sowatamura/kougi/gp2012_5.pdf回転群so(3) 71 5.7 球面調和関数 5.7.1...
TRANSCRIPT
-
SO(3) 91
5.7
5.7.1
1!Lk
!k
!k = −i"kijxi∂j (5.119)
!k SO(3)
!± = !1 ± i!2 = −i(y∂z − z∂y)± (z∂x − x∂z)= (∓x− iy)∂z ± z(∂x ± i∂y) = ∓X±∂z ± 2z∂∓ (5.120)
!z
!z = −i(x∂y − y∂x) =1
2[(x+ iy)(∂x − i∂y)− (x− iy)(∂x + i∂y)] (5.121)
X± = x± iy , ∂± =1
2(∂x ∓ i∂y) , [∂+, X+] = [∂−, X−] = 1 (5.122)
X∗+ = X−.
!± = ∓(x±∂z − 2z∂∓) , !z = X+∂+ −X−∂− (5.123)
highest weight state
!+Ψll(x, y, z) = 0 (5.124)
al
Ψll = alXl+ = al(x+ iy)
l (5.125)
!k− = (X−∂z − 2z∂+)k (5.126)
Ψlm = al
√(l +m)!
(2l)!(l −m)!!l−m− X
l+ (5.127)
-
SO(3) 92
5.7.2
H =1
2m$p2 + V (r) =
−!22m
∇2 + V (r) (5.128)
∆ = ∇2 = 1r2
∂
∂r
(r2∂
∂r
)+
1
r2(−$!2) (5.129)
− $!2 = 1sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
∂2
∂φ2(5.130)
χ(r)ψ(θ,φ)
$!2ψ = αψ (5.131)
!1 = i(sinφ∂
∂θ+ cot θ cosφ
∂
∂φ)
!2 = i(− cosφ∂
∂θ+ cot θ sinφ
∂
∂φ)
!3 = −i∂
∂φ(5.132)
$!2 = −( 1sin θ
∂
∂θsin θ
∂
∂θ+
1
sin2 θ
∂2
∂φ2) (5.133)
!+ = eiφ(
∂
∂θ+ i cot θ
∂
∂φ) (5.134)
!− = e−iφ(− ∂
∂θ+ i cot θ
∂
∂φ) (5.135)
-
SO(3) 93
5.7.3
!3 !2 19
〈θ,φ|Ĵi|l,m〉 = !i〈θ,φ|l,m〉 (5.136)
Ĵi ! θ,φ|jm〉 Ylm(θ,φ) = 〈θ,φ|l,m〉
〈l,m′|Ĵi|l,m〉 =∫
dΩ〈l,m′|θ,φ〉〈θ,φ|Ĵi|l,m〉
=
∫dΩ〈l,m′|θ,φ〉!i〈θ,φ|l,m〉
=
∫dΩYlm′(θφ)!iYlm(θφ) (5.137)
〈l,m′|Ĵi|l,m〉 =∫
dΩYlm′(θφ)!iYlm(θφ) (5.138)
∫dΩ ∫ π
0
∫ 2π
0
sin θdθdφ (5.139)
!i
!iYlm = 〈θφ|Ĵi|l,m〉 =l∑
m′=−l
Ylm′〈lm′|Ĵi|lm〉 (5.140)
5.7.4
Ylm !3
!3Ylm = mYlm (5.141)
19
1. l
2. m
-
SO(3) 94
Ylm(θ,φ) = ψlm(θ)eimφ (5.142)
highest weight state
!3Yll = lYll ⇒ Yll(θ,φ) = ψll(θ)eilφ (5.143)
Y m llowest weight state Yl−l !+
1. lowest weight state lowest weight state φ
Yl−l = ψl−l(θ)e−ilφ (5.144)
lowest weight state
!−Yl−l = ei(−l−1)φ(− ∂
∂θ+ l cot θ)ψl−l(θ) = 0 (5.145)
a
ψl−l(θ) = a(sin θ)l (5.146)
!+
2.
!+ψlmeimφ =
√(l −m)(l +m+ 1)ψlm+1(θ)ei(m+1)φ (5.147)
ψl−l k ψl,k−l
ψlm = a(−1)l+m√
(l −m)!(2l)!(l +m)!
sinm θ(d
d cos θ)l+m sin2l θ (5.148)
m = −l m = k
!+ψlkeikφ = eiφ(
∂
∂θ+ i cot θ
∂
∂φ)ψlke
ikφ
= ei(k+1)φ(∂
∂θ− k cot θ)ψlk
= ei(k+1)φ sink θ∂
∂θ(
1
sink θψlk) (5.149)
-
SO(3) 95
ψlk+1 =1√
(l − k)(l + k + 1)sink θ
∂
∂θ(
1
sink θψlk)
=1√
(l + k)(l − k + 1)sink θ
∂
∂θ(
1
sink θa(−1)l+k
√(l − k)!
(2l)!(l + k)!sink θ(
d
d cos θ)l+k sin2l θ)
= sink+1 θ(a(−1)l+k+1√
(l − k − 1)!(2l)!(l + k + 1)!
(d
d cos θ)l+k+1 sin2l θ) (5.150)
m = k + 1
3. a dΩ = sin θdθdφ
〈l,−l|l,−l〉 =∫ 2π
0
dφ
∫ π
0
dθ sin θψ∗l,−lψl,−l = 2π|a|2∫ π
0
sin2l+1 θ = 2π|a|2(sll!)2 2sl + 1
= 1
(5.151)
a =1
2ll!
√(2l + 1)!
4π(5.152)
! "Ylm(θ,φ) =
(−1)l+m
2ll!
√2l + 1
4π
√(l −m)!(l +m)!
sinm θ(d
d cos θ)l+m sin2l θeimφ (5.153)
# $5.8 SU(2)
5.8.1 dynamical symmetry
SU(2)
H =∑
ρ
(1
2mp2i +
1
2mω2x2i
)(5.154)
O(2)
-
SO(3) 96
m = ω = 1
aρ =1√2!
(xρ + ipρ) , a+ρ =
1√2!
(xρ − ipρ) (5.155)
[(x+ ip), (x− ip)] = 2i[p, x] = 2! ⇒ [aρ, a+σ ] = δρσ (5.156)
H = !∑
(a+ρ aρ +1
2) = !(N + 1) (5.157)
N
N = N1 +N2 , Nρ = a†ρaρ (5.158)
(a1a2
)→
(a′1a′2
)=
(α βγ δ
)(a1a2
)(5.159)
,U =
(α βγ δ
)U †U = 1
dynamicalsymmetry( )
20
1. a′ρ′ = Uρ′ρaρ
[a′ρ′ , a′†σ′ ] = Uρ′ρU
∗σ′σ[aρ, a
†σ] = Uρ′ρU
∗σ′σδρσ (5.160)
UU † = 1 (5.161)
U
2. : N
N =(a+1 , a
+2
)( a1a2
)(5.162)
(a1a2
)
U(2)20
-
SO(3) 97
5.8.2
U(2)
H|ψ〉 = E|ψ〉 (5.163)
|0〉ρ
aρ|0〉ρ = 0 , |p〉ρ =1√p!(a+ρ )
p|0〉 (5.164)
|p〉ρ
〈p|q〉 = δpq , a†ρ|p〉ρ =√
p+ 1|p+ 1〉 , aρ|p〉 =√p|p− 1〉ρ (5.165)
|p〉1|q〉2 =1√p!q!
(a+1 )p(a+2 )
q|0〉1|0〉2 (5.166)
U(2) = SU(2)×U(1)
1. SU(2)a1, a2
[Nρ, a+ρ ] = a
+ρ , [Nρ, aρ] = −aρ (5.167)
J i =1
2σiaba
†aab (5.168)
σi
J1 =1
2(a†1a2 + a
†2a1) , J2 =
−i2(a†1a2 − a
†2a1) , J3 =
1
2(N1 −N2) (5.169)
J± = J1± iJ2J+ = a
†1a2 , J− = a
†2a1 (5.170)
[J+.J−] = [a†1a2, a
†2a1] = a
†1[a2, a
†2]a1 + a
†2[a
†1, a1]a2 = N1 −N2 = 2J3 (5.171)
-
SO(3) 98
[J+, J−] = 2J3 (5.172)
[J3, J+] = J+ , [J3, J−] = −J− (5.173)
Ji SO(3)
2. Casimir Casimir
J2 = J23 +1
2(J+J− + J−J) = J
23 − J3 + J+J−
=1
4(N1 −N2)2 −
1
2(N1 −N2) + a†1a2a
†2a1
=1
4(N1 −N2)2 −
1
2(N1 −N2) +N1(N2 + 1)
=1
4(N1 +N2)
2 +1
2(N1 +N2)
=N
2(N
2+ 1) (5.174)
Ji JiN
[Ji, H] = ![Ji, N ] = 0 (5.175)
J2 J3 H J2N2 j J
2 = j(j + 1) N = 2j
3.
|p〉1|q〉2 =1√p!q!
(a+1 )p(a+2 )
q|0〉1|0〉2 (5.176)
N
2|p〉1|q〉2 =
p+ q
2|p〉1|q〉2 , J3|p〉1|q〉2 =
1
2(p− q)|p〉1|q〉2 (5.177)
|j,m〉 ≡ |p〉1|q〉2 , j =1
2(p+ q) , m =
1
2(p− q) , p = j +m , q = j −m (5.178)
Ji j j12Z + j ≥ 0 j
p + q = 2j p, q ≥ 02j + 1 −j ≤ m ≤ j
-
SO(3) 99
J+|j,m〉 = a†1a2|p〉1|q〉2 =√
p+ 1|p+ 1〉1√q|q − 1〉 =
√(p+ 1)q|j,m+ 1〉 (5.179)
〈j,m+ 1|J+|j,m〉 =√(p+ 1)q =
√(j +m+ 1)(j −m) (5.180)
〈j,m− 1|J−|j,m〉 =√(q + 1)p =
√(j −m+ 1)(j +m) (5.181)
〈j′m′|J3|j,m〉 = mδjj′δmm′ (5.182)
5.8.3 U(1)
SU(2)U(2) U(1) U(1)
U(1) ai
U(1) + eiφ : ai → eiφai (5.183)
U(1) + eiφ : |j,m〉 → e(p+q)iφ|j,m〉 = e2jiφ|j,m〉 (5.184)U(1) φ N = 2j
U(1)
5.8.4 SU(3)
U(2)U(3) N U(N)
5.8.4.1
H =3∑
ρ=1
a†ρaρ +3
2=
3∑
ρ=1
Nρ +3
2(5.185)
,! = 1 m = ω = 1 Nρ
[Nρ, a+ρ ] = a
+ρ , [Nρ, aρ] = −aρ (5.186)
-
SO(3) 100
|k,m, n〉 = (a†1)k(a†2)
m(a†3)n|0〉 (5.187)
k +m+ l = N
aja†i (5.188)
Ta =1
2a†ρλaρσaσ (5.189)
λa = σa , λ4 =
0 0 10 0 01 0 0
, λ5 =
0 0 −i0 0 0i 0 0
,
λ6 =
0 0 00 0 10 1 0
, λ7 =
0 0 00 0 −i0 i 0
, λ8 =1√3
1 0 00 1 00 0 −2
, (5.190)
Tr{(12λa)(
1
2λb)} =
1
2δab (5.191)
[Ta, Tb] =1
4a†[λa,λb]a = fabcTc (5.192)
fabc SU(3)
5.8.4.2
1. N = 1, 3 (4 3C2)
|1, 0, 0〉 (5.193)
2. N = 2, 6 (5 4C2)
|2, 0, 0〉 |1, 1, 0〉 (5.194)
3. N = 3, 5C2 = 10
4. N = 4, 6C2 = 15
-
SO(3) 101
5.8.5
L =1
2m(ṙ2 + r2(θ̇2 + sin2 θφ̇2) (5.195)
U(r) = −kr
(5.196)
H =p2
2m+ U(r) , L = r× p (5.197)
A =1
kmp× L− r
r(5.198)
Aṙ = (1
kmp× L− r
r) · r (5.199)
Ar cos θ =1
mkL2 − r (5.200)
H =k2m
2L2(A2 − 1) (5.201)
∣∣∣∣p+mkA× LL2
∣∣∣∣ =mk
L(5.202)
5.9
5.9.1
5.9.2 weight
j,mJ3 weight j highest weighthighest weight weight
J3 weight
highest weight j−j
-
SO(3) 102
weight (diagram) weightlattice J±
(root)
weight
j = 1 weight
5.9.3
! "highest weight j1, j2
|j1,m1〉 , |j2,m2〉 (5.203)
# $
|j,m− 1〉 = ((j +m)(j −m+ 1))− 12J−|j,m〉 (5.204)
|j,m+ 1〉 = ((j −m)(j +m+ 1))− 12J+|j,m〉 (5.205)
-
SO(3) 103
1.
|j1,m1〉|j2,m2〉 (5.206)
(2j1 + 1)(2j2 + 1)
2.
R̂%θ(|j1,m1〉|j2,m2〉) = (R̂%θ|j1,m1〉)(R̂%θ|j2,m2〉) (5.207)
R̂ = 1− i$"Ĵ
(1− i$"Ĵ)(|j1,m1〉|j2,m2〉) = ((1− i$"Ĵ)|j1,m1〉)((1− i$"Ĵ)|j2,m2〉)= |j1,m1〉|j2,m2〉
−i$"((Ĵ|j1,m1〉)|j2,m2〉+ |j1,m1〉(Ĵ|j2,m2〉)
)+O("2)
(5.208)
1
Ĵ(|j1,m1〉|j2,m2〉) = (Ĵ|j1,m1〉)|j2,m2〉+ |j1,m1〉(Ĵ|j2,m2〉) (5.209)
21
Ĵ = Ĵ(1) + Ĵ(2) (5.212)
J (i) |ji,mi〉J 22
3. J highest weight stateĴ3 = Ĵ
(1)3 + Ĵ
(2)3 (5.213)
J3 J highest weight statem = jmax
|jmax, jmax〉T = |j1, j1〉|j2, j2〉 (5.214)
21
|j1,m1〉 ⊗ |j2,m2〉 (5.210)
J = J(1) ⊗ 1+ 1⊗ J(2) (5.211)
22 !
-
SO(3) 104
J3 m = j1 + j2jmax = j1 + j2 23
Ĵ− = Ĵ(1)− + Ĵ
(2)− (5.215)
2jmax + 1 Vjmax = Vj1+j2
4. Second highest weight state: Ĵ3 m = jmax − 1
|j1, j1 − 1〉|j2, j2〉 , |j1, j1〉|j2, j2 − 1〉 (5.216)
Vj1+j2
J−|j1, j1〉|j2, j2〉 =√
2j1|j1, j1 − 1〉|j2, j2〉+√2j2|j1, j1〉|j2, j2 − 1〉 (5.217)
√2j2|j1, j1 − 1〉|j2, j2〉 −
√2j1|j1, j1〉|j2, j2 − 1〉 (5.218)
Ĵ+ = Ĵ(1)+ +Ĵ
(2)+
0 highest weight state
|jmax − 1, jmax − 1〉T (5.219)
Ĵ− 2(jmax − 1)+ 1 =2(j1 + j2 − 1) + 1 Vj1+j2−1
5.
! "Vj1 ⊗ Vj2 = Vj1+j2 ⊕ Vj1+j2−1 ⊕ · · ·⊕ V|j1−j2| (5.220)# $
|j1 − j2|
= 2(j1 + j2) + 1 + 2(j1 + j2 − 1) + 1 + · · · 2|j1 − j2|+ 1= 2
((1 j1 + j2 )− (1 |j1 − j2|− 1 )
)
= (j1 + j2 + 1)2 − (|j1 − j2|)2
= (j1 + j2 + |j1 − j2|+ 1)(j1 + j2 − |j1 − j2|+ 1)= (2j1 + 1)(2j2 + 1) (5.221)
23 m j1 + j2
-
SO(3) 105
5.9.4 1)j = 12 ⊗ j =12
m (highest weight state)
|1, 1〉 = |12 ,12〉|
12 ,
12〉 (5.222)
J−|12 ,12〉 = |
12 ,−
12〉 , J+|
12 ,−
12〉 = |
12 ,
12〉 (5.223)
J−|1, 1〉 =√2|1, 0〉 , J−|1, 0〉 =
√2|1,−1〉 (5.224)
m (highest weight state) J−
J−|1, 1〉 = J−(|12 ,12〉|
12 ,
12〉) = |
12 ,−
12〉|
12 ,
12〉+ |
12 ,
12〉|
12 ,−
12〉 (5.225)
|1, 0〉 = 1√2
(|12 ,−
12〉|
12 ,
12〉+ |
12 ,
12〉|
12 ,−
12〉)
(5.226)
|0, 0〉 = 1√2
(|12 ,−
12〉|
12 ,
12〉 − |
12 ,
12〉|
12 ,−
12〉)
(5.227)
J−|1, 0〉 =1√2
(|12 ,−
12〉|
12 ,−
12〉+ |
12 ,−
12〉|
12 ,−
12〉)=
√2|12 ,−
12〉|
12 ,−
12〉 (5.228)
|1,−1〉 = |12 ,−12〉|
12 ,−
12〉 (5.229)
|12 ,12〉 = | ↑〉 , |
12 ,−
12〉 = | ↓〉 (5.230)
-
SO(3) 106
2 12! "
1. j = 0
|0, 0〉 = 1√2(| ↓〉| ↑〉 − | ↑〉| ↓〉) (5.231)
2. j = 1
|1, 1〉 = | ↑〉| ↑〉|1, 0〉 = 1√
2(| ↓〉| ↑〉+ | ↑〉| ↓〉)
|1,−1〉 = | ↓〉| ↓〉 (5.232)
3.
V12⊗ V1
2= V0 ⊕ V1 (5.233)
# $5.9.5 2)j = 1⊗ j = 12
|32 ,12〉 =
1√3J−|32 ,
32〉
|32 ,−12〉 =
1
2J−|32 ,
12〉
|32 ,−32〉 =
1√3J−|32 ,−
12〉 (5.234)
J−|1, 1〉 =√2|1, 0〉 , J−|1, 0〉 =
√2|1,−1〉 (5.235)
m (highest weight state)
|32 ,32〉 = |1, 1〉|
12 ,
12〉 ∗ (5.236)
J−
J−|32 ,32〉 = J−(|1, 1〉|
12 ,
12〉) =
√2|1, 0〉|12 ,
12〉+ |1, 1〉|
12 ,−
12〉 (5.237)
|32 ,12〉 =
√23 |1, 0〉|
12 ,
12〉+
1√3|1, 1〉|12 ,−
12〉 ∗ (5.238)
|12 ,12〉 =
√13 |1, 0〉|
12 ,
12〉 −
√23 |1, 1〉|
12 ,−
12〉 ∗ ∗ (5.239)
-
SO(3) 107
J−|32 ,12〉 =
√23(√2|1,−1〉|12 ,
12〉+ |1, 0〉|
12 ,−
12〉) +
√23 |1, 0〉|
12 ,−
12〉
= 2(√
13 |1,−1〉|
12 ,
12〉+
√23 |1, 0〉|
12 ,−
12〉) (5.240)
|32 ,−12〉 =
√13 |1,−1〉|
12 ,
12〉+
√23 |1, 0〉|
12 ,−
12〉 ∗ (5.241)
J−|32 ,−12〉 =
√13 |1,−1〉|
12 ,−
12〉+
√43 |1,−1〉|
12 ,−
12〉
=√3|1,−1〉|12 ,−
12〉
=√3|32 ,−
32〉 (5.242)
|32 ,−32〉 = |1,−1〉|
12 ,−
12〉 ∗ (5.243)
j = 12 J−
J−|12 ,12〉 =
√23 |1,−1〉|
12 ,
12〉+
√13 |1, 0〉|
12 ,−
12〉 −
√43 |1, 0〉|
12 ,−
12〉 (5.244)
|12 ,−12〉 =
√23 |1,−1〉|
12 ,
12〉 −
√13 |1, 0〉|
12 ,−
12〉 ∗ ∗ (5.245)
|32 ,−12〉 J− 0
∗ ∗∗ 2
V1 ⊗ V 12= V 3
2⊕ V 1
2(5.246)
5.9.6
32
12
! "1. V 3
2m m = (−32 ,−
12 ,
12 ,
32) V 12 m 2 m = (−
12 ,
12)
|j,m〉
2. 32 |32 ,
32〉
12 m
12
3.
# $
-
SO(3) 108
1. j = 32 ⊗ j =12
V 32⊗ V 1
2= V2 ⊕ V1 (5.247)
2. 2 j = 1
V1 ⊗ V1 = V2 ⊕ V1 ⊕ V03 × 3 = 5 + 3 + 1 (5.248)
5.10
5.10.1
!" #$$E$E 3
5.10.1.1 EiBi
vi(x)
TRvi(x) = RijRvj = e−iu·(l+L)vi(x) (5.249)
R = D1,
Jk = lk + Lk (5.250)
TRvi(x) = e−iu·Jvi(x) (5.251)
-
SO(3) 109
J 24 |l,m〉|1,m′〉
5.10.1.22
ψ =
(ψ1
ψ2
)= (ψα) (5.252)
ψ1 ψ2
TRψ = e−iu·Jψ (5.253)
J = l + S = (lk +1
2σk) (5.254)
Ylm|12 , s〉
5.11 (Wigner-Eckart theorem)
5.11.1 (Clebsch-Gordan coefficients)
highest weight j1 j2 |j1 − j2| ≤ J ≤ j1 + j2
|J,M〉 =∑
m1,(m1+m2=M)
|j1,m1〉|j2,m2〉〈j1,m1; j2,m2|J,M〉 (5.255)
〈j1,m1; j2,m2|J,M〉
|j1,m1; j2m2〉 = |j1,m1〉|j2,m2〉 (5.256)
1. 〈J ′,M ′|
δJJ ′δMM ′ =∑
m1,m2
〈J ′M ′|j1m1; j2m2〉〈j1m1; j2m2|JM〉 (5.257)
2. |j1−j2| ≤ J ≤ j1+j2 |J,M〉 Vj1⊗Vj2∑
J,M |J,M〉〈J,M | =1
δm1m′1δm2m′2 =∑
J,M
〈j1m1; j2m2|JM〉〈JM |j1m′1; j2m′2〉 (5.258)
24 ! ,!
-
SO(3) 110
5.11.2
|n〉
H|n〉 = En|n〉 (5.259)
H+∆H|n〉+ |∆〉 E +∆E
(H +∆H)(|n〉+ |∆〉) = (E +∆E)(|n〉+ |∆〉) (5.260)
1
∆H|n〉+H|∆〉 = ∆E|n〉+ E|∆〉 (5.261)
|n〉 H 〈m| m = n
〈n|∆H|n〉 = ∆E (5.262)
m 3= n〈m|∆H|n〉 = (En − Em)〈m|∆〉 (5.263)
|m〉 ∆Hz ∆H = Kz
K ∆H j = 1
5.11.3
ψ T̂ R̂
|ψ〉 T̂→ T̂ |ψ〉R̂ ↓ R̂ ↓R̂|ψ〉 R̂T̂ R̂
−1→ R̂T̂ |ψ〉
(5.264)
, T̂
R̂T̂nR̂−1 = T̂mDmn(R̂) (5.265)
DmnT̂n
T̂ j1n1 highest weight j1
R̂(T̂ j1n1 |j2, n2〉) = T̂j1n1 |j2, n2〉D
(j1)m1n1D
(j1)m2n2 (5.266)
-
SO(3) 111
Vj1 ⊗ Vj2
|T : J,M〉 =∑
m1,(m1+m2=M)
〈j1,m1; j2,m2|J,M〉T̂ j1m1 |j2,m2〉 (5.267)
T̂ jn |j, n〉 J|T : J,M〉 M J−
|T : J,M〉 = 〈J ||T ||J〉|J,M〉 (5.268)
T̂ j1m1 |j2,m2〉 =∑
J,m
〈J,m1 +m2|j1,m1; j2,m2〉|T : J,m1 +m2〉 (5.269)
T̂
〈J,M |T̂ j1m1 |j2,m2〉 = 〈J,m1 +m2|j1,m1; j2,m2〉〈J ||T ||J〉δM,m1+m2 (5.270)
highest
weight
〈J ||T ||J〉 = 〈J, J |T |J, J〉 (5.271)
5.12 ( )
H0|n〉 = εn|n〉 (5.272)H ′ , H + λH ′
H0
(H0 + λH′)(|n〉+ λ
∑
m
cm|m〉) = (εn + λε′)(|n〉+ λ∑
m
cm|m〉) (5.273)
λ = 0 eq.(5.272) ε′
〈k| , λ λ
λ〈k|H ′|n〉+∑
m
cmλ〈k|H0|m〉 = λε′〈k|n〉+ λεn∑
m
cm〈k|m〉 (5.274)
-
SO(3) 112
ε′ = 〈n|H ′|n〉 (5.275)
Example : L2
!m |!,m〉 2!+ 1 z
λH ′ = λp ·A = λBLz (5.276)
ε′ = 〈!,m|H ′|!,m〉 = Bm (5.277)m
5.12.1 Example:
Si = 12σi
j = 12 |j, s〉 S3
S3|1
2, s〉 = !s|1
2, s〉 , s = ±1
2(5.278)
| ↑〉 = |12,1
2〉, | ↓〉 = |1
2,−1
2〉 (5.279)
! = 1
S3| ↑〉 =1
2| ↑〉 , S3| ↓〉 = −
1
2| ↓〉 (5.280)
−→S (i)
H = H0 +HIH0 = E(|
−→S (1)|2 + |−→S (2)|2)
HI = 2λ−→S (1) ·−→S (2) (5.281)
Ψ(s, s′) = |s〉1|s′〉2 , s, s′ =↑ , ↓ (5.282)
1.
Ψ(↑, ↑) = | ↑〉1| ↑〉2 (5.283)
-
SO(3) 113
2. H0
HΨ(s, s) = E(|−→S (1)|2 + |−→S (2)|2)Ψs, s′ = E(12(1
2+ 1) +
1
2(1
2+ 1))Ψs, s′ =
3
2Ψ(s, s′)
(5.284)
3. HI −→J =
−→S (1) +
−→S (2) (5.285)
−→J 2 = (
−→S (1) +
−→S (2))2 = (
−→S (1))2 + (
−→S (2))2 + 2
−→S (1) ·−→S (2) (5.286)
2−→S (1)
−→S (2) =
−→J 2 − (−→S (1))2 − (−→S (2))2 (5.287)
4.
HI = λ(j(j + 1)−3
2)) (5.288)
j
5. j = 1, 0(5.289)
6. HB = 2µB(S(1)3 +S
(2)3 ) j = 1
5.12.2
12
−e
i! ∂∂tψ = Hψ, H =
$p2
2me(5.290)
pµ = (H, $p) → (pµ − eAµ) = (H + eφ, $p− e $A) (5.291)
1
2me($p− e $A)2 = 1
2me($p2 − e($p · $A+ $A · $p) + e2 $A2) (5.292)
$A =1
2B(−y, x, 0) (5.293)
$B = rot $A = (0, 0, B) (5.294)
-
SO(3) 114
z $A div $A = 0
$p · $A+ $A · $p = 2 $A · $p = B(xpy − ypx) = BLz = $B · $L (5.295)
H =1
2me$p2 + eφ− e
2me$B · $L+ e
2B2
8me(x2 + y2) (5.296)
$ML =e
2me$L (5.297)
. $ML · $B
12
$M =e
2me($L+ 2$S) (5.298)
12
$M =e
2me($L+ g$S) (5.299)
g gg (g-factor)
g-factor 2 g-2
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