量子力学3-2013

( 3 ) 5 8 5 03 Email: hatsugai.yasuhiro.ge@u.tsukuba.ac.jp

3 5.............................. 5........................ 5........................ 6.............................. 8....................... 8.......................... 8..3...................... 0 5........................... 5............................. 8.3..................... 3.3.................... 3.4.................................. 8.4........................... 8.4....................... 30.4.3......................... 33.4.4............... 34.4.5.............. 36.4.6...................... 4.5............................. 46.5......................... 46.5. :........ 49.5.3 :....... 57.5.4............................ 63.6............ 68.6....................... 68.6..................... 69

.6.3................... 7.6.4................ 74 3 79 3.......................... 79 3.......................... 79 3................... 8 3..3................... 83 3..4.......... 85 3..5............. 86 3............... 87 3.. Boson.............. 87 3.......................... 90 3..3 j /....................... 93 3..4 j l 0,,,......... 95 3..5........................... 99 3..6................. 0 3..7 SU()............. 03 3..8 3j............ 06 Appendices 8 A 9 A...................... 9 A.................... A.3................ A.4........................ 6 B 9 B........................... 9

... x a T a ψ(x) ψ (x) ψ (x + a) ψ(x) x + a T a x x T a x x + a ψ T a ψ ψ (x ) ψ(x) x a ψ T a ψ T a T a ψ(x) ψ (x) ψ(t a x) ψ(x a) ψ(x) aψ () (x) + a ψ() (x) ( a) n ψ (n) (x) n! n0 (a x ) n ψ (n) (x) n! n0 e a x ψ(x) T a e a x e iapx/ħ 5

6 : 03 ( 5 8 5 ) p x iħ x p x p x T a e +iap x/ħ e +iap x/ħ T a T at a T a T a.. U ψ Uψ ψ ψ Uψ, U U O, (O O ) ψ O ψ ψ UOU ψ O O UOU ψ O ψ ψ O ψ O UOU O O [O, U] 0 δλ U λ e iδλg/ħ, G G

: 03 ( 5 8 5 ) 7 U G (δλ G ħ () ) G ( δλ ) U e iδλg/ħ δψ ψ ψ iλgψ δo O O iδλ[g, O]/ħ [G, O] 0 O iħψ Hψ Ψ(t) e iht/ħ Ψ(0) G G t Ψ(t) G Ψ(t) d dt G t d dt Ψ(0) eiht/ħ Ge iht/ħ Ψ(0) (i/ħ) Ψ(0) e iht/ħ [H, G]e iht/ħ Ψ(0) e iδλg/ħ [H, G] 0 G t H e iδλg/ħ [H, G] 0 G G O ( + iδλg/ħ + )O( iδλg/ħ + ) O + iδλ[g, O]/ħ

8 : 03 ( 5 8 5 )... δτ T δτ t t T τ t t + τ ψ(t) ψ (t) U δτ ψ(t) ψ (t ) ψ(t) ψ (t) ψ(t δτ) U δτ ψ(t) δτ δψ ψ(t δτ) ψ(t) δτ t ψ δψ δτ(hψ)/(iħ) iδτ Hψ/ħ U δτ e +ihδτ/ħ.. 3 a T a r r r + a ψ ψ

: 03 ( 5 8 5 ) 9 ψ (r ) ψ(r) ψ (r) ψ(r a) a δψ a ψ ia (iħ ψ)/ħ ia pψ/ħ, p iħ U δa e iδa p ( ) H 0 p m [H, p] 0 ( ) U T a e ia p/ħ δa G δa p δr i i[ δa p/ħ, r i ] iδa j [r i, p j /ħ] a i δr δa r r r δa ψ (r) ψ(r δa) T a r r r + a δp i[δa p, p] 0 ψ x ψ dxψ (x)x ψ (x) dxψ(x a)(x a)ψ(x a) dxψ(x)(x)ψ(x) ψ x ψ

0 : 03 ( 5 8 5 )..3 R R 3 3 r x y z r Rr r rr r RRr RR E 3, R R O(3) (det R) det R SO(3) R R E 3 + δr RR E 3 + δr + δr E 3 δr δω (δr) ij ϵ ijk δω k (δr) i (δrr) i ϵ ijk r j δω k δr δω r r δω δω δω

: 03 ( 5 8 5 ) 3 ψ (Rr) ψ(r), ψ (r) ψ(r r) δψ iδω Lψ/ħ L r p iħ r U δω e iδω L/ħ r G δω L δr i i[ δω L/ħ, r i ] iδω j [r i, L j /ħ] iδω j ϵ jkl r k [r i, p l /ħ] δω j ϵ jkl r k δ il δω j ϵ jki r k ϵ ikj r k δω j δr r δω δω r 3 ψ (Rr) ψ(r), ψ (r) ψ(r r) δψ ψ(r r) ψ(r) ψ(r δrr) ψ(r) ψ(r δr) ψ(r) (δr) ψ (δω r) ψ δω (r )ψ iδω (r p)ψ/ħ iδω Lψ/ħ

: 03 ( 5 8 5 ) δp i i[ δω L/ħ, p i ] iδω j [ L j /ħ, p i ] iδω j ϵ jab [r a p b, p i ]/ħ δω j ϵ jab δ ai p b δω j ϵ jib p b δp δω p 4 [L i, L j ] iħ(r i p j r j p i ) ϵ ijk L k ϵ ijk ϵ kab r a p b (δ ia δ jb δ ib δ ja )r a p b r i p j r j p b [L i, L j ] iħϵ ijk L k δl i i[δω L/ħ, L i ] iδω j [L i, L j /ħ] δω j ϵ ijk L k δl δω L R V 4 V x V y V z 3 () V UV U U e iω L/ħ [L i, L j ] [ϵ iab r a p b, ϵ jcd r c p d ] ϵ iab ϵ jcd [r a p b, r c p d ] ϵ iab ϵ jcd { ra [p b, r c p d ] + [r a, r c p d ]p b } iħϵ iab ϵ jcd { ra δ bc p d + r c δ ad p b } iħϵiab ϵ jbd r a p d + iħϵ iab ϵ jca r c p b iħϵ iab ϵ jdb r a p d iħϵ iba ϵ jca r c p b iħ(δ ij δ ad δ id δ aj )r a p d iħ(δ ij δ bc δ ic δ jb )r c p b iħ(δ ij r p r j p i ) iħ(δ ij r p r i p j ) iħ(r i p j r j p i )

: 03 ( 5 8 5 ) 3 V RV, R SO(3) V (δr) ij ϵ ijk δω k U iδω L/ħ δv i i[δω L, V i ]/ħ δv i ( δr ij V ) j ϵ jik δω k V j ( δω V ) i δv [δω L, V ]/ħ ( δω V ) i δr δω r δp δω r δl δω L r, p, L δω j iδω j [L j, V i ]/ħ ϵ ijk δω j V k [L j, V i ] iħϵ ijk V k [L i, V j ] iħϵ ijk V k

4 : 03 ( 5 8 5 ) V, V V V δ(v V ) i[δω L/ħ, V V )] iv [δω L/ħ, V ] i[δω L/ħ, V ] V V (δω V ) (δω V ) V 0 H 0 p, m H hyd H 0 + Z, 3 4πϵ 0 r H osc H 0 + mω r

. J J x J y J z [J i, J j ] iϵ ijk ħj k J L J J iħj a, b (c- ) [(a J), (b J)] a i b j [J i, J k ] iħϵ ijk a i b j J k iħ(a b) J J [J, J] 0 J J J z ( ϵ ija [J i, J j ] ϵ ija J i J j ϵ ija J j J i ϵ ija J i J j ϵ jia J i J j ϵ ija J i J j (J J) a iϵ ija ϵ ijk J k /ħ iδ ak J k ij a /ħ [J, J i ] [J j J j, J i ] J j [J j, J i ] + [J j, J i ]J j iϵ ijk (J j J k J k J j ) 0 5

6 : 03 ( 5 8 5 ) J x, J y ) J J z J ± J x ± ij y J x (J + + J ) J y i (J + J ) 3 [J, J ± ] 0 [J z, J ± ] ±ħ(j x ± ij y ) ±ħj ± [J +, J ] ħj z 4 J J (J +J + J J +) + J z J z J J J J (J +J + J J + ) + J z J z (J +J J J + ) 3 [J z, J ± ] [J z, J x ± ij y ] iħ(j y ij x ) ħj ± ±ħ(j x ± ij y ) ±ħj ± [J +, J ] [J x + ij y, J x ij y ] i[j x, J y ] + i[j y, J x ] J z 4 J J J xj x + J xj x + J z J z 4 (J + + J )(J + + J ) 4 (J + J )(J + J ) + J z J z (J +J + J J +) + J z J z

: 03 ( 5 8 5 ) 7 J + J J J z (J z ħ) J J + J J z (J z + ħ)

8 : 03 ( 5 8 5 ). [J, J z ] 0 jm J jm j(j + )ħ jm J z jm mħ jm jm j m δ jj δ mm jm jm jm J j(j + ) 0 [J z, J ± ] ±ħj ± jm, jm jm J ± jm (m m )ħ jm J ± jm ± jm J ± jm jm J ± jm 0, m m ± J z J + jm [J z, J + ] jm + J + J z jm ħ(m + )J + jm J + m J + jm jm + J J jm jm jm J ± m

: 03 ( 5 8 5 ) 9 jm + CJ + jm C J J + J J z (J z + ħ) jm + jm + C jm J J + jm ħ [ j(j + ) m(m + ) ] ħ (j m)(j + m + ) C ( ) jm + ħ (j m)(j + m + ) J + jm jm + J + jm ħ (j m)(j + m + ) jm CJ jm C J + J J J z (J z ħ) jm jm C jm J + J jm ħ [ j(j + ) m(m ) ] ħ (j + m)(j m + ) C jm ħ (j + m)(j m + ) J jm jm J jm ħ (j + m)(j m + ) jm + jm ħ (j + m + )(j m) J + jm ħ (j + m + )(j m) J jm + ħ (j + m + )(j m) J J + jm ħ (j + m + )(j m) (J J z (J z + ħ)) jm (j(j + ) m(m + ))) jm (j + m + )(j m) (j m)(j + m + ) jm jm (j + m + )(j m)

0 : 03 ( 5 8 5 ) J x + J y J J z (J +J + J J + ) jm (J J z ) jm ħ [j(j + ) m ] ( J jm + J + jm ) 0 j(j + ) m j(j + ) m J ± m m, m J J + J J z (J z + ) J + jm 0 J j m 0 J + jm /ħ j(j + ) m (m + ) m j J + J J J z (J z ) (j + m )(j m ) + j m (j m )(j + m + ) 0 J + jj 0 J j m /ħ j(j + ) + m ( m + ) m j (j + m )(j m ) + j + m (j + m )(j m + ) 0 J j j 0 jj 0 (J ) k jj jj k 0 k j k j j k 0,,, 3,

: 03 ( 5 8 5 ) j k 0,,, 3, jm J jm ħ (j + m)(j m + ) jm + J + jm ħ (j + m + )(j m) j j + J x, J y, J z jm + ħ (j + m + )(j m) J + jm ħ [ (j + m + )! (j + m)! ] / (j m)! J + jm (j (m + ))! jm + (j + m + )(j + m + )(j m)(j m ) (J + ) jm ħ [ (j + m + )! (j + m)! ] / (j m)! (J + ) jm (j (m + )! jm + k (j + m + ) (j + m + k) (j m) (j m k + ) (J + ) k jm ħ k [ (j + m)! (j + m + k)! ] / (j (m + k)! (J + ) k jm (j m)! jm ħ k [ (j + m k)! (j + m)! ] / (j m)! (J + ) k jm k (j m + k)!

: 03 ( 5 8 5 ) [ ] (j + m)! jm (J + ) m m jm (j m / )! ħ m m, (m > m ) (j + m )! (j m)! [Schwinger(.4)] jm ħ (j + m)(j m + ) J jm jm ħ (j + m)(j + m )(j m + )(j m + )) J jm jm k ħ k (j + m) (j + m k + ) (j m + ) (j m + k) (J ) k jm [ ] / (j + m k)! ħ k (j m)! (J ) k jm (j + m)! (j (m k)! jm ħ k [ (j + m)! (j + m + k)! ] / (j m k)! (J ) k jm + k (j m)! [ ] jm (J ) m m jm ħ m m (j + m / )! (j m)!, m < m (j + m)! (j m )! [Schwinger(.6)]

: 03 ( 5 8 5 ) 3.3 L r p.3. r cos ϕ sin θ r sin ϕ sin θ r cos θ 3 r x y, (Ω θ, ϕ ) Y lm (Ω) z Ω lm z () (e r, e θ, e ϕ ) ( rr h r, θr h θ, ϕr h ϕ ) cos ϕ sin θ cos ϕ cos θ sin ϕ sin ϕ sin θ sin ϕ cos θ cos ϕ cos θ sin θ 0 h r r r, h θ θ r r, h ϕ ϕ r r sin θ e i e j δ ij T T E 3, T T e i e j ϵ ijk e k r θ ϕ x r x θ x ϕ y r y θ y ϕ z r z θ z ϕ diag (h r, h θ, h ϕ ) x y ẽr ẽ θ ẽ ϕ z h rẽ r h θ ẽ θ h ϕ ẽ ϕ x y z x y z diag (h r, h θ, h ϕ ) T T x y z x y z r re r T r r θ r sin θ ϕ e r r + e θ r θ + e ϕ r sin θ ϕ L r p iħr iħ(e ϕ θ e θ sin θ ϕ) ħ i sin ϕ i cos ϕ cot θ i cos ϕ θ + ħ i sin ϕ cot θ ϕ 0 i

4 : 03 ( 5 8 5 ) L + ħ(i sin ϕ + cos ϕ) θ + (i cos ϕ sin ϕ) cot θ ϕ ħe iϕ ( θ + i cot θ ϕ ) L ħ(i sin ϕ cos ϕ) θ + (i cos ϕ + sin ϕ) cot θ ϕ ħe iϕ ( θ + i cot θ ϕ ) L z iħ ϕ ( ) () 5 L Y lm ħ j(j + )Y lm L z Y lm ħmy lm L + Y ll L Y ll 0 Y lm (Ω) Θ lm (θ)φ m (ϕ) L z Y lm ħmy lm π 0 Φ m (ϕ) dϕ Φ m (ϕ) π e imϕ e iπm m j 0 θ f(θ) L + [e imϕ f(θ)] e imϕ ħe iϕ [ df mf cot θ] dθ L [e imϕ f(θ)] e imϕ ħe iϕ [ df + mf cot θ] dθ 5 L + [e imϕ f(θ)] ħe i(m+)ϕ sin m+ d θ d cos θ [sin m θf(θ)] L [e imϕ f(θ)] ħe i(m )ϕ sin (m ) d θ d cos θ [sinm θf(θ)] d dθ d sin θ d cos θ d cos θ dθ d( cos θ) / d cos θ d d cos θ sin θ d d cos θ ( cos θ) / ( cos θ) cot θ

: 03 ( 5 8 5 ) 5 k [ L k +[e imϕ f(θ)] ( ħ) k e i(m+k)ϕ sin m+k θ [ L k [e imϕ f(θ)] ħ k e i(m k)ϕ sin (m k) θ (3) Y l0 L + Y ll 0 Θ ll Θ ll C l sin l θ π 0 Θ ll θ l cot θθ ll 0 ] k d [sin m θf(θ)] d(cos θ) ] k d [sin m θf(θ)] d cos θ dθ sin θ Θ ll (θ) C l d d cos θ [sin m θf(θ)] m sin m θ( cot θ)f(θ) + sin m θ sin θ θf(θ) sin m θ[ d m cot θ]f dθ d d cos θ [sinm θf(θ)] m sin m θ( cot θ)f(θ) + sin m θ sin θ θf(θ) sin m θ[ d + m cot θ]f dθ

6 : 03 ( 5 8 5 ) C l ( ) l C l, C l > 0 6 Y ll (Ω) ( ) l (l + )! π l l! eilϕ sin l θ l0 ħ l l! 7 ll (l)! l! Y l0 (Ω) l + 4π P l(cos θ) 6 P l (t) d l l l! dt l (t ) l l π 0 dθ sin θ Φ C l C l C l C l C l π 0 0 0 dθ sin l+ θ C l dt ( t ) l, d( cos θ)( cos θ) l t s /, (dt s / ds) ds s / ( s) l C l B(/, l + ) Γ(/)Γ(l + ) Γ(l + 3/) C l Γ(/)l! (l + /) (/)Γ(/) l+ l! 3 (l + ) l+ l!( l l!) C l (l + )! C l ( ) l (l + )! l l! 7 Y l0 (Ω) ( ) l (l + )! π (l)! d l l! eilϕ d cos θ sinl θ l + d π l l! d cos θ (cosl θ ) l + P l (cos θ) π

: 03 ( 5 8 5 ) 7 (4) m > 0 Y lm lm ħ m Y lm (Ω) ( ) m l! (l+m)! (l m)! l! l0 [ ] m l + (l m)! d 4π (l + m)! sinm θ P l (cos θ)e imϕ d cos θ (5) m m, m > 0 Y lm Y l m l m ħ m (l m)! l! Y lm (Ω) Y l m (Ω) [ ] m l + (l m)! d 4π (l + m)! sinm θ P l (cos θ)e imϕ d cos θ l! l0 (l+m)! (6) m > 0 Θ lm Θ l m Y lm (Ω) Θ lm ( ) m Θ l m Y lm ( ) m Y l m

8 : 03 ( 5 8 5 ).4.4. B H m (p ea) + eϕ A rot A B A B r eϕ ϕ (rot A) i ϵ ijk j ϵ kab B a r b ϵ ijkϵ kaj B a ϵ ijkϵ ajk B a δ ia B a B i rot A B H H 0 + H P + H D H 0 p m H P e (p A + A p) m H D e m A

: 03 ( 5 8 5 ) 9 H D e (B r) (B r) 4m (B r) (B r) 4m ] e [ (B r ) (B r) e 4m 4m B r r r sin θ e θ B r H P p A idiv A + A p A p H P e m A p e e e (B r) p B (r p) m m m B L L ħ l(l + ) l H P B L B z L z l + H P H P e B (L + gs), g m H P + H S H S µ B µ g e m S gµ BS/ħ µ B eħ m S, S ħ (/ + )/, S / () µ B

30 : 03 ( 5 8 5 ).4. j j, 3, ( ) ( ) ( ) j S / S / J S S Sm S(S + ) Sm, S / S z S + +ħ S + S z S ħ S S + S S ħ ( + )( + ) ħ S + S + S ( ħ + )( + + ) ħ S ħ σ + σ z + σ z S ± S ± σ z ( ) ψ ( +, ) ψ σ z ψ ( + σ z + σ z + + σ z σ z ) ( 0 0 )

: 03 ( 5 8 5 ) 3 + σ + σ + + σ + + σ + 0 + σ + σ 0 ψ σ + ψ ψ σ ψ ( ( 0 0 0 0 0 0 ) ) ( ) ψ σ x ψ (ψ σ ψ + ψ 0 σ ψ) 0 ( ) ψ σ y ψ i (ψ σ ψ ψ 0 i σ ψ) i 0 ( ) ( ) ( ) 0 0 i 0 σ x, σ y, σ z 0 i 0 0 Trσ i 0 { σ i σ j σ 0 iϵ ijk σ k i j i j σ 0 ( 0 0 )

3 : 03 ( 5 8 5 ) σ i σ 0 {σ i, σ j } σ i σ j + σ j σ i 0, i j {σ i, σ j } δ ij σ 0 A A 3 A i σ i, i0 A i TrAσ i 3 u, v 8 (u σ)(v σ) (u v)σ 0 + i(u v) σ H µb S, µ gµ B /ħ TrH µb TrS 0 E, E E +E 0 E E E H E µ B /4 E ± ±µ B / 8 (u σ)(v σ) u i σ i v j σ j ( ij + i j )u i v j σ i σ j i u i v i σ i + ϵ ijk u i v j σ i σ j (u v)σ 0 + i ijk u i v j ϵ ijk σ k

: 03 ( 5 8 5 ) 33.4.3 ( ) ψ(r, +) ψ(r) ψ(r, ) τ (r, σ), σ ± ψ(τ) ψ(r, σ) H 0 (r) [ H0 (r) µ B ] ψ(r) [ H 0 (r) (gµ B /ħ)b S ] ψ(r) E ψ(r) ψ(r) ψ(r) χ ψ(r, σ) ψ(r)χ(σ) χ S B µ B χ ± ±(gµ B /ħ) χ ± H 0 ψ(r) E j ψ(r) H 0 [ H0 (r) µ B ] ψ j± (τ) E j± ψ(τ) E j± E j ± gµ B /ħ

34 : 03 ( 5 8 5 ).4.4 B 0 Θ iσ K JK, J [ 0 0 K σ (Anti-Unitary) ] r r ΘpΘ J(r) J r p p ΘpΘ J(p) J ( iħ ) iħ p L L ΘLΘ (ΘrΘ ) (ΘpΘ ) r ( p) L ( ) S S (iσ )(σ ) ( iσ ) (σ )(σ )(σ ) σ iσ 3 i σ S S S (iσ )(σ ) ( iσ ) (σ )( σ )(σ ) σ3 σ S S 3 S 3 (iσ )(σ 3 ) ( iσ ) (σ )(σ 3 )(σ ) iσ σ i σ 3 S 3 S S ΘSΘ S B S B L H SO f(r)s L f : H SO ΘH SO Θ f (r)( S) ( L) H SO

: 03 ( 5 8 5 ) 35 Θ KJKJ J ΘHΘ JH J H [H, Θ] ΘH HΘ 0 [ 0 0 ] [ 0 0 ] 0 H(r) ψ(r) E ψ(r), ψ(r) ψ ΘH(r) ψ(r) H(r)Θ ψ(r) EΘ ψ(r) ψ Θ Θ ψ H ψ Θ E ψ Θ ψ ψ Θ (ψ +, ψ )(iσ )K ( ψ + ψ [ ] ( (ψ+, ψ ) 0 0 ( ) (ψ +, ψ ) ψ ψ + ) ψ+ ψ ) ( ψ + ψ ) ψ +ψ ψ ψ + 0

36 : 03 ( 5 8 5 ) ψ ψ Θ N ( ) σ, σ,, σ N σ σ σ N, σ i ± Θ K(iσ )(iσ ) (iσ N ) Θ K (iσ ) (iσ ) (iσ N ) ( ) N A UK U ψ, ϕ Θψ Θϕ KUψ KUϕ [(U ai ψ i ) ] (U aj ϕ j ) U ai ψ i Uajϕ j (UaiU aj ) ψ i ϕ j (U U) ijψ i ϕ j δ ij ψ i ϕ j ϕ ψ N ψ N ψn Θ ψ N ψn Θ ψ N Θψ N Θ ψ N Θψ N ( ) N ψ N Θψ N ( ) N ψ N ψn Θ N ψ N ψn Θ ψ N ψn Θ ψ N ψn Θ 0.4.5 S, S [S iµ, S jν ] 0, i j S S + S

: 03 ( 5 8 5 ) 37 [S µ, S ν ] iϵ µνλ ħs λ S S, S m,(m ±) m,(m ±) S z m ħm m, m ±/ S z m ħm m, m ±/ m m m m 4 { + m i,,, 4 S S SM ħ S(S + ) SM S z SM ħm SM S + S + + S + S + (S + + S + ) ( S + ) + ( S+ ) 0 S z S z + S z S z (S z + S z ) ( S z ) + ( Sz ) ( ħ + ħ ) S ħs, M ħm s, m

38 : 03 ( 5 8 5 ) (-/,/) m+ m m (/,/) m (-/,-/) (/,-/).: S ħ/, S ħ/ 0 ħ (s + m)(s m + ) S, s, m ħ S ħ (S + S ) ( + ) ħ (s + m)(s m + ) S 0, s, m 0 ħ S ( + ) ħ (S + S ) ( + ) ( )

: 03 ( 5 8 5 ) 39 ψ m () ( 0 ψ m0 ψ m () ) ψ (, ) ( ) ψ 0 (,,, ) (, ) ψ (, ) ( ) m m + m 0 ψ 0, 0 ( ) t ψ 0 ( ) m S + t 0 s 0, S + t (S + + S + )(,,, ) 0 t 00 sm m m m m sm (m, m ) m m sm 0, m + m m,,,,, 0,,, 0 +,,, 0 0, 0,, 0, 0 +,, 0, 0,,,,

40 : 03 ( 5 8 5 ), ψ (,, ) ( (, 0, 0, 0 ) ψ,, 0, 0, 0 0,, 0, 0, 0 ), ψ (,, ) (,,, 0, 0, 0,, ) (ψ, ψ 0, ψ ) [ m, m sm ] (ψ, ψ 0, ψ ) m m sm m, m m, m δ m m δ m m sm s, 0, m s,, s sm sm m m sm sm m m m m m m m m m m H JS S, J > 0

: 03 ( 5 8 5 ) 4 (S + S ) S + S + S S ( + ) + ( + ) + S S [ H J (S + S ) 3 ] 4 s 0m 0 ( ) 3 E 0 J ( 0(0 + ) 3 ) 3 4 4 J, 0, E J ( ( + ) 3 ) 4 4 J ()() ()().4.6 J L + S J jm ħ j(j + ) J z jm ħm jm

4 : 03 ( 5 8 5 ) m (L+/,/) m (L+/,-/).: L S ħ/ J J z m m lm, m lm m jm m m m m jm m m + n jm k ħ k m m jm jm m m J + l 0 l +, l + l, (j + m k)! (j + m)! (j m)! (j m + k)! J jm k

: 03 ( 5 8 5 ) 43 l +, l + k ħ k L k ll ħ k (l)!k! l k (l k)! ħ k ħ k S ħ (l + k)! (0)! (l + )! k! J l k +, l + (l + k)! (0)! (l + )! k! (L + S ) k l, (l + k)! (L k + kl k S ) l, (l + )!k! (l + k)! (l + )!k! ( (l)!k! (l k)! l k, + k l + k l + ) (l)!(k )! (l k + )! l + k, l k, + k l + l + k, J l +, M l + k { j l, m M m l k, m l + k, m j, m ± j, m m m,, m JM : ( p43) J m m j + j +M+/ j + j M+/ j + m L / l +, l l l + l, + l + l,

44 : 03 ( 5 8 5 ) m l ( ) l, l l, l l + l, l + l + l, J + L + l, m ħ (l + m + )(l m) lm + J + l, l l + L + l l + l + S + l l ħ (l)() l l + + ħ l + l 0 j l m l l, l k (l k)! (0)! ħ k (l )! k! J l k, l ħ k (l k)! (0)! (l )! ( (L + S ) k ħ k (l k)! (l )! ( k! l + l, + (0)! k! ) l l + l, l + (Lk + kl k S ) l, + l l + Lk l, L k ll ħ k (l )! (l k)! L k ll ħ k (l )! (l k)! L k ll ħ k (l)!k! l k (l k)! ( + k)! l k ()! (k)! l k ()! )

: 03 ( 5 8 5 ) 45 l, l k + k l + l k, k (l + )(l k) l k, + (l) (l + )(l k) l k, k + l + l k, l k + l + l k, J l, M l k { j l, m M m l k, m l k, m j, m ± j, m m m,, m JM : ( p43) J m m j j M+/ j + j +M+/ j + J j + / J + j + l + J j / J + j l 4l + (l + ) ( + )(l + )

46 : 03 ( 5 8 5 ).5 J J [J ai, J aj ] iħϵ ijk J ak, a, [J i, J j ] 0 J J + J J [J i, J j ] iħϵ ijk J k.5. J j m ħ j (j + ) j m J z j m ħm j m J j m ħ j (j + ) j m J z j m ħm j m (j + )(j + ) J jm ħj(j ) jm J z jm ħm jm jm jm j m j m A m m,jm j m j m jm B jm,m m j m j m j m j m jm A m m,jm

: 03 ( 5 8 5 ) 47 jm j m j m B jm,m m j m j m jm jm j m j m ((Clebsch-Gordan ) j m j m jm jm j m j m jm j m j m j m j m jm j m j m jm jm j m j m jm j m j m j m j m j m j m jm j m j m j m j m j m j m jm jm j m j m j m j m j m j m j m jm δ j jδ m m j m j m jm jm j m j m δ m m δ m m J z jm ħm jm (J z + J z ) j m j m j m j m jm ħ(m + m ) j m j m j m j m jm j m j m jm 0 : if m m + m

48 : 03 ( 5 8 5 ) j j m + m j + j j j + j J m j,, j m + m j + j (m, m ) (j, j ), (j, j ) j j + j m + m j + j (m, m ) (j, j ), (j, j ), (j, j ) j j m + m j + j s j j + j s j j s j j j + j s j j j j + j,, j j j j j < j j, j > j + j j > jj < (j + ) (j < + j > )(j > j < + ) + (j > j < + ) (j > + j < + )(j > j < + ) (j + )(j + )

: 03 ( 5 8 5 ) 49 m (j, j ) m (-j, j ).3: J, J ( ) j j j j j j + j + j.5. : J + J + + J + jm j m j m j m j m jm J + jm J + j m j m j m j m jm + J + j m j m j m j m jm [ (j + m + )(j m) ] / jm + [ (j + m + )(j m ) ] / j m + j m j m j m jm + [ (j + m + )(j m ) ] / j m j m + j m j m jm

50 : 03 ( 5 8 5 ) j m j m [ (j + m + )(j m) ] / j m j m jm + [ (j + m + )(j m ) ] / j m j m j m + j m j m j m jm + [ (j + m + )(j m ) ] / j m j m j m j m + j m j m jm [ (j + m )(j m + ) ] / j m j m jm + [ (j + m )(j m + ) ] / j m j m jm J J + J J jm J j m j m j m j m jm + J j m j m j m j m jm j m j m [ (j m + )(j + m) ] / jm [ (j m + )(j + m ) ] / j m j m j m j m jm + [ (j m + )(j + m ) ] / j m j m j m j m jm [ (j m + )(j + m) ] / j m j m jm [ (j m + )(j + m ) ] / j m j m j m j m j m j m jm + [ (j m + )(j + m ) ] / j m j m j m j m j m j m jm [ (j m + )(j + m) ] / j m j m jm [ (j m )(j + m + ) ] / j m + j m jm + [ (j m )(j + m + ) ] / j m j m + jm

: 03 ( 5 8 5 ) 5 I j m j m jm + [ ] / (j + m )(j m + ) j m j m jm (j + m + )(j m) [ ] / (j + m )(j m + ) + j m j m jm (j + m + )(j m) [ ] / (j m )(j + m + ) j m j m jm j m + j m jm (j m + )(j + m) [ ] / (j m )(j + m + ) + j m j m + jm (j m + )(j + m) j j + j ψ j +j ( j j, j j ) Ψ j +j ( j + j, j + j ) ψ j +j () ψ j +j Ψ j +j () j j + j ψ j +j ( j j, j j, j j, j j ) Ψ j +j ( j + j, j + j, j + j, j + j ) ψ j +j j + j, j + j Mψ j +j j + j, j + j m j, m j, j j + j, m m + m j + j j j j j j + j j + j m m j, m j, j j + j, m j + j [ (j + j ) ] / j j j j j + j j + j [ j ] / j j j j j + j j + j [ j ] /

5 : 03 ( 5 8 5 ) j j j j j + j j + j j j j j j + j j + j j j + j j j + j j + j j + j ψ j j +j j j +j ψ ( j j j j, j j j j ) ψ ψ ψψ E P j + j, j + j j + j, j + j ψ j j (, )ψ j + j j + j P ψ( ψ C > 0 j j +j j j +j ( j j j +j j j +j j j j j +j j +j ( j j +j j j j +j j j j +j j j +j ) ) ϕ j j j j )ψ )ψ j + j, j + j C( P )ϕ ( j j Cψ +j j j j +j Cψ ( j j j j +j j +j j j j +j j j +j ) ) ) ( 0 )

: 03 ( 5 8 5 ) 53 j + j, j + j j + j, j + j C (j +j )j (j +j ) j C +j j j + j, j + j ψ j j +j j j +j j j, j j j + j, j + j j j, j j j + j, j + j C > 0 k ϕ j j, j j k j j + j j j + j j j, j j k j + j k, j + j k > 0 m +m j +j m j +j j j + j j m, j + m j + j m j, m j j j, j j j + j, j + j (j ) (j + j ) j j, j j j + j, j + j j (j ) (j + j )(j + j ) j j, j j j + j, j + j m j, m j j j, j j j + j, j + j j (j + j ) j j, j j j + j, j + j j + (j + j ) j j, j j j + j, j + j 4j j (j + j )(j + j )

54 : 03 ( 5 8 5 ) m j, m j j j, j j j + j, j + j (j ) (j + j ) j j, j j j + j, j + j j (j ) (j + j )(j + j ) j j + j m j + j j m 0, j + m j + j m j, m j j j, j j j + j, j + j (j ) (j + j ) j j, j j j + j, j + j j (j ) (j + j )(j + j ) m j, m j j j, j j j + j j + j j (j + j ) j j, j j j + j, j + j j + (j + j ) j j, j j j + j, j + j j (j + j )(j + j ) j (j + j )(j + j ) m j, m j j j (j + j )(j + j ) j j, j j j + j, j + j (j ) (j + j ) j j, j j j + j, j + j j (j ) (j + j )(j + j )

: 03 ( 5 8 5 ) 55 ψ ( j j, j j, j j, j, j, j j, j j ) ψ ψ ψψ E 3 j + j, j + j ψ (j + j )(j + j ) j + j, j + j ψ (j + j )(j + j ) j (j ) 4j j j (j ) j (j ) j j j (j ) P j + j, j + j j + j, j + j + j + j, j + j j + j, j + j ψ (j + j )(j + j ) j (j ) j j (j ) j j (j )(j ) j j (j ) 4j j j j (j ) j j (j )(j ) j j (j ) j (j ) ψ +ψ (j + j )(j + j ) j (j ) (j j ) j (j ) j j (j )(j ) (j j ) j (j ) (j j ) (j j ) j (j ) j j (j )(j ) (j j ) j (j ) j (j ) ψ

56 : 03 ( 5 8 5 ) m + m j + j ϕ j j, j j j + j, j + j C( P )ϕ Cψ[ 0 0 j j (j )(j ) j j (j ) (j + j )(j + j ) j (j ) j j (j )(j ) (j j ) ] j (j ) (j + j )(j + j ) j (j ) [ j j (j )(j ) ] j (j +j ) (j j )(j +j ) Cψ (j +j )(j +j )(j +j ) j (j ) + j (j +j )+j (j +j ) (j +j )(j +j )(j +j ) (j ) Cψ[ j j (j )(j ) ] j (j +j )(j +j ) j (j ) j (j ) (j +j )(j +j ) j (j ) j (j ) Cψ (j )(j ) (j + j )(j + j ) j (j ) j (j ) + (j )(j ) + j (j ) (j + j )(j + j ) j + j, j + j ψ (j + j )(j + j ) j (j ) (j )(j ) j (j ) j j, j j j + j, j + j j (j ) (j + j )(j + j ) j j, j j j + j, j + j (j )(j ) (j + j )(j + j ) j j, j j j + j, j + j j (j ) (j + j )(j + j )

: 03 ( 5 8 5 ) 57.5.3 : j j + j ψ j +j ( j j, j j ) Ψ j +j ( j + j, j + j ) ψ j +j () ψ j +j Ψ j +j () jm ħ (j + m)(j m + ) J jm j + j, j + j ħ (j + j ) J j + j, j + j ħ (j + j ) (J j, j j, j + j, j J j, j ) [ ħ ħ j j, j j, j + j, j (ħ ] j ) j, j (j + j ) j j j j, j j + j j, j j j + j j + j j j + j ψ j +j ( j j, j j, j j, j j ) Ψ j +j ( j + j, j + j, j + j, j + j ) Ψ j +j Ψ j +j ψ j +j ψ j +j E Ψ j +j ψ j +j M M (ψ, ψ) ( j j +j j j +j, ψ)

58 : 03 ( 5 8 5 ) ϕ ψ P E MM ψ ψ + ψψ ψψ E ψ ψ E P ψ (E P )ψ ψ (E P )ϕ ψ ψ / ψ E P E j j +j j j +j ( j j j (, ) j + j j + j j j +j j j +j j j j j +j j +j ( j j +j j j j +j j j j +j j j +j ϕ ( 0 ) ) ) ) ) ψ (E P )ϕ ( j j +j j j j +j ( j j j j +j j +j j j j +j j j +j ) ) ) ( 0 ) ψ (j + j )j (j + j ) ψ ψ / ψ j j +j j j +j

: 03 ( 5 8 5 ) 59 Ψ j +j ψ j +j j j +j j j +j j j +j j j +j ϕ ( j j, j j j + j, j + j j j, j j j + j, j + j 0 ) k ϕ j j, j j k j j + j j j + j j j, j j k j + j k, j + j k > 0 jm ħ (j + m)(j + m )(j m + )(j m + )) J jm [ ] j + j, j + j ħ J + J J + J j j, j j (j + j )(j + j ) (j + j )(j + j ) ( j (j ) j j, j j + j j j j, j j + j (j ) j j, j j ) (j + j )(j + j ) ( j (j ) j j, j j + 4j j j j, j j + j (j ) j j, j j ) ψ j +j ψ

60 : 03 ( 5 8 5 ) ψ j +j ( j j, j j, j j, j j, j j, j j ) j (j ) ψ 4j j (j + j )(j + j ) j (j ) j j + j j + j, j + j ħ (j + j ) J j + j, j + j ħ (j + j ) J ψ j +j j j +j j j +j J ψ j +j ((J + J ) j j, j j, (J + J ) j j, j j ) ( (j ) j j, j j + j j j, j j, j j j, j j + (j ) j j, j j ) (j ) 0 ħ( j j, j j, j j, j j, j j, j j ) j j 0 (j ) (j ) 0 ħψ j +j j j 0 (j ) j + j, j + j (j + j ) ψ j +j (j ) 0 j j 0 (j ) j j +j j j +j ψ j +j ψ ψ (j )j j j (j + j )(j + j ) j (j )

: 03 ( 5 8 5 ) 6 j j + j 3 Ψ j +j ( j + j, j + j, j + j, j + j, j + j, j + j ) Ψ j +j Ψ j +j ψ j +j ψ j +j E 3 Ψ j +j ψ j +j M M (ψ, ψ, ψ) (ψ C, ψ) E 3 MM ψ C ψ C + ψψ ψψ E 3 ψ C ψ C E 3 P ψ (E 3 P )ψ ϕ ψ (E 3 P )ϕ ψ ψ / ψ ψ P (j + j )(j + j ) j (j ) j j (j ) j j (j )(j ) j j (j ) 4j j j j (j ) j j (j )(j ) j j (j ) j (j ) + (j + j )(j + j ) j (j ) (j j ) j (j ) j j (j )(j ) (j j ) j (j ) (j j ) (j j ) j (j ) j j (j )(j ) (j j ) j (j ) j (j ) ϕ 0 0

6 : 03 ( 5 8 5 ) ψ (E 3 P )ϕ 0 0 0 0 j + j + ( j +j j ( j +j + j j j ( j +j + j j j (j )(j ) (j +j )(j +j ) (j ) j (j ) (j +j )(j +j ) (j +j )(j ) (j +j )(j +j ) (j + j )(j + j ) j (j ) (j + j )(j + j ) j +j ) j j (j )(j ) j +j ) j (j ) j +j )(j ) j j (j )(j ) (j ) j (j ) j (j ) j (j )) (j )(j ) j (j ) j (j ) + (j )(j ) + j (j ) j j + 4j j j j + + j j j 3j 3j + j + 4j j + (j + j ) 3(j + j ) + [(j + j ) ](j + j ) ψ / ψ (j + j )(j + j ) j (j )) (j )(j ) j (j ) j + j, j + j ψ (j + j )(j + j ) j (j ) (j )(j ) j (j )

: 03 ( 5 8 5 ) 63 j j, j j j + j, j + j j (j ) (j + j )(j + j ) j j, j j j + j, j + j (j )(j ) (j + j )(j + j ) j j, j j j + j, j + j j (j ) (j + j )(j + j ).5.4,,,,, ħ 4 J,,, ħ (J,, +, J, ) (, 0, +, 0, ) (, 0,, +,,, 0 ) (, 0,,,,,, 0 ) ( ), 0 ħ 4 3 J, ħ 4 (J + J J + J ),,, (,,, +, 0,, 0 +,,, ) 4 6 (,,, +, 0,, 0 +,,, ) (,,,,, 0,, 0,,,, ) 6 6 6

64 : 03 ( 5 8 5 ), ħ 4 3 3 J, 3 ħ 3 (J 3 + 3J J + 3J J + J ), 3,, ħ 3 (3J J + 3J J ),,, (3,,, 0 + 3, 0, ) (,,, +,,, ), ħ 4 4 3 3 4 J, 4 ( ) ħ 4 4 (J 4 4 + J 3 J + ħ 4 4 6J J,,, 6( ),,, 4,,, ( 4 ) J J + ( 4 3 ) J J 3 + J 4 ),,, m,, (,,, ) (, 0,,,,,, 0 )M (, 0,,,,,, 0 ) ( ) (,,, ) (, 0,,,,,, 0 ) ( )

: 03 ( 5 8 5 ) 65, 0, ħ J, ħ (J + J )(, 0,,,,,, 0 ) ( (,,, +, 0,, 0,, 0,, 0 ) +,,, ) (,,,,, 0,, 0,,,, ) (,,,,, 0,, 0,,,, ) ħ J, 0 0 0 0 ( ħ (J,,,, (J + J ), 0,, 0, J,,, ) (,,, 0,,,, 0 +, 0,,,, 0,, ) ( ) 0 (,,, 0,, 0,, ) 0 ( ) (,,, 0,, 0,, ) 0 ) 0 ) 0 ( ) m 0 3 J, (J + J ), 0 (, 0,, 0, 0, 0 ) (,,,,, 0,, 0,,,, )M 6 M 6 0 6

66 : 03 ( 5 8 5 ) M 6 3 6 0 3 3 6 0, 0 (,,,,, 0,, 0,,,, ) 3 3 3 M M (ψ C, ψ) ψ C 6 6 0 6 E 3 MM ψ C ψ C + ψψ P + ψψ ψ (E 3 P )ψ (E 3 P ) E 3 P ϕ ψ (E 3 P )ϕ ψ ψ / ψ ψ (E 3 P )ϕ

: 03 ( 5 8 5 ) 67 ϕ P E 3 P 0 0 6 6 0 6 ψ 3 3 3 3 3 3 3 3 3 3 3 3 ( 6 6 6 0 ) ψ 3 3 3 3 3 3 3 3 3 3 3 3

68 : 03 ( 5 8 5 ).6.6. Y lm, f L z (Y lm f) (L z Y lm )f + Y lm L z f ħmy lm + Y lm f L ± (Y lm f) (L ± Y lm )f + Y lm L ± f ħ (l m)(l ± m + )Y lm± f + Y lm L ± f l [L z, Y lm ] ħmy lm [L ±, Y lm ] ħ (l m)(l ± m + )Y lm± [L ±, Y m ] ħ ( m)( ± m)y m± [L +, Y ] 0, [L, Y ] ħ Y 0 [L +, Y 0 ] ħ Y, [L, Y 0 ] ħ Y [L +, Y ] ħ Y 0, [L, Y ]0 J ([J i, J j ] iħϵ ijk J k ) [J +, J + ] 0, [J, J + ] ħj z [J +, J z ] ħj +, [J, J z ] ħj [J +, J ] ħj z, [J, J ]0 [J z,±, ] Y J + Y 0 J z Y J

: 03 ( 5 8 5 ) 69 0 k T (k) q, q k, k +, k, k k + k [J z, T q (k) ] ħkt q (k) [J ±, T (k) q ] ħ (k q)(k ± m + )T (k) q± k V V x V y V z L [L i, V j ] iħϵ ijk V k r, r, J T () (J x + ij y ) T () 0 J z T () (J x ij y ).6. k i T k q T (k ) q T (k ) q k q, k q kq k q, k q kq [J, T (k ) q T (k ) q ] [J, T (k ) q ]T (k ) q + T (k ) q [J, T (k ) q ] [J z, T q (k) ] [J z, T (k ) q T (k ) q ] k q, k q kq [ ] [J z, T (k ) q ]T (k ) q + T (k ) q [J z, T (k ) q ] k q, k q kq ħ(q + q )T (k ) ħ(q + q )T (k) q q T (k ) q k q, k q kq

70 : 03 ( 5 8 5 ) [J ±, T q (k) ] [J ±, T (k ) q T (k ) q ] k q, k q kq [ [J ±, T (k ) q ]T (k ) q + T (k ) q [J ±, T (k ) [ (k ħ q )(k ± q + )T (k ) ] q ] q ±T (k ) q + (k q )(k ± q + )T (k ) q T (k ) q ± k q, k q kq ] k q, k q kq [ (k ħt (k ) q T (k ) q (q ))(k ± (q ) + ) k q, k q kq + ] (k (q ))(k ± (q ) + ) k q, k q kq [ (k ħt (k ) q T (k ) q q + )(k ± q ) k q, k q kq + ] (k q + )(k ± q ) k q, k q kq jm ± ħ (j m)(j ± m + ) J ± jm ħ (k q + )(k ± q ) k q J k q ħ (k q + )(k ± q ) k q J k q [J ±, T (k) q ] T (k ) T (k ) q T (k ) q q T (k ) q k q, k q (J + J ) kq k q, k q J ± kq ħ (k q)(k ± q + )T (k ) q T (k ) q k q, k q kq ± ħ (k q)(k ± q + )T (k) q± T q (k) k k + k k +k k k k q k k kq k q, k q T q (k) T (k ) q T (k ) q

: 03 ( 5 8 5 ) 7, Clebsch-Gordan T (k ) q T (k ) q T (k) q kq k q, k q k q, k q kq T (k) q.6.3 U, V (UV ) ij,i j j U ii V jj CG + 0 j m m m j m, j m jm j m m m j m, j m jm 0 / 0 / 0 / 0 / 0 - / 0 - / 0 0 0 0 6 0 0 0 / 0 - / 6 0 - / - - 0 / 6 - - 0 / - 0 - / - 0 - / 0 0 - / 3 0 0 0 0 / 3 - - - 0 0 - / 3 U (U x + iu y ) U 0 U z U (U x iu y )

7 : 03 ( 5 8 5 ) V 0 T (0) 0 U m V m m, m 00 U V, 00 + U 0 V 0 0, 0 00 + U V, 00 U + V (/ 3) + U z V z ( / 3) U V + (/ 3) ( / 3)( (U +V + U V + ) + U z V z ) 3 U V ( ) T () U m V m m, m U 0 V 0, + U V 0, 0 U z ( / )V + ( / ) + ( / )U + V z (/ ) ( ) Uz (V x + iv y ) (U x + iu y )V z ( (Uz V x U x V z ) i(u y V z U z V y ) ) ( (U V )y i(u V ) x ) i (U V ) + i (U V ) T () 0 U m V m m, m 0 U V, 0 + U V, 0 (/)U V + ( / ) (/)U + V (/ ) ( (Ux iu y )(V x + iv y ) (U x + iu y )(V x iv y ) ) (iu x V y iu y V x ) i (U V ) z i (U V ) 0

: 03 ( 5 8 5 ) 73 T () U m V m m, m U 0 V 0, + U V 0, 0 U z (/ )V (/ ) + (/ )U V z ( / ) ( ) Uz (V x iv y ) (U x iu y )V z ( (Uz V x U x V z ) + i(u y V z U z V y ) ) ( (U V )y + i(u V ) x ) i (U V ) i (U V ) T () q i (U V ) q T () U m V m m, m U V, U V T () U m V m m, m U V, U V T () U m V m m, m U 0 V 0, + U V 0, 0 U 0 V (/ ) + U V 0 (/ ) (U 0 V + U V 0 ) T () U m V m m, m U 0 V 0, + U V 0, 0 U 0 V / ) + U V 0 (/ ) (U 0 V + U V 0 )

74 : 03 ( 5 8 5 ) T () 0 U m V m m, m 0 U V, 0 + U 0 V 0 0, 0 0 + U V, 0 (U V + U 0 V 0 + U V ) 6.6.4 { j m } jm T (k ) q j m k q, j m jm q m q m Ω k j q,m k q, j m jm Ω k j q,m T (k ) q j m Ω k j q,m J J Ω k j q,m JT (k ) q j m J z Ω k j q,m ħ(q + m ) Ω k j q,m [J, T (k ) q ] j m + T (k ) q J j m J ± Ω k j q,m ħ (k q )(k ± q + )T (k ) q ± j m +ħ (j m )(j ± m + )T (k ) q j m ± ħ (k q )(k ± q + ) Ω k j q ±,m +ħ (j m )(j ± m + ) Ω k j q,m ±

: 03 ( 5 8 5 ) 75 J z jm ħ(q + m ) Ω kj q,m k q j m jm q,m ħ(q + m ) jm q,m ħm jm J ± jm [ ħ (k q )(k ± q + )T (k ) q ± j m k q j m jm q m +ħ ] (j m )(j ± m + )T (k ) q j m ± k q j m jm [ ħ (k (q ))(k ± (q ) + )T (k ) j m k q, j m jm q m +ħ ] (j (m ))(j ± (m ) + )T (k ) q j m k q, j m jm [ ħ (k q + ))(k ± q )T (k ) j m k q, j m jm q m q +ħ ] (j m + ))(j ± m )T (k ) q j m k q, j m jm [ ] T (k ) q j m J k q, j m ) jm + T (k ) j m (J k q, j m ) jm q m T (k ) q j m k q, j m (J ± + J ±) ) jm q m T (k ) q j m k q, j m J ± jm q m ħ (j m)(j ± m + ) q m T (k ) q j m k q, j m jm ± ħ (j m)(j ± m + ) jm 9 0 jm 9 q q ħ (j ± m)(j m + ) jm J jm 0

76 : 03 ( 5 8 5 ) Ω k j q,m jm jm jm k q, j m j m j m Ω k j q,m j m T (k ) q j m jm j m jm jm k q, j m jm δ jj δ mm jm jm jm k q, j m j m j m j m k q, j m jm Ω k j q,m jm jm jm k q, j m jm jm m jm jm ħ(j + m)(j m + ) jm J jm ħ(j + m)(j m + ) (J + jm ) jm ( jm ) jm jm jm jm jm j T k q j j + jm T (k ) q j m jm jm jm k q, j m j T (k ) q j j + jm k q, j m m

: 03 ( 5 8 5 ) 77 m 0, m 0 jm 0 T (k ) q j m 0 j T (k) q j j + jm 0 k q, j m 0 j T (k ) q j jm0 T (k ) q j m 0 j + jm 0 k q, j m 0 m, m jm T (k ) q j m j T (k) q j j + jm k q, j m CG O S jm O S j m 0, j j 0 O V jm O V j m 0, j j 0, ±

3 ħ 3. 3.. R( ) r r Rr R SO(3) RR E 3 det R r r r r Rr (x, y, z ) (x, y, z)r det(r E 3 ) det( R E 3 ) det(r E 3 ) det R det(e 3 R) det(r E 3 ) 79

80 : 03 ( 5 8 5 ) 3.: ( - - ) det(r E 3 ) 0 v Rv v v v v α R α (v) v α v β R R β (v )R α (v ) RR R α Rβ R β R α E 3 det R det R β det R α v 3 γ R R R R (R 3 R )(R ) R 3 (R R )

: 03 ( 5 8 5 ) 8 R e er R R R R R e ˆv v R(v) R(v)v v Q QR(v)Q Qv Qv R QRQ Qv R Qv QRQ R R QRQ Q R R R QRQ R R class( ) 3.. R. z α R α (z) (x, y, z z),. y β R β (y ) (x, y y, z ) R β (y ) R α (z)r β (y)[r α (z)] 3. z γ R γ (z ) R γ (z ) R β (y )R γ (z )[R β (y )] R γ (z ) R α (z)r γ (z)[r α (z)] R γ (z)

8 : 03 ( 5 8 5 ) 3.: ( - - ) R(α, β, γ) R γ (z )R β (y )R α (z) R γ (z ) R β (y ) R α (z) [R β (y )R γ (z )[R β (y )] ] R β (y ) R α (z) R β (y )R γ (z ) R α (z) R α (z)r β (y)[r α (z)] R γ (z) R α (z) R α (z)r β (y)r γ (z) α, β, γ 0 α < π 0 β < π 0 γ < π ˆn θ R θ (ˆn) R θ (ˆn) e iˆn J

: 03 ( 5 8 5 ) 83 ( ) R(α, β, γ) e ijzα e ijyβ e ijzγ n, θ α, β,γ 3..3 R ψ(r) Rψ(r) ψ(r r) ψ ψ R Rψ ψ R ψ ψ R ψ R ψ R R ψ ψ ψ R R O ψ R O R ψ R ψ R O R R ψ ψ O ψ O R ROR ROR H RHR H [H, R] 0

84 : 03 ( 5 8 5 ) R E d H ψ i ψ i E, i,, d R RH ψ i HR ψ i R ψ i E R ψ i { ψ i } R ψ i ψ j D ji (R) R D ji (R) R ψ ( ψ,, ψ) d ) {D(R)} ji D ji (R) Rψ ψd(r) (Rψ) ψ R D ψ ψ R Rψ ψ ψ E d D ψ ψd D D D d R R R R R ψ R R ψ R ψd(r ) ψd(r )D(R 3 ) ψd(r ) R d D(R) D(R R ) D(R )D(R ) {D(R)} R d ψ ψ R d D(R) D(R)

: 03 ( 5 8 5 ) 85 3..4 R e iθn J Rψ ψd R iθn J J J z ψ m m ψ m J ± ψ m (j m)(j ± m + ) ψ m± J z ψ ψd j (J z ), J z ψ m ψ m D j m m (J z) J ± ψ ψd j (J ± ), J ± ψ m ψ m D j m m (J ±) D j m m (J z) δ m mm D j m m (J ±) δ m,m± (j m)(j ± m + )) j j + R(α, β, γ) e ij zγ e ij yβ e ij zα R(α, β, γ)ψ m ψ m [D j (R(α, β, γ))] m m jm R jm jm e ijzα e ijyβ e ijzγ jm e im γ jm e ijyβ jm e imα [D j (R(α, β, γ))] m m J y α γ 0 [D j (R(α, β, γ))] m m e im α d j m m e imγ d j m m jm e ij yβ jm d j m m

86 : 03 ( 5 8 5 ) 3..5 j ψ j ψ j ( j, j, j, j +,, j, j ) Rψ j ψ j D j (R) j m, j m jm jm j m, j m R j m, j m j m, j m D j m m Dj m m jm D j m m jm j m, j m D j m m Dj m m j m, j m jm D j m m jm j m, j m CG D j m m Dj m m jm j m, j m jm j m, j m D j m m j m, j m jm j m, j m jm D j m m

: 03 ( 5 8 5 ) 87 3. J. Schwinger On angular momentum p.9 in Quantum theory of angular momentum Ed. L. C. Biedenharn and nad H. van Dam, Academic press (965) 3.. Boson Boson a ± a ( a + a ) [a ζ, a ζ ] δ ζζ, [a ζ, a ζ ] 0, [a ζ, a ζ ] 0 n J n n + + n a a a ζ a ζ a +a + a a J a σa, J + J x + ij y a +a J J x ij y a +a + J z (n + n )

88 : 03 ( 5 8 5 ) i j [J i, J j ] 4 [a α(σ i ) αβ a β, a γ(σ j ) γδ a δ ] [a α(σ i ) αβ [a β, a γ](σ j ) γδ a δ + a γ(σ j ) γδ [a α, ] a δ ](σ i ) αβ a β 4 [a α(σ i ) αβ δ βγ (σ j ) γδ a δ a γ(σ ] j ) γδ δ αδ (σ i ) αβ a β 4 [a α(σ i ) αβ (σ j ) βδ a δ a γ(σ ] j ) γα (σ i ) αβ a β 4 [a α(σ i σ j ) αδ a δ a γ(σ ] j σ i ) γβ a β 4 4 a (σ i σ j σ j σ i )a iϵ ijka σ k a iϵ ijk J k J J (J +J + J J + ) + Jz (a +a a a + + a a + a +a ) + 4 (n + n ) ( n+ (n + ) + n (n + + ) ) + ) (n+ + n ) 4( 4n + n n + 4 n n( n + ) S σ/ σ σ (σ + σ + σ σ + ) + σ z σ z σ σ ± ± ± ± σ σ ± ± ± ( + σ σ ) ± ± ) ± ± P ± ± ( + σ σ ) ± ± P ± P P ij ji

: 03 ( 5 8 5 ) 89 σ σ P (σ ) αβ (σ ) γδ δ αδ δ βγ δ αβ δ γδ J 4 a α(σ ) αβ (σ ) γδ a β a γ(σ ) γδ (σ ) δ a δ 4 (δ αδδ βγ δ αβ δ γδ )a αa β a γa δ a αa β a β a α 4 a αa α a γa γ a αa β (a α a β + δ αβ) 4 n n(n + ) 4 n n( n + ) j, m J j(j + ) j n (n + + n ) m (n + n ) n + j + m n j m j, m jm n + n n+!n! (a +) n + (a ) n 0 [(j + m)(j m)] / (a +) n + (a ) n 0

90 : 03 ( 5 8 5 ) 3.. D j m m dj m m e ijyβ jm [(j + m)(j m)] / e ijyβ (a +) n + (a ) n 0 [(j + m)(j m)] / (e ij yβ a +e +ijyβ ) n + (e ijyβ a e +ijyβ ) n e ijyβ 0 [(j + m)(j m)] / (e ij yβ a +e +ijyβ ) n + (e ijyβ a e +ijyβ ) n 0 n n n σ P + P P ± (E ± n σ) P± P ± P ± P + + P E P + P 0 ψ ± P ± ψ ± ψ ± U (ψ +, ψ ) (n σ)u Udiag (+, ) a ( a + a ) U a, a Ua [a α, (a β) ] δ αβ

: 03 ( 5 8 5 ) 9 n J a n σ a (n + n ) n ± a ±a ± e iθa a ae +iθa a e iθ a e iθn J ae iθn J Ue iθn J a e iθn J Udiag (e i θ, e i θ )a Udiag (e i θ, e i θ )U a (ψ +, ψ )diag (e i θ, e i θ ) ( ψ + ψ ) a (e i θ P+ + e i θ P )a (ei θ (E + (n σ)) + e i θ (E (n σ)))a (E cos θ + in σ sin θ )a n 0 0 e iβj y ae iβj y e iβj y ( ( a + a ) [ e iβj y cos β sin β sin β cos β ) a + cos β + a sin β a + sin β + a cos β ] ( a + a )

9 : 03 ( 5 8 5 ) n + j + m, n j m e ijyβ jm [(j + m)(j m)] / (a + cos β + a sin β )n + ( a + sin β + a cos β )n 0 ( ) ( ) n + n [(j + m)(j m)] / k l kl (a +) n + k (a ) k cos n + k β sink β (a +) n l (a ) l ( ) n l sin n l β β cosl ( ) ( [(j + m)(j m)] / kl j + m k j m l ) ( ) j m l (a +) j k l (a ) k+l cos j+m k+l β sinj m l+k β jm a + n + j + m, a n j m j k l j + m, j k l m k + l j m l j k m l j k l j k j + k + m j + m k + l j m j m l j m j + k + m k m + m j + m k + l j + m k + j k m j k + m m j m l + k j m + k j + k + m k m + m

: 03 ( 5 8 5 ) 93 d j m m jm e ijyβ jm ( ( ) k m+m [(j + m)(j m)] / kl cos j k+m m β sink m+m β j + m k ) ( j m j k m ) (j + m )!(j m )! k ( ) k m+m (j + m)!(j m)!(j + m )!(j m )! (j + m k)!k!(k m + m )!(j k m )! cos j k+m m β sink m+m β 3..3 j / d /, j / 0 ( + )!( )!( + )!( )! ( ) k ( + k)!k!(k + )!( k cos k+ )! k0 cos β β sink + β d /, 0 k0 ( ) k+ ( )!( + )!( + )!( )! ( k)!k!(k + + )!( k cos k )! β sink+ + β sin β d /, k ( ) k ( + )!( )!( )!( + )! ( + k)!k!(k )!( k + cos k+ + )! β sink β sin β d /, 0 ( )!( + )!( )!( + )! ( ) k ( k)!k!(k + )!( k + cos k + )! k0 β sink+ β cos β

94 : 03 ( 5 8 5 ) ( ) d / D / ( ( cos β sin β sin β cos β ) ( e i α 0 0 e i α cos β sin β sin β cos β ) ( ) e i γ 0 0 e i γ ) e i (α+γ) cos β e i (α γ) sin β e i (α γ) sin β e i (α+γ) cos β e i γ σ z e i β σ y e i α σ z ˆn θ D / u(ˆn, θ) e i θ ˆn σ E cos θ iˆn σ sin θ SU() [D / ] [D / )] det D / SU() a, b C ( ) D / a b b a a + b ( ) a e i (α+γ) cos β 0 β/ π/ b e i (α γ) sin β 0 cos β/, sin β/

: 03 ( 5 8 5 ) 95 a b 0 α < π 0 γ < π 0 α < π 0 γ < 4π SO(3) SU() j / 4 Re α x, Im α x, Re β x 3, Im β x 4 x + x + x 3 + x 4 (x, x, x 3, x 4 ) 4 3 S 3 SU() S 3 ( )S z e iθ z U() U() S 3..4 j l 0,,, j l m 0, m M d l M0 l!l!(l + M)!(l M)! ( ) k+m β β (l k)!k!(k + M)!(l k M)! cosl k M sink+m k DM0(αβγ) l e l!l!(l + M)!(l M)! imα ( ) k+m β β (l k)!k!(k + M)!(l k M)! cosl k M sink+m k

96 : 03 ( 5 8 5 ) M l D l l0 e imα k0 ( ) k+l l!l!(l)!0! e ilα ( ) l cos l β β l!0!l!0! sinl (l)! ( ) l e ilα sin l β l l! Y lm l!l!(l)!0! β β (l k)!k!(k + l)!(l k l)! cosl k sink+l Y ll (β, α) l + 4π [Dl l0(α, β)] (α 0, β 0, γ 0 ) R(α 0, β 0, γ 0 ) (α, β, γ) R(α, β, γ) Q Q R(α, β, γ) R(α 0, β 0, γ 0 ) (α 0, β 0, γ 0 ) Q (α, β, γ) ẑ R(α, β, γ) ψ(r(α, β, γ)ẑ) Qψ(R(α, β, γ)ẑ) ψ(q R(α, β, γ)ẑ) ψ(r(α 0, β 0, γ 0 )ẑ) N ψ M (R(α, β, γ)ẑ) [D l MN(R(α, β, γ))] D l Qψ M (R(α, β, γ)ẑ) ψ M (R(α 0, β 0, γ 0 )ẑ) [D l MN(R(α 0, β 0, γ 0 ))] [D l MN(Q R(α, β, γ))] [D l MK(Q )] [D l KN(R(α, β, γ))] [[D l (Q)] ] MK[D l KN(R(α, β, γ))] [D l KN(R(α, β, γ))] D l KM(Q) ψ K (R(α, β, γ)ẑ)d l KM(Q)

: 03 ( 5 8 5 ) 97 Y ll N 0 Y lm (β, α) Y lm (R(α, β, 0)ẑ) l + 4π [Dl m0(α, β)] l + ( ) k+m l! (l + m)!(l m)! 4π (l k)!k!(k + m)!(l k m)! l+ Y l0 (β, α) P 4π l(cos β) k cos l k m β sink+m β e+imα D l l0(r(α, β, γ) P l (cos β) Y lm (β, α) l + 4π [Dl m0(α, β)] P l (cos β) Dl0(R(α, l β, γ) CG D j m m Dj m m jm j m, j m jm j m, j m D j m m j l,j l,j l m m 0 [D l m 0Dl m 0] lm l m, l m lm l 0, j 0 [D l m m] lm l m, l m l0 l 0, j 0 [D l m 0] 4π (l + )(l + ) Y l m Y l m 4π lm l m, l m l0 l 0, j 0 Y lm l + Y l m Y l m (l + )(l + ) lm l m, l m l0 l 0, j 0 Y lm 4π(l + )

98 : 03 ( 5 8 5 ) 3 dωylm(ω)y l m (Ω)Y l m (Ω) [ ] (l + )(l + ) l0 l 0, j 0 lm l m, l m 4π(l + ) [ ] m, m, m m, m, m CG (ϕ 0, θ 0, 0), (ϕ, θ, 0) R 0, R R 0 R R (Φ, Θ, Γ) D l 00(R 0 R ) m D l 0m(R 0 )D l m0(r ) m [D l m0(r 0 )] D l m0(r ) D l 00(R ) 4π l + m Y lm (θ 0, ϕ 0 )Y lm(θ, ϕ ) P l (cos Θ) ẑ 0 R 0 ẑ, ẑ R ẑ, ẑ R ẑ R ẑ R 0 R ẑ ẑ 0 R 0 ẑ ẑ R 0 ẑ ẑ 0 ẑ ẑ ẑ Θ ẑ 0 ẑ (θ 0, ϕ 0 ), (θ, ϕ ) Θ (θ 0, ϕ 0 ) (θ, ϕ )

: 03 ( 5 8 5 ) 99 4π l + m Y lm (ˆΩ)Y lm(ˆω ) P l (ˆΩ ˆΩ ) 3..5 P l (t) d l l l! dt l (t ) l l! l l! πi [ ] ξ l dξ πi C t (ξ t) ξ t C dξ (ξ ) l (ξ t) l+ C + t t ζ ξ (ξ t) ζ 0 C + z ξ ± R ζ R tζ + ζ ) R tζ + ζ, ξ ± ( tζ + ζ ) ζ f(z) z n f(z) πi C z + f (n) (z) n! πi C z + ξ dξ f(ξ) ξ z f(ξ) (ξ z) n+ ζξ ζ ξ t, ζξ ξ + t ζ 0, ξ ( ± tζ + ζ )/ζ

00 : 03 ( 5 8 5 ) ξ R ζ ζ 0 ξ t ζ ζ C+ 0 ξ C+ t R ξζ dξ dζ ζ ( R) t + ζ ζ R dζ R + R + tζ ζ dζ ζ R R tζ + dζ ξ t ζ R ζr dζ P l (t) πi C + 0 dζ Rζ d l l+ l! dζ l R d l ζ0 l! dζ l tζ + ζ ζ0 R ζ 0 R P l (t)ζ l tζ + ζ r r l0 r r r r r > ( r < r > ) cos θ + ( r < l0 r< l r> l+ P l (cos θ) r r θ r > r r r < r > ) r r 4π lm l + r< l r> l+ Y lm ( ˆΩ)Y lm( ˆΩ ) ρ(r) ϕ(r) ϕ(r) d 3 r ρ(r ) 4πϵ 0 r r

: 03 ( 5 8 5 ) 0 ρ(r) 0, r > R r < r, r > r 3 ϕ(r) q lm 4π q lm 4πϵ 0 l + r Y lm(ˆr) l+ lm d 3 r (r ) l ρ(r )Ylm(ˆr ) q lm 3..6 ( ) x + x x ϕ jm (x) x j+m + x j m (j + m)!(j m)! jm ϕ jm (a ) 0 j m j ϕ jm (x) jm j m j ϕ jm (x)ϕ jm (a ) 0 (j)! (a +x + + a x ) j 0 (a x) j 0 (j)! j m j (a +x + ) j+m (a x ) j m 0 (j + m)!(j m)! 3 ϕ(r) q lm d 3 r ρ(r )r Y 4πϵ lm(ˆr 4π ) 0 l + r l+ Y lm(ˆr) 4π q lm 4πϵ 0 l + r l+ Y lm(ˆr) lm d 3 r (r ) l ρ(r )Ylm(ˆr )

0 : 03 ( 5 8 5 ) e (a x) 0 jm ϕ jm (x) jm af(a ) 0 f(a ) 0 (e a f(x) f(x + a)) a ϕ jm (x)e λj + jm e λa + a ϕ jm (x) jm jm jm e λa + a e (a x) 0 e λa + a e (a x) 0 e a + x ++(λa + +a )x 0 e a + (x ++λx )+a x 0 jm j m e λj + j m ϕ j m (x) ϕ jm(x + + λx, x ) ϕ jm ((x + + λx, x )) jm (x + + λx ) j+m x j m (j + m)!(j m)! ( ) j+m λ k j + m (j + m)!(j m)! k k0 ( ) j j + m λ m m m m jm J m m + jm m m jm J m m jm m m k j m m j m m j x j+m + x j m x j+m k + x j m+k (j + m )!(j m λ )! (j + m)! m m (j + m)!(j m)! (m m )!(j + m )! ϕ jm (x) λ m m (j + m)!(j m )! (m m )! (j + m )!(j m)! ϕ jm (x) (j + m)!(j m )! (j + m )!(j m)!, (m m ) (j + m )!(j m)! (j + m)!(j m )!, (m m)

: 03 ( 5 8 5 ) 03 3..7 SU() ( ) x + x x x x x + + x SU() ( ) u (x, iσ y x ) x + x x x + SU() a (a +, a ) a u (a +x + + a x, a +x + a x +) [a α, a β ] δ αβ [a α, a β] 0 [a α, a β ] 0 ( ) ( ) x + y + x, y x y (xy) (yx) x + y + + x y [xy] [yx] x + y x y + (xx) a ((xa ), [x a ]) a ua ( x + x x x + ) ( (xa ) [xa ] )

04 : 03 ( 5 8 5 ) jj (xa ) j (j)! 0 (a + ) j 0 (j)! (a +x + + a x ) j (j)! 0 (j)! m (j)! m (j)! m (a +x + ) j+m (a x ) j m 0 (j + m)!(j m)! x j+m + x j m (j + m)!(j m)! jm jm ϕ jm (x) (j)! ϕ jm(x)ϕ jm (x) m m ( x + + x ) j x x (j)! (j + m)!(j m)! x + (j+m) x (j m) j / SO(3) SO(3) SU() u e iθn S e i θn σ SU() n a a u a u a a ua J a σa a σ a

: 03 ( 5 8 5 ) 05 σ uσu (σ α) E α β σ ασ β iϵ αβγ uσ γ u iϵ αβγ σ γ σ Trσ Truσu Tru uσ Trσ 0 σ α Q αβ σ β Q αβ R {σ α, σ β} Q αα Q ββ {σ α, σ β } Q αα Q ββ δα β Q αγ Q βγ δ αβ Q αγ Q βγ δ αβ Q Q E 3 Q det Q Q SO(3) SU() SO(3)

06 : 03 ( 5 8 5 ) σ σ σ 3 uσ σ σ 3u uiu ie σ σ σ 3 Q α Q β Q 3γ σ α σ β σ γ [ α Q α )Q 3γ σ γ + γ α(β)(q ] Q α Q α Q 3γ σ α σ β σ γ α β Q α Q α Q 3γ σ α σ β σ γ (α,β,γ)p (3) (α,β,γ)p (3) ie det Q Q α Q α Q 3γ ie ϵ αβγ det Q 3..8 3j 4 J a σa J b σb J + a b (a b) J b a (b a) J + J z (a a b b) J z n a n b [J +, J ] [a b, b a] [a αb α, b β a β] [a α, b β a β]b α + a α[b α, b β a β] b β [a α, a β ]b α + a α[b α, b β ]a β b b + a a J z [a α[a ] α, a β ]b β a β [b α, b β ]b α [J z, J + ] [a αa α b αb α, a β b β] a b J + [J z, J ] ([J, J z ]) ([J +, J z ]) J + J

: 03 ( 5 8 5 ) 07 [J +, J + ] [a +a, a b] [a +a, a b ] a +[a, a ]b a +b [J +, J + ] [b +b, a b] [b +b, a +b + ] a +[b +, b + ]b a +b J J + J [J +, J + ] 0 ([J +, J + ]) [J, J ] 0 [J +, J ] [a +a, b a] [a +a, b +a + ] b +[a +, a + ]a b +a [J +, J + ] [b +b, b a] [b +b, b a ] b +[b, b ]a b +a [J +, J ] 0 ([J +, J ]) [J +, J ] 0 [J z, J + ] [n a+ n a, a b] [n a+ n a, a +b + + a b ] [n a+, a +b + ] [n a, a b ] (a +b + a b ) [J z, J + ] [n b+ n b, a b] [n b+ n b, a +b + + a b ] [n b+, a +b + ] [n b, a b ] ( a +b + + a b ) [J z, J + ] 0 ([J z, J + ]) [J z, J ] 0 [J z, J z ] 4 [n + n, n a n b ] 0 [J +, J z ] [a +a, n a n b ] [a +a, a +a + + a a ] [a +a, a +a + ] + [a +a, a a ] a +[a +, a + ]a + a +[a, a ]a 0 [J +, J z ] [b +b, n a n b ] [b +b, b +b + + b b ] [b +b, b +b + ] [b +b, b b ] b +[b +, b + ]b + b +[b, b ]b 0

08 : 03 ( 5 8 5 ) [J+, J z ] 0 ([J+, J z ]) [J, J z ]) 0 [J i, J j ] 0 J J a αa β b γb δ (δ αδ δ βγ δ αβ δ γδ ) a αa β b β b α a αa α b γb γ a αa β (b α b β δ αβ) a αa α b γb γ a αb α b β a β n a n an b J + J n a n an b J n a( n a + ) + n b( n b + ) + J + J n a n an b (n a + n b ) n a + 4 (n a + n b n a n b ) + J + J (n a n b ) n a + 4 (n a n b ) + J + J J + J + J z (J z ) J J + + J z (J z + ) ( ) 0 J iσ y 0 K + a J b a +b a b + [a b ] K K + [ab] K 3 (a a + b b) + (n a + n b ) + n +

: 03 ( 5 8 5 ) 09 [K +, K ] [a +b a b +, b a + b + a ] [a +b, b a + ] [a +b, b + a ] [a b +, b a + ] + [a b +, b + a ] [a +b, b a + ] + [a b +, b + a ] b [a +, a + ]b + a +[b, b ]a + + b + [a, a ]b + + a [b +, b + ]a b b a +a + b + b + a a (n a + n b + ) K z [K z, K + ] [n a + n b, a +b a b +] [ ] [n a, a +b ] + [n b, a +b ] [n a, a b +] [n b, a b +] [ ] [a +a +, a +b ] + [b b, a +b ] [a a, a b +] [b +b +, a b +] (a +b + b a + a b + a b +) a +b a b + K + [K z, K ] K K + K (a +b a b +)(b a + b + a ) a +a + b b a a + b +b a +a b b + + a a b +b + n a n b a +a + b +b + a a b b a a + b +b a +a b b + n a n b a αa β b β b α J J a αa β b β b α n an b K + K + n an b

0 : 03 ( 5 8 5 ) J n a( n a + ) + n b( n b + ) K + K + n an b (n a + n b ) + 4 (n a + n b + n a n b ) K + K n + 4 (n a + n b ) K + K n( n + ) K +K (K z )K z K + K (K z + )K z K K + J, J z, J z, K z J jmµν j(j + ) jmµν J z jmµν m jmµν J z jmµν µ jmµν K z jmµν ν jmµν µ j j (j, j ) (µ, ν) J z ν j + j + j µ + ν j µ + ν j µ j j j j

: 03 ( 5 8 5 ) K K + K j(j + ) + ν(ν ) 0 ν j + j + j j K z K ± jmµν (ν ± )K ± jmµν K ± jmµν C ν± jmµν ± C ν± jmµν K K ± jmµν jmµν [(K z ± )K z j(j + )] jmµν ν(ν ± ) j(j + ) (ν + j)(ν j) ± (ν j) (ν j)(ν ± j ± ) C ν 0, ν j + K jmµj + 0 ν C ν+ (ν j)(ν + j + ) jmµj + jmµj + 3 (j + ) K + jmµj + (j + 3)(j + ) K + jmµj +. jmµν (j + )! (ν j )!(j + ν)! Kν j + jmµj + ω jν (K + ) jmνj +

: 03 ( 5 8 5 ) ω jν (λ) (j + )! (ν + j)!(ν j )! λν j 4 [(j + )!] / νj+ χ jν (λ) jmµν e λk + jmµj + χ jν (λ) λ ν j (j + ν)! (ν j )! ν j + ν j + j + j j + j j j µ j j + µ j j µ j m j + j m + m j m, j m jjµj + (a +) j +m (b +) j +m (j + m )!(j + m )! 0 (a +) j+µ (b +) j µ (j + µ)!(j µ)! 0 jjµj + (a +) j+µ (b +) j µ 0 (j + µ)!(j µ)! (j)! 4 m [(j + )!] / jmµj + ϕ jm (x) (xa ) j+µ (xb ) j µ (j + µ)!(j µ)! 0 νj+ λ ν j (j + ν)! (ν j )! jmµν νj+ ( : Schwinger(3.8)) (ν j )! (λk +) ν j jmµj + e λk + jmµj +

: 03 ( 5 8 5 ) 3 ϕ jµ (ξ) µ (j)! jmµj + ϕ jm (x)ϕ jµ (ξ) mµ (j)! µ (ξ + a x) j+µ (ξ b x) j µ (j)! (j + µ)!(j µ)! 0 (j)! (ξ +a x + ξ b x) j 0 j (j)! jmµj + ϕ jm (x)ϕ jµ (ξ) e ξ +(xa )+ξ (xb ) 0 jmµ e λk + e λ[a b ] e λ[a b ]+ξ + (xa )+ξ (xb ) 0 (j)! jmµ e λk + jmµj + ϕ jm (x)ϕ jµ (ξ) (j)! jmµν j + jmµν (*) ω jν (K + ) (j + )! jmµν ϕ jm(x)ϕ jµ (ξ)χ jν (λ) jmµν ϕ jm (x)ϕ jµ (ξ)χ jν (λ) (Schwinger(3.35)) (j)! jmµν ϕ jm (x) [a b ] ν j (xa ) j+µ (xb ) j µ (j + )! 0 (ν + j)!(ν j )!(j + µ)!(j µ)! m µ j j, ν j + j + (j, j ) j j jm ϕ jm (x) m x + z, x z +, [ ] / j + [a b ] j +j j (xa ) j+j j (xb ) j j +j (j + j + j + )! (j + j j)!(j + j j )!(j j + j )! 0 j j jm jmµν (xa ) x + a + + x a z a + z +a [a z ] (xb ) [b z ]

4 : 03 ( 5 8 5 ) ( j j jm ϕ jm ( m z z + ) ) [ ] / j + [a b ] j +j j [a z] j+j j [b z] j j +j (j + j + j + )! (j + j j)!(j + j j )!(j j + j )! 0 m m j m j m ϕ j m (x)ϕ j m (y) e (xa )+(yb ) 0 j j 3, m m 3 5 6 7 j j j 3 m 3 j m j m ( ) j3+m3 ϕ j m (x)ϕ j m (y)ϕ j3 m 3 (z) m m m 3 [ ] / ( ) j +j j 3 j 3 + [yx] j +j j 3 [xz] j 3+j j [yz] j 3 j +j (j 3 + j + j + )! (j + j j 3 )!(j 3 + j j )!(j 3 j + j )! 5 [a b ] a +b a b + [a b ] b a + b + a [ba] 6 ( z ϕ jm ( z + ))) (z ) j+m ( z + ) j m ( ) j m (z +) j m (z ) j+m ( ) j m ϕ j m (z) (j + m)!(j m)!) (j + m)!(j m)!) 7 0 [ba] j +j j 3 [az] j 3+j j [bz] j 3 j +j e (xa )+(yb ) 0 0 [ba] j +j j 3 [az] j 3+j j [ b e (xa )+(yb ) 0 z]j3 j+j 0 [ba] j+j j3 [az] j3+j j [yz] j3 j+j e (xa )+(yb ) 0 0 [ba] j +j j 3 [xz] j 3+j j [yz] j 3 j +j e (xa )+(yb ) 0 [yx] j+j j3 [xz] j3+j j [yz] j3 j+j 0 e (xa )+(yb ) 0 [yx] j +j j 3 [xz] j 3+j j [yz] j 3 j +j

: 03 ( 5 8 5 ) 5 jm j m j m j j jm j m j m j + ( ) j j +m X(j j j; m m m) { } j + ( ) j j +m j j j m m m X 3j m m m 3 X(j j j 3 ; m m m 3 )ϕ j m (x)ϕ j m (y)ϕ j3 m 3 (z) [yx] j +j j 3 [xz] j 3+j j [yz] j 3 j +j (j + j + j 3 + )! (j + j j 3 )!(j 3 + j j )!(j 3 j + j )! [yz] j 3 j +j [zx] j 3+j j [xy] j +j j 3 (j + j + j 3 + )! (j + j j 3 )!(j 3 + j j )!(j 3 j + j )! X (J j + j + j 3 ) X(j j j 3 ; m m m 3 )ϕ j m (x)ϕ j m (y)ϕ j3 m 3 (z) m m m 3 X(j; m) 3 m m m 3 i [(j i + m i )!(j m i )!] xj +m + x j m y j +m + y j m [yz] j 3 j +j [zx] j 3+j j [xy] j +j j 3 (j + j + j 3 + )! (j + j j 3 )!(j 3 + j j )!(j 3 j + j )! n n n 3 ( [(J + )!(J j )!(J j )!(J j 3 )!] / ) ( ) ( ) J j J j J j 3 n n n 3 z j 3+m 3 + z j 3 m 3 (y + z ) J j n ( y z + ) n (z + x ) J j n ( z x + ) n (x + y ) J j 3 n 3 ( x y + ) n 3 ( ) n 3 [(J j i )!] / [(J + )!] / (J j n n n i n i )!n i! 3 i x J j 3 n 3 +n + x J j n +n 3 y J j n +n 3 + y J j 3 n 3 +n z J j n +n + z J j n +n ( )

6 : 03 ( 5 8 5 ) n n + n + n 3 x + j + m J j 3 n 3 + n j + j j 3 n 3 + n n n 3 m j + j 3 n n 3 m j + j 3 n 3 n m j 3 + j n n m 3 j + j X(j; m) ( ) n [(J + )!] / n n n 3 X X(j; m)ϕ j m (x)ϕ j m (y)ϕ j3 m 3 (z) m 3 [(j i + m i )!(j i m i )!(J j i )!] / (J j i n i )!n i! i [yz] J j [zx] J j [xy] J j 3 (J + )!(J j )!(J j )!(J j 3 )! Φ j j j 3 (αβγ) j j j 3 (J j )+(J j ) + (J j 3 ) J Φ j j j 3 (αβγ) α J j β J j γ J j 3 (J + )! (J j )!(J j )!(J j 3 )! X(j; m)ϕ j m (x)ϕ j m (y)ϕ j3 m 3 (z)φ j j j 3 (αβγ) jm J j j j 3 (α[yz])j j (β[zx])j j (γ[xy])j j3 (J j )!(J j )!(J j 3 )! j +j +j 3 (α[yz])j (β[zx]) j (γ[xy]) j 3 (j J + j + j 3)! (α[yz] + β[zx] + γ[xy]) J J! J e α[yz]+β[zx]+γ[xy]

: 03 ( 5 8 5 ) 7 X X 3j (j, j, j 3 ), (m, m, m 3 ), (α, β, γ), (x, y, z) ( j j j 3 m m m 3 ) ( j j 3 j m m 3 m ) ( j 3 j j m 3 m m j j, m m, α β, γ γ, x y, Φ j j j 3 ( β, α, γ) ( ) J Φ j j j 3 (α, β, γ) ( ) ( ) ( ) ( j j j 3 j 3 j j j j 3 j ( ) J j j j 3 m m m 3 m 3 m m m m 3 m m m m 3 x + x, y + y, z + z, α α, β β, γ γ [xy] [xy] Φ j j j 3 ( α, β, γ) ( ) J Φ j j j 3 (αβγ), ϕ jm ϕ j m ( j j j 3 m m m 3 ) ( ) J ( j j j 3 m m m 3 ) ) ) m m m 3 0 ( ) j j j 3 0 0 0 0 : J j + j + j 3 odd jm j m, j m ( ) j + ( ) j j +m j j j m m m ( ) j + ( ) j j +m+j j j j m m m ( ) ( ) j +j j j + ( ) j j +m j j j m m m ( ) j +j j jm j m, j m θ x ± x ± e ±iθ, y ± y ± e ±iθ, z ± z ± e ±iθ e iθ(m +m +m 3 ) ( ) j j j 3 0 : m + m + m 3 0 m m m 3

A A. f x y f : x y y f(x) A f g A : f g g Af g(x) (Af)(x) I f y I : f y y I[f(x)] [a, b] ψ(x) ψ(x) ψ ψ (x) ψ 9

0 : 03 ( 5 8 5 ) ψ ψ ψ(x) φ(x) (ψ, φ) (ψ, φ) dx ψ (x)φ(x) ψ φ ( ψ ) ψ ( ψ ) ψ A ψ(x) Aψ(x) Aψ A ψ A A (ψ, Aφ) (A ψ, φ) A ψ ( A ψ ) ( A ψ ) ψ (A ) ψ A ψ Aφ A ψ φ ψ A φ ψ A φ ψ (A φ) ( ψ A) φ ψ A (A ψ ) ( ψ A φ ) ( ψ Aφ ) ( A ψ φ ) dx ( A ψ(x) ) φ (x) φ (A ψ ) φ A ψ

: 03 ( 5 8 5 ) δ(x a) a ψ(x) a ψ dx (δ(x a)) ψ(x) ψ(a) ψ(x) ψ(x) x ψ ψ(x) δ(x a) ψ a x a δ(x a) x x y δ(x y) {φ n (x)} φ n φ m δ nm φ n (x)φ n(y) δ(x y) n x φ n φ n y x y n φ n φ n n

: 03 ( 5 8 5 ) A. [a, a ] a 0 0 0 a 0 n n n! (a ) n 0 ˆn n n n ˆn a a e iλˆn 0 0 A.3 A a a A a a a a A a A a a A A x, y x (A y ) A x y x (A y ) x A y x A A x ( A x )

: 03 ( 5 8 5 ) 3 a a A a ( a A) a a a a a a a a a, b (a b) A a a a, A b b b a A b a (A b ) a b b ( a A) b a a b (a b) a b 0, a b 0, (a b) n n n ( ), v v i a i v i, a i a j δ ij a i v i a i v v i a i a i v ( a i a i ) v i

4 : 03 ( 5 8 5 ) v a i a i i ( ) a i P i a i a i P i a i a i a i a i a i a i P i P i i i, i,, i j δ ij, i i i U U i U i a i i i a i UU i U U i i i a i a i AU i A a i i i a i a i i

: 03 ( 5 8 5 ) 5 U A i a i a i i i ij a i a i i j a j i a i a i a i i a i P i N A V V AV V diag (a,, a N ) V ( a, a, a N ) V V E N (V V ) ij a i a j δ ij V V E N V V ( a, a N ) P i i i a. a N i P i A V diag (a,, a N )V i a i P i

6 : 03 ( 5 8 5 ) A.4 [AB, C] A[B, C] + [A, C]B [A, BC] B[A, C] + [A, B]C A, B [A, B] AB BA 0 A a, a (A a a a,a a a a ) a (AB BA) a a a B a a B a a (a a ) a A a 0 ( ) () A, B [A, B] 0 A a, a, B a B a 0, a a A a B B B a a a a B a a a B a a B a B a A B n M a a + an an n a a A[B, C] + [A, C]B ABC ACB + (ACB CAB) ABC CAB [AB, C] [A, BC] [BC, A] B[C, A] [B, A]C B[A, C] + [A, B]C

: 03 ( 5 8 5 ) 7 an B an ( an B an ) ( an B an ) B b an ab, n,, M M an B an an ab an ab b n ab M an an ab n B ab ( a a + n a a ) an an B nn an an B an an ab M an an ab n n an an ab b ab b B b [A, B] AB BA 0 (A A, B B) A a, b a, b a B a, b a, b b

B B. [a, a ], a 0 0 j j! (a ) j 0 ˆn a a ˆn j j j n f(n) af(n) f(n + )a an aa a (a a + )a na + a (n + )a an an n (n + )a n (n + ) a an k (n + ) k a 9

30 : 03 ( 5 8 5 ) [a k, a ] ka k [a, (a ) k ] k(a ) k [a, f(a )] a f(a ) af(a ) 0 a f(a ) 0 [a, f(a)] ( [a, f(a )]) a f(a) n 3 [a k, n] ka k, k,, [n, (a ) k ] k(a ) k, k,, [a, a ] a[a, a ] + [a, a ]a a [a 3, a ] a[a, a ] + [a, a ]a 3a [a k, a ] a[a k, a ] + [a, a ]a k a(k )a k + a k ka k [a, (a ) k ] k(a ) k 3 [a, n] [a, a a] a [a, n] a[a, n] + [a, n]a a [a 3, n] a [a, n] + [a, n]a 3a 3 [a k, n] ka k

........................ 95............................4... 76 3j............ 5...................... 6 SO(3)........................... 04 SU()............................ 94 X.......................... 5................. 0.......................... 9....................... 8.......................... 0, 79......................... 79............................8................ 90............................80....................... 4......................... 5......................................46..................................... 76.......................................................9........................................... 3.............. 69..........69..................... 7..... 98, 00............................3.......................... 8............... 34, 36..................... 99 40, 4, 47, 49, 69, 76................................ 8......................... 84.. 94, 03................. 0..................... 40.............. 99 3........................ 95............................4.......................... 8 3