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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