:用SU(2)去證SO(3)... 10/24 22:23
推 ed78617:感謝，不過有沒有SO(3)的證法啊？ 10/24 22:28
※ 引述《ed78617 (雞爪)》之銘言：
: [領域]古典力學 (題目相關領域)
: [來源] Goldstein (課本習題、考古題、參考書...)
: [題目]
: Show that a rotation about any given axis can be obtained as the product of
: two successive rotations, each through 180°
: [瓶頸] (寫寫自己的想法，方便大家為你解答)
: 請教一下要如何下手
: 是利用球面三角形的餘弦律和正弦律嗎？
: 先謝謝各位~
太久沒碰古力了 我用量子力學證明
Take two 3-d unit vector n1 and n2.
in the case of n1˙n2 = cos(γ) and n1×n2=n3*sin(γ),
γ the angle between two vectors and n3 is the unit vector orth. to n1 and n2.
A vector V after a rotation around n1 by π
becomes exp(-iσ˙n1*π) V exp(iσ˙n1*π/2), where σ is the Pauli matrix
By using exp{iσ˙n1*π}=cos(π/2)+i*sin(π/2)*(σ˙n1)=i*(σ˙n1)
(-1)*exp{iσ˙n1*π}*exp{iσ˙n2*π} = (σ˙n1)(σ˙n2)
=1 (n1˙n2) + i σ˙(n1×n2) = cos(γ) + i (σ˙n3) sin(γ)
= exp{i(σ˙n3)*γ},
which stands for a rotation about the n3-axis by 2*γ.
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