S+ = ̄h(
0 1
0 0
)
S−χ− = 0
S−χ+ =√
s(s+ 1)−m(m−1) ̄hχ−= ̄hχ−S− = ̄h(
0 0
1 0
)
We can now calculateSxandSy.
Sx=1
2
(S++S−) =
̄h
2(
0 1
1 0
)
Sy=1
2 i(S+−S−) =
̄h
2(
0 −i
i 0)
These are again Hermitian, Traceless matrices.
18.11.7Derive Spin^12 Rotation Matrices*.
In section 18.11.3, we derived the expression for the rotation operator for orbital angular momentum
vectors. The rotation operators for internal angular momentumwill follow the same formula.
Rz(θ) = eiθSz
h ̄ =eiθ 2 σzRx(θ) = eiθ 2 σxRy(θ) = ei
θ 2 σyeiθ 2 σj
=∑∞
n=0(iθ
2)nn!σnjWe now can compute the series by looking at the behavior ofσjn.
σz =(
1 0
0 − 1
)
σ^2 z=(
1 0
0 1
)
σy =(
0 −i
i 0)
σy^2 =(
1 0
0 1
)
σx =(
0 1
1 0
)
σ^2 x=(
1 0
0 1
)
Doing the sums
Rz(θ) = eiθ 2 σz
=
∑∞
n=0(iθ 2 )n
n!^00∑∞
n=0(− 2 iθ)n
n!
=
(
ei
θ 2
0
0 e−i
θ 2)
Ry(θ) =
∑∞
n=0, 2 , 4 ...(iθ 2 )n
n! −i∑∞
n=1, 3 , 5 ...(iθ 2 )n
n!i∑∞
n=1, 3 , 5 ...(iθ 2 )n
n!∑∞
n=0, 2 , 4 ...(iθ 2 )n
n!
=
(
cosθ 2 sinθ 2
−sinθ 2 cosθ 2