文档介绍：1
−→−→σ 0 1 0 −i 1 0 0 1
s = 2 ;σ1 = ;σ2 = ;σ3 = s+ = ;
1 0 ! i 0 ! 0 −1 ! 0 0 !
0 0 1 0 1 0
s−= ;
s+ = 0;s+ = ;s−= 0;
1 0 ! 0 ! 1 ! 0 ! 1 !
1 0
s−= ;
0 ! 1 !
2 3
exp(−in·s) Taylor exp(−in·s) = 1+(−in·s)+(−in·s) /2!+(−in·s) /3!+ · · ·;
2 1 2 m
℄
(n·s) =ninj /2(sisj +sj si) = ninj /2(δi,j/2) = ( 2 ) ; m=2j+1 (−in·sθ) /m! =
j 2j+1 j 2j
(−1) (θ/2) m (−1) θ
(−2i) (2j+1)! n · s; m=2j
(−in · sθ) /m! = (2j)! ; −2in · s sin(θ/2);
cos(θ/2); exp(−in · s) = cos(θ/2) − in · σ sin(θ/2); n=ey ;
1/2 cos(θ/2) − sin θ/2
′
dm′,m = cos(θ/2)δm ,m − iσy sin(θ/2) =
sin(θ/2) cos(θ/2) !
Æ
℄
exp(−iJnθ)|ψi = λ(n, θ)|ψi; θθ= 0 Jn|ψi = κn|ψi; κn
Æ
n
x y [Jx, Jy]|ψi = iJz|ψi = [κx, κy]|ψi = 0|ψi; Jx|ψi = Jx|ψi = 0|ψi;
|ψi J = 0 ; λ(n, θ) = 1.
e = sin(θ) cos(ϕ)i + sin(θ) sin(ϕ)j + cos(θ)k e = g(ϕ, θ, 0)k; g(ϕ, θ, 0)
+ +
(ϕ, θ, 0)
e α D(e, α) = D(g)D(k, α)D (g); Je = D(g)JkD (g);
′ j
′
Je D(g)ψj,m, m. D(g)ψj,m = m′ exp(−im ϕ)ψj,m dm′,m(θ).
|ψi |n, jmi , n P
g ′ j
|ψi = |n, jmihn, jm|ψi; |ψ i = D(g)|ψi = D(g) |n, jmihn, jm|ψi = |n, jm iDm′,m(g)hn, jm|ψi,
n,j,m n,j,m ′
X X n,j,m,mX
∗∗′ j 2 ′
g I I ′′