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SO(3) 71 5.7 5.7.1 1 ħ L k l k l k = iϵ kij x i j (5.117) l k SO(3) l z l ± = l 1 ± il = i(y z z y ) ± (z x x z ) = ( x iy) z ± z( x ± i y ) = X ± z ± z (5.118) l z = i(x y y x ) = 1 [(x + iy)( x i y ) (x iy)( x + i y )] (5.119) X ± = x ± iy, ± = 1 ( x i y ), [ +, X + ] = [, X ] = 1 (5.10) X + = X. l ± = (x ± z z ), l z = X + + X (5.11) highest weight state a l 5.7. l + Ψ ll (x, y, z) = 0 (5.1) Ψ ll = a l X l + = a l (x + iy) l (5.13) l k = (X z z + ) k (5.14) (l + m)! Ψ lm = a l (l)!(l m)! ll m X+ l (5.15) H = 1 m p + V (r) = ħ m + V (r) (5.16)

SO(3) 7 = = 1 ( r ) + 1 r r r r ( l ) (5.17) l = 1 ( sin θ ) + sin θ θ θ ϕ (5.18) χ(r)ψ(θ, ϕ) l ψ = αψ (5.19) l 1 = i(sin ϕ θ l = i( cos ϕ θ l 3 = i ϕ + cot θ cos ϕ ϕ ) + cot θ sin ϕ ϕ ) (5.130) l 1 = ( sin θ θ sin θ θ + 1 sin θ ϕ ) (5.131) l + = e iϕ ( θ + i cot θ ϕ ) (5.13) 5.7.3 l = e iϕ ( θ + i cot θ ϕ ) (5.133) l 3 l 17 θ, ϕ Ĵi l, m = l i θ, ϕ l, m (5.134) 17 1. l. m

SO(3) 73 Ĵi l θ, ϕ jm Y lm (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 l i θ, ϕ l, m = dωy lm (θϕ)l i Y lm (θϕ) (5.135) l, m Ĵi l, m = dωy lm (θϕ)l i Y lm (θϕ) (5.136) dω π π l i l i Y lm = θϕ Ĵi l, m = 0 0 sin θdθdϕ (5.137) l m = l Y lm lm Ĵi lm (5.138) 5.7.4 Y lm l 3 highest weight state l 3 Y lm = my lm (5.139) Y lm (θ, ϕ) = ψ lm (θ)e imϕ (5.140) l 3 Y ll = ly ll Y ll (θ, ϕ) = ψ ll (θ)e ilϕ (5.141) Y m l lowest weight state Y l l l + 1. lowest weight state lowest weight state ϕ Y l l = ψ l l (θ)e ilϕ (5.14)

SO(3) 74 lowest weight state l Y l l = e i( l 1)ϕ ( θ + l cot θ)ψ l l(θ) = 0 (5.143) a ψ l l (θ) = a(sin θ) l (5.144) l +. l + ψ lm e imϕ = (l m)(l + m + 1)ψ lm+1 (θ)e i(m+1)ϕ (5.145) ψ l l k ψ l,k l ψ lm = a( 1) l+m (l m)! d (l)!(l + m)! sinm θ( d cos θ )l+m sin l θ (5.146) m = l m = k ψ lk+1 = l + ψ lk e ikϕ = e iϕ ( θ + i cot θ ϕ )ψ lke ikϕ 1 sin k θ (l k)(l + k + 1) θ ( 1 sin k θ ψ lk) = e i(k+1)ϕ ( θ k cot θ)ψ lk = e i(k+1)ϕ sin k θ θ ( 1 sin k θ ψ lk) (5.147) 1 = sin k θ (l + k)(l k + 1) θ ( 1 (l k)! d sin k θ a( 1)l+k (l)!(l + k)! sink θ( d cos θ )l+k sin l θ) = sin k+1 θ(a( 1) l+k+1 (l k 1)! (l)!(l + k + 1)! ( d d cos θ )l+k+1 sin l θ) (5.148) m = k + 1 3. a dω = sin θdθdϕ l, l l, l = π 0 dϕ π 0 dθ sin θψ l, lψ l, l = π a π 0 sin l+1 θ = π a (s l l!) sl + 1 = 1 (5.149)

SO(3) 75 a = 1 (l + 1)! l l! 4π (5.150) Y lm (θ, ϕ) = ( 1)l+m l + 1 (l m)! d l l! 4π (l + m)! sinm θ( d cos θ )l+m sin l θe imϕ (5.151) 5.8 SU() 5.8.1 dynamical symmetry SU() H = ( 1 m p i + 1 ) mω x i (5.15) ρ O() m = ω = 1 a ρ = 1 (x ρ + ip ρ ), a + ρ = 1 (x ρ ip ρ ) (5.153) ħ ħ [(x + ip), (x ip)] = i[p, x] = ħ [a ρ, a + σ ] = δ ρσ (5.154) N H = ħ (a + ρ a ρ + 1 ) = ħ(n + 1) (5.155) N = N 1 + N, N ρ = a ρa ρ (5.156)

SO(3) 76 ( ) ( ) ( ) ( ) a1 a 1 α β a1 = (5.157) γ δ a a ( ) α β,u = U γ δ U = 1 dynamical symmetry( ) 18 1. a ρ = U ρ ρa ρ U [a ρ, a σ ] = U ρ ρu σ σ[a ρ, a σ] = U ρ ρu σ σδ ρσ (5.158) a UU = 1 (5.159). : N N = ( ) ( ) a + 1, a + a 1 (5.160) a ( a1 a ) U() 5.8. U() H ψ = E ψ (5.161) 0 ρ a ρ 0 ρ = 0, p ρ = 1 p! (a + ρ ) p 0 (5.16) 18

SO(3) 77 p ρ p q = δ pq, a ρ p ρ = p + 1 p + 1, a ρ p = p p 1 ρ (5.163) p 1 q = 1 p!q! (a + 1 ) p (a + ) q 0 1 0 (5.164) U() = SU() U(1) 1. SU() a 1, a [N ρ, a + ρ ] = a + ρ, [N ρ, a ρ ] = a ρ (5.165) σ i J i = 1 σi aba aa b (5.166) J 1 = 1 (a 1a + a a 1 ), J = i (a 1a a a 1 ), J 3 = 1 (N 1 N ) (5.167) J ± = J 1 ± ij J + = a 1a, J = a a 1 (5.168) [J +.J ] = [a 1a, a a 1 ] = a 1[a, a ]a 1 + a [a 1, a 1 ]a = N 1 N = J 3 (5.169) [J +, J ] = J 3 (5.170) [J 3, J + ] = J +, [J 3, J ] = J (5.171) J i SO(3)

SO(3) 78. Casimir Casimir J = J3 + 1 (J +J + J J ) = J3 J 3 + J + J = 1 4 (N 1 N ) 1 (N 1 N ) + a 1a a a 1 = 1 4 (N 1 N ) 1 (N 1 N ) + N 1 (N + 1) = 1 4 (N 1 + N ) + 1 (N 1 + N ) = N (N + 1) (5.17) J i J i N [J i, H] = ħ[j i, N] = 0 (5.173) J J 3 H J N j J = j(j + 1) N = j 3. p 1 q = 1 p!q! (a + 1 ) p (a + ) q 0 1 0 (5.174) N p 1 q = p + q p 1 q, J 3 p 1 q = 1 (p q) p 1 q (5.175) j, m p 1 q, j = 1 (p + q), m = 1 (p q), p = j + m, q = j m (5.176) J i j j 1 Z j 0 j p + q = j p, q 0 j + 1 j m j J + j, m = a 1a p 1 q = p + 1 p + 1 1 q q 1 = (p + 1)q j, m + 1 (5.177) j, m + 1 J + j, m = (p + 1)q = (j + m + 1)(j m) (5.178) j, m 1 J j, m = (q + 1)p = (j m + 1)(j + m) (5.179) j m J 3 j, m = mδ jj δ mm (5.180)

SO(3) 79 5.8.3 U(1) SU() U() U(1) U(1) U(1) a i U(1) e iϕ : a i e iϕ a i (5.181) U(1) e iϕ : j, m e (p+q)iϕ j, m = e jiϕ j, m (5.18) U(1) ϕ N = j U(1) 5.9 5.9.1 weight j, m J 3 weight j highest weight highest weight weight J 3 weight highest weight j j weight (diagram) weight lattice J ± (root)

SO(3) 80 weight 5.9. j = 1 weight highest weight j 1, j j 1, m 1, j, m (5.183) j, m 1 = ((j + m)(j m + 1)) 1 J j, m (5.184) j, m + 1 = ((j m)(j + m + 1)) 1 J+ j, m (5.185) 1. j 1, m 1 j, m (5.186) (j 1 + 1)(j + 1)

SO(3) 81. ˆR θ ( j 1, m 1 j, m ) = ( ˆR θ j 1, m 1 )( ˆR θ j, m ) (5.187) ˆR = 1 i ϵĵ (1 i ϵĵ)( j 1, m 1 j, m ) = ((1 i ϵĵ) j 1, m 1 )((1 i ϵĵ) j, m ) = j 1, ( m 1 j, m ) i ϵ (Ĵ j 1, m 1 ) j, m + j 1, m 1 (Ĵ j, m ) + O(ϵ ) 1 19 (5.188) Ĵ( j 1, m 1 j, m ) = (Ĵ j 1, m 1 ) j, m + j 1, m 1 (Ĵ j, m ) (5.189) Ĵ = Ĵ(1) + Ĵ() (5.19) J (i) j i, m i J 0 3. J highest weight state Ĵ3 = Ĵ (1) 3 + Ĵ () 3 (5.193) J 3 J highest weight state m = j max j max, j max T = j 1, j 1 j, j (5.194) J 3 m = j 1 + j j max = j 1 + j 1 Ĵ = Ĵ(1) + Ĵ() (5.195) j max + 1 V jmax = V j1 +j 19 j 1, m 1 j, m (5.190) J = J (1) 1 + 1 J () (5.191) 0 ħ 1 m j 1 + j

SO(3) 8 4. Second highest weight state: Ĵ3 m = j max 1 j 1, j 1 1 j, j, j 1, j 1 j, j 1 (5.196) V j1 +j J j 1, j 1 j, j = j 1 j 1, j 1 1 j, j + j j 1, j 1 j, j 1 (5.197) j j 1, j 1 1 j, j j 1 j 1, j 1 j, j 1 (5.198) Ĵ+ = Ĵ(1) + +Ĵ() + 0 highest weight state j max 1, j max 1 T (5.199) Ĵ (j max 1) + 1 = (j 1 + j 1) + 1 V j1 +j 1 5. V j1 V j = V j1 +j V j1 +j 1 V j1 j (5.00) j 1 j = (j ( 1 + j ) + 1 + (j 1 + j 1) + 1 + j 1 j + 1 ) = (1 j 1 + j ) (1 j 1 j 1 ) = (j 1 + j + 1) ( j 1 j ) = (j 1 + j + j 1 j + 1)(j 1 + j j 1 j + 1) = (j 1 + 1)(j + 1) (5.01) 5.9.3 1)j = 1 j = 1 m (highest weight state) 1, 1 = 1, 1 1, 1 (5.0)

SO(3) 83 J 1, 1 = 1, 1, J + 1, 1 = 1, 1 (5.03) J 1, 1 = 1, 0, J 1, 0 = 1, 1 (5.04) m (highest weight state) J J 1, 1 = J ( 1, 1 1, 1 ) = 1, 1 1, 1 + 1, 1 1, 1 (5.05) 1, 0 = 1 ( 1, 1 1, 1 + 1, 1 1, 1 ) (5.06) 0, 0 = 1 ( 1, 1 1, 1 1, 1 1, 1 ) (5.07) J 1, 0 = 1 ( 1, 1 1, 1 + 1, 1 1, 1 ) = 1, 1 1, 1 (5.08) 1, 1 = 1, 1 1, 1 (5.09) 1, 1 =, 1, 1 = (5.10) 1 1. j = 0 0, 0 = 1 ( ) (5.11). j = 1 1, 1 = 1, 0 = 1 ( + ) 1, 1 = (5.1) 3. V 1 V 1 = V 0 V 1 (5.13)

SO(3) 84 5.9.4 )j = 1 j = 1 3, 1 = 1 3 J 3, 3 3, 1 = 1 J 3, 1 3, 3 = 1 3 J 3, 1 (5.14) J 1, 1 = 1, 0, J 1, 0 = 1, 1 (5.15) m (highest weight state) J 3, 3 = 1, 1 1, 1 (5.16) J 3, 3 = J ( 1, 1 1, 1 ) = 1, 0 1, 1 + 1, 1 1, 1 (5.17) 3, 1 = 3 1, 0 1, 1 + 1 3 1, 1 1, 1 (5.18) 1, 1 = 1 1, 0 1, 1 1, 1 1, 1 (5.19) 3 3 J 3, 1 = ( 1, 1 1, 1 + 1, 0 1, 1 ) + 1, 0 1, 1 3 3 1 = ( 1, 1 1, 1 + 1, 0 1, 1 ) (5.0) 3 3 3, 1 = 1 1, 1 1, 1 + 1, 0 1, 1 (5.1) 3 3 J 3, 1 = 1 1, 1 1, 1 + 4 1, 1 1, 1 3 3 = 3 1, 1 1, 1 = 3 3, 3 (5.) 3, 3 = 1, 1 1, 1 (5.3) j = 1 J J 1, 1 = 1 4 3 3 3 (5.4)

SO(3) 85 1, 1 = 1, 1 1, 1 1 1, 0 1, 1 (5.5) 3 3 3, 1 J 0 V 1 V 1 = V 3 V 1 (5.6) 5.9.5 3 1 1. V 3 m m = ( 3, 1, 1, 3) V 1 m m = ( 1, 1) j, m. 3 3, 3 1 m 1 3. 1. j = 3 j = 1

SO(3) 86 V 3 V 1. j = 1 = V V 1 (5.7) V 1 V 1 = V V 1 V 0 3 3 = 5 + 3 + 1 (5.8) 5.10 5.10.1 E E 3 E i B i v i (x) T R v i (x) = R ij Rv j = e iu (l+l) v i (x) (5.9) R = D 1, J k = l k + L k (5.30) T R v i (x) = e iu J v i (x) (5.31) J l, m 1, m ħ,ħ

SO(3) 87 ( ) ψ 1 ψ = = (ψ α ) (5.3) ψ ψ 1 ψ T R ψ = e iu J ψ (5.33) J = l + S = (l k + 1 σ k) (5.34) Y lm 1, s 5.11 (Wigner-Eckart theorem) 5.11.1 (Clebsch-Gordan coefficients) highest weight j 1 j j 1 j J j 1 + j J, M = j 1, m 1 j, m j 1, m 1 ; j, m J, M (5.35) m 1,(m 1 +m =M) j 1, m 1 ; j, m J, M j 1, m 1 ; j m = j 1, m 1 j, m (5.36) J, M δ JJ δ MM = m 1 J M j 1 m 1 ; j m j 1 m 1 ; j m JM (5.37) M JM JM V J j 1 j J j 1 + j δ m1 m 1 δ m m = M j 1 m 1 ; j m JM JM j 1 m 1; j m (5.38) 5.11. n H n = E n n (5.39)

SO(3) 88 H + H n + E + E (H + H)( n + ) = (E + E)( n + ) (5.40) 1 H n + H = E n + E (5.41) n H m m = n m n n H n = E (5.4) m H n = (E n E m ) m (5.43) m H z H = Kz K H j = 1 5.11.3 ˆT ˆR ˆT n ˆR 1 = ˆT m D mn ( ˆR) (5.44) ψ ˆR ˆR ψ ˆT ˆR ˆT ˆR 1 ˆT ψ ˆR ˆR ˆT ψ (5.45) ˆT j 1 n 1 highest weight j 1 ˆR( ˆT j 1 n 1 j, n ) = ˆT j 1 n 1 j, n D (j 1) m 1 n 1 D (j 1) m n (5.46) V j1 V j T : J, M = j 1, m 1 ; j, m J, M ˆT j 1 m 1 j, m (5.47) ˆT j n m 1,(m 1 +m =M) j, n J T : J, M M J T : J, M = J T J J, M (5.48)

SO(3) 89 ˆT j 1 m 1 j, m = J,m J, m 1 + m j 1, m 1 ; j, m T : J, m 1 + m (5.49) ˆT J, M ˆT j 1 m 1 j, m = J, m 1 + m j 1, m 1 ; j, m J T J δ M,m1 +m (5.50) highest weight J T J = J, J T J, J (5.51) 5.1 ( ) H 0 n = ε n n (5.5) H, H + λh H 0 (H 0 + λh )( n + λ m c m m ) = (ε n + λε )( n + λ m c m m ) (5.53) λ = 0 eq.(5.5) ε k, λ λ λ k H n + c m λ k H 0 m = λε k n + λε n c m k m (5.54) m m ε = n H n (5.55) Example : L l m l, m l + 1 z λh = λp A = λbl z (5.56) m ε = l, m H l, m = Bm (5.57)

SO(3) 90 5.1.1 1 e iħ p ψ = Hψ, H = (5.58) t m e p µ = (H, p) (p µ ea µ ) = (H + eϕ, p e A) (5.59) 1 m e ( p e A) = 1 m e ( p e( p A + A p) + e A ) (5.60) A = 1 B( y, x, 0) (5.61) B = rot A = (0, 0, B) (5.6) z A div A = 0 p A + A p = A p = B(xp y yp x ) = BL z = B L (5.63) H = 1 p + eϕ e B e B L + (x + y ) (5.64) m e m e 8m e M L = e m e L (5.65). M L B 1 M = e ( L m + S) (5.66) e 1 M = e m e ( L + g S) (5.67) g g g (g-factor)

SO(3) 91 g-factor g-factor g-

9 p File: miniexam13.txt 1. Ĵ1, Ĵ, Ĵ3 [Ĵi, Ĵj] = iϵ ijk Ĵ k Ĵ ± = Ĵ1 ± iĵ (5.68) [Ĵ3, Ĵ±] = ±Ĵ±, (5.69) (a) z ϕ ˆR ϕ (b) [Ĵ+, Ĵ ] = Ĵ3 (5.70) (c) Ĵ3 m Ĵ 3 m = m m, m m = δ mm (5.71) m = m ± 1 m Ĵ± m (d) 1 Ĵ1,Ĵ,Ĵ3,Ĵ±

93 6 S = dtl(q, q) (6.1) p i = L q i, H = p i q i L(q, q), q = q(q, p) (6.) S = dt(p i q i H(q, p), (6.3) δs = dtδp i ( q i H p i ) + δq i ( ṗ i H q i ) (6.4) q i = H p i, ṗ i = H q i (6.5) f = {f, H} = f H f H (6.6) q a p a p a q a p i dq i Hdt = P i dq i H dt + d(w (Q, p, t) + p i q i ) (6.7) (p, Q) q i dp i Hdt = P i dq i H dt + W Q i dqi + W p i dp i + W t q i = W p i, P i = W Q i, H = H W t dt (6.8) (6.9) (q, p) (Q, P ) {f(q, p), g(q, p)} p.q = {f(q(q, P ), p(q, P )), g(q(q, P ), p(q, P ))} P,Q (6.10)

94 W = p i f i (Q) (6.11) q i = f i (Q), P i = p j f j Q i (6.1) W = p i a i jq j (6.13) q i = a i jq j, P i = p j a j i (6.14) a i j,z cos ϕ sin ϕ 0 a i j = sin ϕ cos ϕ 0 (6.15) 0 0 1 W ϕ (p, Q) = p 1 (cos ϕq 1 sin ϕq ) p (sin ϕq 1 + cos ϕq ) p 3 Q 3 (6.16) 6.0. ϕ 1 ϕ W ϕ = W 0 + W ϕ ϕ δϕ = p 1Q i (p 1 Q p Q 1 )δϕ = W 0 + L 3 δϕ (6.17) δq i = W p i Q i = L 3 p i = {q i, L 3 }δϕ, δp i = {P i, L 3 } (6.18) δf = {f, L 3 }δϕ = f ϕ = {f, L 3} (6.19) δ ϕ if = {f, L i }δϕ i (6.0) W = W 0 + K a δϕ a (6.1)

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